Berkeley physics laboratory

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mathematics and statistics

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Copyright 1971 by McGraw-Hill, Inc.

All rights reserved. Printed in the United States of America.

No part of this publication may be reproduced, stored in a

retrieval system, or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher.

Library o/Congress Catalog Card Number 79-12~10S:'

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07-050481-4

1234567890 BABA 79876543210

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The first edition of the Berkeley Physics Laboratory

copyright © 1963, 1964, 1965 by Education Development

Center was supported by a grant from the National Science

Foundation to EDC. This material is available to publishers

and authors on a royalty-free basis by applying to the

Education Development Center.

This book was set in Times New Roman, printed on

permanent paper, and bound by George Banta Com­

pany, Inc. The drawings were done by Felix Cooper;

the designer was Elliot Epstein. The editors were Bradford

Bayne and Joan A. DeMattia. Sally Ellyson

supervised production.

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• contents

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3 9091 01132322 1

berkeley physics laboratory, 2d edition complete contents of the 12 units

. mathematics and statistics MS·' MS·2 MS·3 MS·4 MS·S MS·6

derivatives and integrals

trigonometric and exponential functions

loaded dice

probability distributions

binomial distribution

normal distribution

acoustics and fluids AF-' AF·2 AF-3 AF-4 AF-S AF·6

acoustic waves acoustic diffraction and Interference acoustic interferometry fluid flow viscous flow turbulent flow

mechanics

microwave optics

M·'

MO·' MO·2 MO-3 MO·4

velocity and acceleration

M·2 cOllisions

M-3 dissipative forces

M·4 periodic motion

M·S forced OSCillations

electronic Instrumentation EI·' EI·2 EI·3 EI·4 E'·S

voltage, current. and resistance measurements measurement of ac voltage and current waveform measurements comparison of variable voltages transducers

F·2 F-3 F-4 F-S

laser optics LO-' LO-2 LO-3 LO-4 LO-5

reflection and refraction of light polarization of light diffraction of light interference of light holography

atomic physics

fields

F·'

microwave production and reflection Interference and diffraction the klystron microwave propagation

radial fields

image charges

field lines and reciprocity

the magnetic field

magnetic coupling

AP-, AP-2 Ap·3 AP-4 AP·S

atomic spectra photoelectric effect the photomultiplier and photon noise ionization by electrons electron diffraction

nuclear physics electrons and fields EF-' acceleratIon and deflection of electrons

EF-2 focusing and intensity control

EF·3 magnetic deflection of electrons

EF-4 helical motion of electrons

EF-S .vacuum diodes and the magnetron condition

Np·, Np·2 Np·3 Np·4 NP·S

geiger·mueller tube radioactive decay the scintillation counter beta and gamma absorption neutron activation

semiconductor electronics electric circuits EC-' EC-2 EC-3 EC·4 EC-S

resistance-capacitance circuits

resistance-Inductance circuits

LRC circuIts and OSCillations

coupled oscillators

periodic structures and transmission lines

SE·' SE·2

semiconductor diodes tunnel diodes and relaxation oscillators SE-3 the transistor SE·4 transistor amplifiers SE-tt pOSitive feedback and oscillation SE-6 negative feedback

354835

preface

In developing this revised version of the Berkeley Physics Laboratory, we have tried to make the original material more useful for beginning students, many of whom will be taking their first college-level physics course con­ currently with the use of this laboratory course. At the same time, we have tried very hard to preserve the essential characteristics and flavor of the original version, particularly with respect to its use of contemporary instru­ mentation and its frequent contact with current or recent research in physics. These qualities, we feel, were largely responsible for the fairly wide acceptance of the first edition, and we have tried to preserve them in this revision. Most of the original experiments have been retained; the order and organization have been changed, and the discussions have all been com­ pletely rewritten with the aim of making them more readable and self­ contained. In addition, a large number of completely new experiments have been developed, so that the total number of experiments is nearly double that of the original version. Specifically, the experiments are organized into twelve units, with four to six experiments in each unit. Most units begin with rather elementary experi­ ments and conclude with more challenging ones. Usually the same basic equipment is used for all experiments within a unit, with minor changes in accessories for the individual experiments. This scheme has the considerable pedagogical advantage that a student does not have to familiarize himself with a completely new setup for every experiment. Each experiment is sub­ divided into sections, each with a numbered paragraph of discussion. Thus, an instructor who wishes to assign only part of an experiment can refer to the sections by number. Our hope is that this scheme will make a sufficiently flexible system so that instructors with various objectives can use this material as a basic resource to construct their own individualized courses, selecting those units, and those experiments or parts of experiments within units, which meet their needs. It is not essential to go straigh\ through this course from beginning to end. Some experiments, however, do have desirable prerequisites. For example, a student should be familiar with the experiments on Electronic Instru­ mentation before continuing with Electric Circuits or Electrons and Fields. In most cases the experiments have been designed so that they can be carried out reasonably thoroughly by an average student in one 3-hour laboratory period. In some cases it will be desirable to omit some sections of certain experiments or to allow more than one laboratory period. The organization of the material is, we feel, very suitable for an "open-ended" laboratory in which students work at their own pace, each according to his ability and motivation. In the revised edition we have used the MKS system of units throughout, with occasional references to CGS or British units. Although the esthetic qualities of the MKS system can be debated, one overwhelming advantage

,

r

i

of the MKS system is the universal use of this system in practical electrical measurements. In addition, most new elementary texts now use this system. A table of conversion factors is included for the benefit of those readers who were brought up on CGS units. Finally, we wish to repeat the statement from the Preface to the first edition that this laboratory course may make greater demands on the average student than more conventional laboratory activities. We have tried very hard to avoid making the new material a "cookbook," and we are aware that as a result some students will have to struggle. This struggle is an essential part of the learning process, and from it will come greater strength.

Alan M. Portis Hugh D. Yonng

i constants

A list of physical constants which may be needed for your laboratory work is given in the table of Fundamental Physical Constants. The fundamental constants are given in MKSA units. In practical calculations, other units such as electron volts or atomic mass units are sometimes more convenient to use than the basic MKSA units. Some of the constants and combinations of constants that frequently occur are given with various units in the table of Other Useful Constants. A few commonly used conversion factors are also given. FUNDAMENTAL PHYSICAL CONSTANTS Symbol

Name Speed of light Charge of electron Mass of electron Mass of neutron Mass of proton Planck's constant

c

e m m. mp

h h

Permittivity of free space

= h/2n

Eo 1/4nl:o flo

Permeability of free space Boltzmann's constant Gas constant Avogadro's number Mechanical equivalent of heat Gravitational constant

k R No J G

Value 2.998 x 108 m/sec 1.602 x 10- 19 coul 9.109 x 10- 31 kg 1.675 x 10- 27 kg 1.672 x 10- 27 kg 6.626 x 10- 34 joule sec 1.054 x 10- 34 joule sec 8.854 x 10- 12 farad/m 8.988 x 10 9 m/farad 4n x 10- 7 weber/amp m 1.380 x 10 23 joule/K 8.314 joules/mole K 6.023 X 10 23 molecules/mole 4.186 joules/cal 6.67 x 10- 11 N_m 2 /kg 2

OTHER USEFUL CONSTANTS Symbol

Name Planck's constant Boltzmann's constant Coulomb constant Electron rest energy Proton rest energy Energy equivalent of I amu Electron magnetic moment Bohr radius Electron Compton wavelength Fine-structure constant Classical electron radius Rydberg constant

h k 2 e / 4nB o

me 2 M p c2

loJc 2 fl a

=

Ac

=

r1.

=

re

=

ell/2m 4nBo1l2/ me 2 h/mc e2/4rr.Eohc e2/4rr.somc1

R" Conversion factors I eV 1.602 x 10- 19 joule I A 10- 10 m 1 amu = 1.661 x 10- 27 kg ..... 931.5 MeV

Value 4.136 x 1O- 15 eVsec 8.617 x 10- 5 eV/K 14.42 ev A 0.5110 MeV 938.3 MeV 931.5 MeV 0.9273 x 10- 23 joule m 2 /weber 0.5292 x 10- 10 m 2.426 x 1O- 12 m 1/137.0 2.818 x 1O- 1S m 1.097 x 10- 7 m

mathematics and statistics

INTRODUCTION The first two experiments in this unit review some of the mathematics that

you will use in your introductory physics Course. This review will take the form of laboratory activities in which you will develop certain relations empirically. We begin with the calculus and introduce differentiation and int~gration in a~ operational way. Next we consider several special functions whIch are partIcularly useful in physics-trigonometric and exponential functions. In the remaining experiments in this unit you will learn a few basic concepts in probability and statistics, and you will See some applications of these con­ cepts to physical measurements. (These experiments need not be performed at the beginning of the laboratory sequence but can be introduced at any time.) Because of the central role of measurements in all of science, these concepts are of great importance. In all branches of science we deal with numbers which originate in experimental observations. In fact, the very essence of science is discovering and using correlations among quantitative observations of physical phenomena. Statistical considerations are important for two reasons. First, measure­ ments are never exact; the numbers which result are ofvery little value unless we have some idea of the extent of their inaccuracy. If several numbers are used to compute a result, we need to know how the inaccuracies of the indi­ vidual numbers influence the inaccuracy of the final result. In comparing a theoretical prediction with an experimental result, we need to know some­ thing about the accuracies of both if anything intelligent is to be said about whether or not they agree. By considering the statistical behavior of errors of observation we can deal with these problems systematically to obtain results which are as precise as possible and whose remaining uncertainties are known. A second reason for the importance of statistical concepts is that some physical laws are intrinsically statistical in nature. A familiar example is the radioactive decay of unstable nuclei. In a sample ofa given unstable element, we have no way of predicting when any individual nucleus will decay, but we can describe in statistical terms how many are likely to decay in a given time interval, how many will probably be left after a certain time, and so forth. Thus, in this case, we deal not with precise predictions of events but with probabilities of various combinations of events. In the development of quan­ tum mechanics, probability theory is of even more fundamental importance.

1

experiment

M S-1

derivatives and integrals

introduction Although the ideas of the calculus can be introduced without reference to any particular physical situation, we prefer to show the physical usefulness of the basic concepts by discussing a particular laboratory situation.

k I

experiment We shall consider the motion of a cart along a straight track. The position of the cart is described at any instant by giving its distance from some refer­ ence point on the track. We call this distance x; clearly, it varies with time (I) when the cart moves, so x is a/unction of I. We now tilt the track slightly and release the cart at time 1 0 from the reference point x = 0, which we take near the top of the track. Then we measure the position at a succession of times, using a multiple~fiash photo­ graph, a spark timer, or some other means. The spark timer, to be discussed in more detail in Experiment M-I, uses a high-voltage pulse which causes a spark to jump from cart to track at a succession of equally spaced time inter­ vals. The spark positions are recorded as holes in a strip of paper laid along the track, thus providing a permanent record of the successive positions. In a certain experiment, the data obtained were as shown in Table 1. This table also includes additional columns for calculations to be described later. Plot the data given in Table I on a sheet of graph paper. (K & E 46-1320, which has 10 x 10 divisions to the half inch, is suitable.) Plot time along the long direction and displacement along the short direction. Draw a smooth curve through the data points.

AVERAGE VELOCITY The average velocity during a time interval between 11 and 12 in which the

displacement has changed from

Xl

to x 2 is defined as x2

-

12 -

Xl

(1)

11

From the data given in Table 1 find the average velocity during the first second; during the first 10-sec interval; during the first 20-sec interval; during the second lO-sec interval. INSTANTANEOUS VELOCITY The instantaneous velocity may be thought of as the value of the average

velocity when the time interval becomes extremely short. As an example let us 3

mathematics and statistics

TABLE 1

I

Time ,.

,~

Displacement x, m !

!ix, m

Velocity v,

0.000

0.0000

1.000

0.0064

2.000

0.0249

3.000

0.0544

4.000

0.0937

5.000

0.1420

6.000

0.1984

7.000

0.2621

8.000

0.3324

9.000

0.4088

~~~~~~~

.,

Acceleration a, m/sec2

mjsec

~

i

I

I I i

I

I I

10.000

0.4905

11.000

0.5772

12.000

0.6683

13.000

0.7633

14.000

0.8621

15.000

0.9641

16.000

1.0680

17.000

1.1769

18.000

1.2871

19.000

1.3994

20.000

1.5137

!

!

I

I I

I

I

-~~ ~~~~~~\

~

attempt from the data given in Table I to find the instantaneous velocity at t = 10 sec. We use the shorthand notation Ax == X 2 Xl and At == t2 t l , where the symbol A is the Greek letter delta. The composite symbol Ax can be called "change in x"; it is not the product of A and x! Fill in Table 2 for v. Make a plot of the average velocity vas a function of the time interval At. Extrapolate your data to llt = O. What is your estimate of the instantaneous lO sec? What we have done graphically is to find the value velocity at t which v approaches as At approaches zero; this is called the limit of v as At 4

derivatives and integrals

MS-1

l1t

TABLE 2

o

20

5

15

8

12

9

11

approaches zero and is the mathematical definition ofinstantaneous velocity. This defines the instantaneous velocity:

. Ax v = I1m At-O At

(2)

This expression is also called the derivative of x with respect to t. It may seem odd and even inconsistent that we have used velocities over definite time intervals to define instantaneous velocity at a single point, where no time interval is involved. Yet we know intuitively that instantaneous velocity at a point is a sensible concept. The concept of derivative which we have just discussed provides a sound mathematical basis for the idea of an instantaneous velocity (or any other instantaneous rate of change), and this is its most fundamental significance. Because the velocity is changing slowly, the average velocity for an interval At = 1 sec and the instantaneous velocity at the center of the interval should be reasonably close. Using At = 1 sec as a time interval, fill in the velocity column in Table 1 from t = 0.500 to 19.500 sec. With reference to your plot ofTable 1 the average velocity between 0 and 20 sec is just the slope of the chord drawn through the displacement data points at t = 0 and 20 sec. Note that the slope of a line is not equal to the tangent of the angle the line makes with the horizontal, as it would be if the vertical and horizontal scales were the same. Here the scales are different, and have different units; to find the slope of the line we choose two points, find the differences x 2 Xl and t2 tI> and take their quotient. Draw this chord. Also draw chords for the other intervals given in Table 2. Draw the tangent to your curve at t = 10 sec. Compute the slope of the tangent and compare with your extrapolated value of average velocity in the limit At goes to zero. What is the relation between instantaneous velocity and the slope of the tangent?

ACCElERATION Using the same graph on which you plotted the displacement data ofTable ],

also plot the velocity data, using a new coordinate scale on the right side of the paper. What can you say about the velocity as a function of time? The rate of change of velocity is called the acceleration. The average acceleration t 1, during which the velocity changes by an during a time interval At = t2 amount Av = V 1 - VI' is defined as (3) 5

mathematics and statistics

What is the average acceleration in the interval between t = 0 and 20 sec? Fill in Table 2 for a. The instantaneous acceleration is defined as the limit of the average acceleration as the time intervaillt approaches zero: . Ilv a= hm­ II.t .... O Ilt

(4)

Assuming that an interval of I sec is sufficiently short to give a good approx­ imation of the instantaneous acceleration a, complete Table I from t = 1.000 to 19.000 sec. Note that the average acceleration between t = 0 and 20 sec is just the slope of the chord drawn through these velocity data points. Draw chords through your velocity data for the other time intervals in Table 2. Note that as the time interval becomes shorter the slope ofthe chord approach­ es the slope of the velocity curve. What is the relation between the slope of the tangent and the instantaneous acceleration? Plot your acceleration data on the same sheet of graph paper, showing a new scale for acceleration.

DIFFERENTIATION

The limit indicated in Eq. (2) is called the derivative ofx with respect to time. The process of determining the instantaneous velocity if x is known as a function oftime for all times is called differentiation. This operation is written symbolically as dx dt

= lim Ilx = v 1I.t-0

Ilt

(5)

Similarly, the instantaneous acceleration is expressed as dv dt

rImA=a Ilv 1I.t-0

ilt

(6)

ACCELERATION DATA

We introduce the process of integration by considering again the cart on an inclined track. Let us imagine that this cart has mounted on it an accelerom­ eter'" and that we are able to obtain directly instantaneous values of the acceleration of the cart. We shall now see how it is possible from the accelera­ tion alone to determine the velocity as a function of time (knowing that the cart started from rest) and the position data (knowing that the cart was at x = 0 at t = 0). The accelerometer data for the cart are given in Table 3. This table also includes additional columns for calculations to be described below. Using a new sheet of graph paper of the same kind as used earlier, plot the data for acceleration as a function of time.

VELOCITY

We may use Eq. (3) to find the change in velocity during any time interval: (7)

That is, the velocity V2 at the end of a time interval (12 - (1) is equal to the velocity VI at the beginning of the interval plus the average acceleration aover * 6

A device for measuring instantaneous acceleration.

mathematics and statistics

just 0.01206 m/sec. Similarly, the velocity change in the interval from 1.500 to 2.500 sec is 0.01192 m/sec, and so on. The first interval (I = 0.000 to 0.500 sec) requires special treatment, being only half as long as the others. The instantaneous acceleration at t = 0.500 sec is approximately equal to the average of the values at 0.000 and 1.000 sec, which is !<,0.01333 m/sec 2 + 0.01206 m/sec 2). Then the instantaneous acceleration at the center of the interval (i.e., 1 = 0.250 sec) is approximately the average of this value for 1 = 0.0500 sec and the value for t 0.000 sec. The end result of all this is that for the first interval,

a~

(~)0.01333 m/sec 2

+ (*)0.0126 mjsec 2

= 0.01302 m/sec 2

This is equivalent to taking a weighted average of the values of a at 0.000 and 1.000 sec, weighting the former three times as much as the latter because the center of the interval (0.250 sec) is "three times as close" to 0.000 as to 1.000 sec. Thus we find that the velocity change from 0.000 to 0.500 sec is 0.00651 m/sec. Since v = 0 at t = 0, this is also the actual velocity at 0.500 sec. Now, using the successive changes in v, we can compute the actual values at t = 1.500 sec, 2.500 sec, and so on. The interval between 19.500 and 20.000 sec is handled the same way as for the first interval. Compute these velocities and record the results in Table 3. Plot your velocity data on the same sheet of graph paper used for the accelerometer data, and compare this plot with your earlier plot of velocity as determined from the displacement data. Note that the velocity of the cart at any time t is just the area under the acceleration curve from 1 = 0 to the final time I. If the initial velocity at time t = 0.000 is not zero but some initial value vo, this area still gives the total change during the interval, and the final velocity is then the sum of Vo and this change.

DISPLACEMENT

From Eq. (1) we have for the displacement

x2

=

Xl

+

V(t2

t1)

(8)

This equation is analogous to Eq. (7) and states that the displacement at the end of an interval is equal to the displacement at the beginning of the interval plus the product of the average velocity over the interval and the duration of the interval. If the interval is sufficiently short, the average velocity should be just the velocity at the midpoint. As an example we compute the displacement at t 1.000 sec. We take the average velocity between t = 0.000 and 1.000 sec to be the value computed at t = 0.500 sec, namely, v = 0.00651 m/see. Then the displacement at t = 1.000 sec will be 0.00651 m. Similarly, the dis­ placement at t = 2.000 sec may be computed to be X =

0.00651

+ 0.01857 ilt

0.02508 m

In this way determine the displacement as a function of time. Note that the dis­ placement at time t is the area under the velocity curve from tOto time t. Compare your calculated displacement data with the direct data given in Table I.

INTEGRATION

8

From the above discussion we see that the total velocity change in any time interval can be obtained by dividing this interval into many smaller intervals,

derivatives and integrals

MS-1

which we may call1lt i , multiplying each interval by the average value of a in that interval, denoted by til , and summing these products. Symbolically, N

Vr

=

Vo

+

L

tii

(9)

Iltl

1= 1

If the acceleration is known continuously for every instant of time, the time intervals can be made arbitrarily small, and we speak of the limit of this expression as all the Illi -'> 0 and N -'> 00. The usual notation is

It:~o

itl

til

t

Ilt;

a

(10)

dt

and the expression is called the integral of a. Thus we have

I

"'I

Vr

=

Vo

+

a

(II)

dt

",0

Similarly,

questions 1 Although the displacement data appeared to lie on a smooth curve and the

computed velocity data appeared to lie on a smooth curve, some scatter appears in the acceleration data. Explain the origin of this scatter. 2 What would happen to the computed velocity data if larger time intervals

were taken? What would happen to the computed accelerations? 3 In determining velocity from acceleration, what would be the deviation of the computed velocity if larger intervals were taken? What would be the devia­ tion of the computed displacement? Does this explain any discrepancy that you obtain between direct displacement data and the displacement as computed from accelerometer data?

9

experiment

MS -2

trigonometric and exponential functions

introduction In this experiment we introduce trigonometric and exponential functions and their differential and integral relations. As an application of exponential functions we discuss the operation of the slide rule and show how a slide rule may be constructed from logarithmic graph paper.

experiment TRIGONOMETRIC FUNCTIONS

In Fig. I we show an x-y coordinate system with a circle of radius r drawn about the origin. We designate by () the angle between the horizontal axis and

FIGURE 1

the diagonal shown. If s is the length of the subtended arc as shown, then the angle () expressed in radians is given by ()=s

(12)

r

What is the angle between the x and y axes expressed in radians? What is the angle between the + x and - x axes? What is the total angle about the origin? We define the trigonometric functions sine (abbreviated sin) and cosine (abbreviated cos) of () as follows: sin ()

y

r

cos ()

x r

= -

(13) 11

mathematics and statistics

Since by the Pythagorean theorem the square of the hypotenuse is equal to the sum of the squares of the adjacent sides, (14)

it follows by substitution from Eq. (12) that sin 0 and cos 0 are related as follows: (I 5)

We note that since 0

2n corresponds to a complete revolution, 0 and

o + 2n represent the same angle. Thus it must be true that sin (0

+ 2n) =

sin 0

and

cos (0

+ 2n) =

cos 0

(16)

In addition, if sin 0 is known for angles in the first quadrant (0 to n/2), it may be computed for any angle in any other quadrant by using a relation obtained from Fig. 1. For example, sin (n - 0) = sin 0, sin ( - 0) = - sin and so on. Similar though not identical relations can be derived for the cosine function. These relations are also exhibited by the graphs ofsin and cos for all angles from 0 to 2n, shown in Fig. 2.

e,

e

DIFFERENTIATION OF TRIGONOMETRIC FUNCTIONS

e

e

The general behavior of the derivative of sin 0 with respect to can be ob­ tained from Fig. 2 by looking at the slope of the curve at various points. We

FIGURE 2

iI

sin 0

i·

cos

0

i'

,

;

e

note that the maximum slope occurs at = 0, where the function itself is zero. The slope then decreases until at 0 n/2 (where the function itself has the value unity) it is zero. It becomes more and more negative until 0 = n, and so on. In short, the ups and downs of the derivative of sin correspond to the actual values of cos 0, suggesting that perhaps

e

!

sin 0 = cos 0

(17)

A similar investigation shows that the general shape of the derivative ofcos 0 is similar to that of sin 0 except for a change of sign. The derivatives of sin 0 and cos can be explored in more detail by numerical operation on a table ofvalues ofsin 0 and cos 0, as given in Table 4. The difference in values of sin 0 for adjacent values of e, divided by the dif­ ference in 0, which is 0.01 in each case, gives the approximate value of the derivative at each point. Complete the fourth column in the table, and

e

12

MS-2

trigonometric and exponential functions

e

TABLE 4

sin

e

cos

e

L\ sin

e

L\ cos -~-

M

e

....-­

M

L cos

e

xM

I

L sin

e

xM

1\\\\\\\\\\\\\\\\\ ~\\\\\\\\\\\\\\\\l\\\\\\\\\\\\\\\\\~\\\\\\\\\\\\\\\\

0.00

.00000

1.00000

I 0.01

.01000

.99995

0.02

.02000

.99980

I 0.03

.03000

.99955

I 0.04

.03999

.99920

0.05

.04998

.99875

0.06

.05996

.99820

, 0.07

.06994

.99755

0.08

.07991

.99680 '

I 0.09

.08988

.99595

0.10

.09983

.99500

0.11

.10978

.99396

0.12

.11971

.99281

0.13

.12963

.99156

0.14

.13954

.99022

0.15

.14944

.98877

0.16

.15932

.98723

0.17

.16919

.98558

: 0.18

.17903

.98384

0.19

.18886

.98200

I 0.20

.19867

.98007

0.21

.20846

0.22

.21823

.97590 '

0.23

.22798

.97367

0.24

.23770

.97134

0.25

.24740

.96891

,

I

i !

i

I !

I

I

.97803

~\\\\\\\\R R\\\\\\W ~\\\\\\\\\\\\\'1\\\\\\\\\\\\\\\\\ 13

mathematics and statistics

compare the values with the corresponding values of cos O. Similarly, com­ pute values for the derivative of cos 0 by completing the fifth column of the table, and compare with corresponding values of sin O. By reversing the above process, we can find the area under the graph of sin 0 from 0 = 0 to any final value of O. The element of area corresponding to an interval 110 is given by sin 0 110. The total area is the sum ofthe values of sin 0 in successive intervals, each multiplied by 110, which has the value 0.01 in each case except the first. Each value of 0 represents the center of an interval, so the first interval is only from 0.00 to 0.005. Complete the sixth column of the table and compare with the second column. What do you con­ clude about the integral of cos 0 ? Repeat these calculations for the area under the curve of sin O. Fill in the last column in the table; the sum is subtracted from unity to facilitate com­ parison with cos 0 because cos 0 = I when 0 = O. Finally, we derive analytically the results we have already discovered empirically. FIGURE 3

In Fig. 3 we show our original triangle and a second triangle with the angle at the origin increased from 0 to 0 + 110. We shall assume that the incremental angle 110 is small. From Fig. 3 we note the following relationships: sin 0 = ~

cos 0

r

\

sin (0

y+b

+ 110)

cos (0

r

x

(18)

=-

r

+ 110) =

x

a r

(19)

Now if the incremental angle 110 is small, the diagonal and the length along the arc s are very nearly equal and we can write 110 = c r

In addition, the angle between c and b is approximately equal to 0 so that we have b

=

c cos 0

=

r 110 cos 0

a = c sin 0 = r 110 sin 0 14

(20)

trigonometric and exponential functions

MS-2

Substituting Eqs. (20) into Eqs. (19), we obtain sin (0

+ 1l0)

cos (0

+ 1l0)

sin 0

+

110 cos 0

= cos 0 - 110 sin 0

(21)

From Eqs. (21), the changes in the functions divided by the corresponding change 110 are given approximately by sin (0

Il(sin 0)

+ 1l0)

- sin 0

110

cos 0

cos(O + 1l0) - cos 0 - - ' - - - -.. - - == -sin 0

Il(cos 0)

110

(22)

110

In the limit that 110 goes to zero, we obtain the derivatives . Il(sin (}) _ 11m A(} - cos

d(sin 0)

dO d(cos (})

dO

40"'0

Ll

v

L1

. Il(cos 0) . () 11m = -sm 40"'0 Il(}

(23)

Alternatively, we may sum Eqs. (22) over increments Il(} from 0 to (J to obtain

L Il(sin ()) = L cos 0 MJ cos () ;: 1 + L Il( cos (}) = 1 - L sin () Il(} sin (J =

(24)

In the limit that the incremental angles go to zero, we obtain the expressions "9

sin (J =

J cos () dO 0

cos 0 = 1

EXPO N ENTIAL FUN CTI ONS

J:

(25)

sin 0 dO

Each ofthe trigonometricfunctions sin 0 and cos () has the property that when

it is differentiated twice, the result is the negative of the original function. Now we investigate a function having the property that for each value of x the derivative of the function is equal to the value of the function itself. That is, y is a function of x having the property that

dy dx = y

'(26)

In addition, we require that at x 0 the function must have the value unity. The function that satisfies these requirements can be constructed graphical­ ly by the following procedure. Starting at the point x = 0, y = 1, construct a line having a slope of unity, for an interval up to x 0.1. Read from the graph the value of y for this point; it will be 1.1. Next, construct another line starting at this point, with a slope of 1.1, extending to x = 0.2. Continue this way, with the value of the function at the end of each segment determining the slope of the next segment. Continue at least to x = 1.0. Repeat the con­ struction using intervals of 0.01, in the range from 0 to 0.1. This function is usually called the exponential function, y = tr. A few 15

mathematics and statistics

values are tabulated in Table 5. Complete the third column of this table, just as for the sine and cosine functions, to verify that the derivative at each point TABLE 5

x

fiX

.00

1.0000

.01

1.0101

.02

1.0202

.03

1.0305

.04

1.0408

.05

1.0515

.06

1.0618

.07

1.0725

.08

1.0833

.09

1.0942

.10

l.l052

.11

l.l163

!J.e"'j!J.x

1+l:e"'!J.x

\\\\\\\\\\\\\\\\\l\\\\\\W

\\

1.1275

.13

l.l388

.14

l.l503

.15

l.l618

.16

l.l735

.17

l.l853

I

i

l.l972 1.2092

16

.21

1 """'1 ,

.22

1.2461

.23

1.2586

.24

1.2712

.25

1.2840

\\\\\\\\\\\\\\\\\ ~\\\\\\W

trigonometric and exponential functions

MS-2

is equal to the value of the function at the corresponding point. Also complete the last column to obtain values of the integral of e" from x = 0 to an arbitrary value of x, and compare with the value of the function at the corresponding end point, to verify that the integral of the function is equal to the function itself. That is,

= 1+

eX

J:

(27)

eX dx

The number e is a fundamental constant; its/ approximate value is 2.71828. The formulas for the derivative and integral can also be derived from the fact that the exponential function can be defined as the number e raised to the x power, without making use 9f Eq. (26). These derivations are given in most elementary calculus texts.

LOGARITH MS

Suppose we want to multiply two numbers a and b which may be expressed as a

=

eX

(28)

Using the law of exponents, we can write the product ab

=

(eX)(e Y ) = e"+ Y

(29)

Now by using Table 5 we can avoid having to multiply a and b directly. As an example, let a = 1.1052 and b 1.1618 (two values which are in the table.) These values correspond to x 0.10 and y = 0.15, respectively. Then the product of a and b should correspond to x + y = 0.25 or to 1.2840. Check this by direct multiplication. Although we have used the number e as a "base" in the above example, any base could have been used. For convenience lOis commonly used. If we have U

v

= lOX

lOy

the logarithm gives the inverse relation between x and write

x = 10glo U

y =

U

or y and v, and we

IOg10 V

where we have indicated that the logarithm is taken to the base 10. The product UD lOx+y

.

implies that log UD

X

+y

log U

+ log v

Then to find the product of U and D, we first find the logarithm of U and the logarithm of D. We add the two logarithms to obtain the logarithm of UD. We then use the logarithm table again to find out what number has the logarithm x + y; this number is the product of U and v. Simple? Logarithms to the base e are often convenient in analysis because of the simplicity of their derivatives. The usual notation is In u. Thus if x = In u, then U = eX. This relation can be used to compute the derivative ofll\,u. From Eq. (26), we have du

dx

..

U

and

dx

du

U

17

III

I,

mathematics and statistics

so we obtain the simple result d

-In u du

1 u

=­

(30)

We can also derive the additional relations e1n " = In e" = u

(31)

Their derivation is left as an exercise.

LOGARITHMIC SCALES AND THE In addition to the ordinary graph paper used in Experiment MS-I, there are SLIDE RU LE available graph papers in which one of the grids gives the logarithm directly

while the other grid is linear. This is called a semilogarithmic grid. There are also papers which have two logarithmic grids. These papers are said to have full logarithmic grids, or log-log grids. A commonly available one-cycle semilogarithmic grid has dimensions of 7 by 10 in. This means that the logarithm (to the base of 10) of the number tabulated on the left is just equal to the distance (in inches) along the vertical scale divided by 10. To check this, draw a diagonal at 45° starting from the lower left-hand comer and using the graph paper complete Table 6. Compare your values with a standard logarithm table. What are you checking?

,

I ,I

:

1:11 'f TABLE 6

N

logN

N

1.00

3.00

1.50

3.50

2.00

4.OC

2.50

4.50

log N

I,

I

:

:,i

,C

r,.,

5.00

,

') By cutting a vertical strip off the right side of the paper you can make a slide rule for performing multiplication and division. We leave this to your ingenuity. How would you make a slide rule for squares and square roots? Cubes and cube roots? Semilog paper is useful in investigating the relation between two variables if it is suspected that there is a logarithmic or exponential relation. For exanwle, suppose the voltage V(t) across a discharging capacitor is thought to vary as (32) 18

"

trigonometric and exponential functions

MS-2

where Vo is the initial voltage at time t = 0, and ':1. is an unknown constant. We take natural logarithms of both sides of Eq. (32) to obtain In V(t)

= In

(33)

Vo - At

Since In V(t) is directly proportional to t, ifwe plot the data on semilog paper, using the log scale for V(t) and the linear scale for t, the result should be a straight line. Furthermore, regarding the t axis as the horizontal axis, the slope of the line is equal to the constant A (except for a change of sign) and the intercept on the logarithmic axis at t = 0 gives the value of Vo. Thus the exponential relation can be verified and the constants in Eq. (32) determined without having to look up logarithmic or exponential functions. Of course, if this equation is not obeyed, then the semilog plot will not be a straight line. A similar application of logarithmic scales on graphs involves log-log paper, on which both scales are logarithmic. A common application of this paper is the situation in which one variable is thought to be proportional to a power of the other, such as (34) where A and n are constants. Taking natural logs of both sides of this equation and using the properties of logarithms, we obtain Iny

=

InA

+ nlnx

(35)



Thus, In y is directly proportional to In x, and a plot ofln y versus In x should be a straight line with slope n. Alternatively, we plot y as a function of x on log-log paper. The slope of the line gives the value of the exponent n in Eq. (34). In this case A cannot be obtained from an intercept, since In x is never zero for any finite value of x. If the two variables are not related by a power law of the form of Eq. (34), then the log-log plot will not be a straight line.

questions 1 For ()

0.10, by how much does the approximate value of the derivative, calculated from the differences, differ from the true value 1 Does this error increase or decrease as () increases 1 For example, for () = 0.20 is the error greater than or less than it is for () = 0.101

2 Derive an expression for the derivative with respect to x of sin ax. 3 Find a differential equation analogous to Eq. (26) to define the function e- X • What is the derivative of e - x 1 4

For what value of x does eX

=

01

5 Derive Eq. (31). 6 In the logarithmic scales used in graph paper, is some particular number

chosen as the base of logarithms 1 Explain.

19

experiment

M S-3

loaded dice

introduction The last four experiments in this unit deal with probability and statistics. As mentioned in the INTRODUCTION for this unit, there are two basic reasons for the importance of statistics in physics. One has to do with the analysis of physical measurements containing random and unpredictable experimental errors; the other is concerned with statistical descriptions ofphysical systems, such as a gas containing a very large number of molecules, and with phenom­ ena which are intrinsically statistical, such as radioactive decay and quantum­ mechanical descriptions of systems. The present experiment poses a few questions about a simple physical system which, while it is not very directly related to basic physics, is never­ theless of some practical interest. We shall not be able to answer completely all the questions we raise, and some of the answers will be intuitive and imprecise. Nevertheless, this beginning will point the way to be followed in the remaining experiments. In the present experiment we have a pair of dice; one has been loaded by the insertion of lead slugs, and the other is unloaded. The problem is to determine which die is loaded and on which side. Now if a problem like this were presented to a physicist, he might suggest various ways of making this determination. One way would be to find a liquid with about the same densi ty as the die, immerse the die in the liquid, and note whether one particular side always faces up. Alternatively, the die might be permitted to fall in a viscous liquid that is less dense than the die. Another method (somewhat less accurate) would be to suspend the die from threads attached in several dif­ ferent ways and in this way locate the center of mass. In this experiment we take the gambler's approach, which may appear to a physicist to be the worst possible method. We simply toss each die repeatedly, recording each time which face comes up. The problem with this method is that there are likely to be large errors. Which face comes up depends on the position when the die is released, the amount of spin that it has and how it hits the backboard and table, as well as how it is loaded. However, if we assume that all these factors are reasonably random and we toss the die a large enough number of times, we may expect that we should be able to detect whether the die is loaded, and how. A much more difficult question is the following: How sure are we that the die is unloaded or loaded in a particular way? In the analysis of this experi­ ment we shall quote without proof the result of statistical analysis of prob­ lems of this sort. In later experiments we shall develop the theory needed for this experiment. But you may find it more interesting to do a real experiment first! 21

mathematics and statistics

experiment You should have two dice, numbered I and 2. Let N be the number of times a die is tossed in a particular experiment and n be the number that comes up. That is, n ranges from I to 6. The number of times each number comes up is written F(n) and called the frequency. Thus in a given experiment, if the number I appeared 7 times, the number 2 appeared 5 times, and so on, we would have F( I) = 7, F(2) = 5, and so on. Toss each die 10 times (N = 10) and record the frequency with which each face comes up, for each die. Estimate the probability for each face by com­ puting F(n)!Nfor each frequency, and plot histograms of the data, similar to that shown in Fig. 4. FIGURE 4

0.4

F(n) ~

0.3 0.2 0.1

n

We may now compare this with the theoretical probability, which we denote by fin). What should we expectf{n) to be? For an unloaded die there are six equally likely events, corresponding to the six faces, so we expect fin) = i = 0.1666· ... If we observe deviations from this value, it is either because N was too small and random fluctuations were significant or because there was a systematic difference between faces (the die was loaded). The statistical problem is to determine from an analysis of the data whether the observed deviation fromf{n) = i is significant. For a die we may expect on physical grounds that if a given side comes up more often than random, it does so at the expense of the opposite side. Thus we may want to look at differences in probability for opposite sides. In Fig. 5 we replot the histogram of Fig. 4 in this way. Can you guess from your data which die is loaded and which is the heavy face? Make a tentative guess. Now toss each die 100 times recording the frequencies. Plot new probability histograms similar to Figs. 4 and 5. Now which die do you think is loaded? Which do you think is the heavy face? Were you right the first time? The chances are that you were fooled. Now we can ask the following questions:

• How sure (based on your data) did you have a right to be the first time-after 10 tosses?

• How sure did you have a right to be after 100 tosses? • How many tosses do you have to make in order that you have significant data? This is, ofcourse, a very useful thing to know because there may be little 22

loaded dice

point in continuing to toss the die if you can already identify the heavy face with 95% certainty.

It es

MS-3

FIGURE 5

p. is Ie ie

:h 1­ ,0

e e

:t

r

e e e

-0.3

THE CHI-SQUARE TEST If we make N observations and each one has v possible results (v

6 for a die), we can make a prediction about the deviation to be expected between the observed frequency F(n) of a given event and the predicted frequency Nf(n). For 60 trials we may find that the frequency for a given n is 12 instead of the lOwe expect; for 600 we might find 94 instead of the 100 we expect. The important point though. is that as the number of trials N increases, the difference between predicted and observed frequencies does not increase as rapidly as the predicted frequency itself. In fact, there is reason to believe that on the average this difference is likely to increase only as the square-root of the predicted frequency. We are not yet in a position to justify this statement in detail, so for the moment we take it on faith, anticipating more thorough discussion in later experiments. According to the above statement, the quantity F(n) - Nj(n)

[Nj(n)r I2

for any given value of n ought to have a magnitude the order of unity. To eliminate the possibility of negative differences, we square the above expres­ sion; then we add these terms for the v different values of n (again, for a die v = 6). The result is usually denoted by X2 : ,2 _

X

"

-7

[F(n) - Nf(n)J2

Nf(n)

(36)

We expect that this sum will be order of v. If it is appreciably larger than v; i.e., if the observed frequencies differ from the predicted ones by an un­ expectedly large amount, on the average, then we begin to suspect that the system we are observing does not in fact follow the ideal distribution we have predicted. If the ideal system is an unloaded die, for whichf(n) = i, then a value of X2 much larger than 6 indicates that the die is loaded. 23

mathematics and statistics

For a situation for which all events are equally probable, we havef(n) = N. In this case we can simplify Eq. (36), obtaining

I/v. We also may use the fact that L F(n)

(37) For example, for the data shown in Fig. 4 we have

X2 = 10{6(0.24)

I}

= 4.4

(38)

Since this is of the order v, we may reasonably suppose that the observeQ deviations are not significant. But how do we judge significance? Suppose we repeat the sequence of 10 tosses a large number of times, computing a value of chi-square for each set of 10 tosses. We may expect a normalized distribution for l something like that shown in Fig. 6. We may characterize a given value of X2 by the proba­

o--~--------------~------~X2 bility that it is exceeded in the distribution. This is just the percentage of the area under the curve for values of X2 which exceed the particular value. This is called the level of confidence P. A small confidence level means that the chance that the original distribution of events is strictly random is quite small. Thus, in our case a loaded die should have a large chi-square and a correspond­ ingly small level of confidence. In Table 7 we give the values of X2 at various

TABLE 7

v = 6 Confidence level, P (%)

99 98 95 90 80 20 10 5 2 1 0.1

2

X

0.872 1.134 1.635 2.204 3.070 8.558 10.645 12.592 15.033 16.812 22.457 .

confidence levels for six events. Then, for a value of chi-square of 4.4 we can be almost 80% certain that the distribution is random. This is not to say that deviations would not show up for larger N but simply to say that with 10 tosses the deviations are not significant. 24

loaded dice

MS-3

Compute chi-square for 10 tosses of dice I and 2 and for 100 tosses. Which die do you think is loaded? What is your confidence level (that you are observing a statistical fluctuation)? If you observe significant deviations from randomness for one of your dice, which face (or faces) do you think are loaded? By taping a piece of thread in succession to each of three orthogonal faces, check your determination.

questions 1 For an unloaded die the theoretical probabilitiesfi:n) must satisfy the relation 6

L. f(n)

=

I

n=l

Why? If the die is loaded, is this relation still satisfied? Explain. 2 Suppose a penny is flipped 100 times, and the result is 54 heads and 46 tails. What can you say about whether or not the penny is lopsided? 3 In view of the relation stated in Question I, not all the six values offi: n) are independent; if any five are known, the sixth may be computed. Does this suggest that in using the X2 test we should take v = 5 instead of v = 6? Explain. 4 When an ideal (symmetric) coin is flipped a large number of times, the ratio

of heads to tails must approach the value unity. Does this also mean that the difference between number of heads and number of tails must approach zero? That is, might this difference grow larger and larger with the number of trials, and still give a ratio that approaches unity? Explain. 5

Could the X2 test, or some variation of it, be used to determine which side of a loaded die is the heavy side? Describe how this might be done.

25

experiment

M S-4

probability distributions

introduction We saw in Experiment MS-3 how notions of probability could be used in the analysis of a simple experimental measurement and, particularly, how such notions could be used in the design of experiments. In the remaining experi­ ments in this unit we shall develop these notions somewhat more systematical­ ly. We begin with probability considerations involved in various classical games of chance. These have very little to do directly with science; they do, however, have considerable intrinsic (and perhaps practical) interest, and in addition provide a useful framework for introducing basic ideas. We begin with the construction of a table of random numbers, which is then used in a variety of experiments. In a list of one-digit random numbers, the numbers 0 through 9 appear with equal probability. This means that, for example, if we count the number of 7's in a very large list, the total number of 7's will be very nearly one-tenth of the total number of digits. As the total number increases, the ratio approaches one-tenth more and more closely_ This, in fact, is precisely what we mean when we say that the probability of occurrence of a 7 is lo. Similarly, in flipping a coin we say that the probability of "heads" is t, which means that in a very large number of flips the ratio of the number of heads to the total number of flips is t (assuming the coin is not lopsided). We shall generate a three-digit random-number table by using a set of three icosohedral (twenty-sided) dice. If the dice are symmetric and no attempt is made to manipulate them, the number on each die should be random. Later we will learn to test whether these numbers are truly random, using a tech­ nique similar to that used in Experiment MS-3. The random-number table can be used for experimental study of various probability distributions, and the results of the experiments can be compared with theoretical predictions. In characterizing a set of numbers, especially when these numbers are associated with an experimental result such as a measurement or an examina­ tion score, several properties of the set are of interest. The most obvious one is the arithmetic mean, usually called simply the mean or average. The questions "What was the average on the exam? Was my score above or below average?" are heard in every classroom. To compute the mean we simply add all the numbers and divide by the total number of numbers. Formally, if we have N numbers denoted by nt, n 2 , • •• , nN , or a typical one by n i , where i 1, 2, ... , N, and if we denote the mean by ii, then its definition is _ n1 n =

+

1

N

L

ni (39) N i=1 Another interesting question, after the mean has been found, is how much the various numbers differ from the mean, on the average. If the average =

27

mathematics and statistics

exam score is 70 but most of the scores fall in the range between 65 and 75, the "spread" is not very great, but if they are sprinkled from 20 to 99, the spread is greater. Clearly, the significance of a score of 60 is different in the two cases. Thus we need a quantitative measure of this spread or dispersion, as it is usually called. One possibility is to simply take the difference between each number and the mean and take the average of these differences. This gives rise to some difficulty, since some differences are positive and others negative. In fact, it is fairly easy to prove that the average of the differences is always zero. To circumvent this difficulty, we square each difference, obtaining numbers which are always positive. Then we average the squares by adding them and dividing by N, and finally take the square root. The result is sometimes called the "root-mean-square deviation," but the usual name is standard deviation. This measure of dispersion is usually denoted by (1. Translating the above word definition into symbols, we have I N

N

(n;

n?

(40)

The square of the standard deviation (12 is often called the variance. Often we need to distinguish between the mean and variance for a particular set of numbers and the mean and variance of a hypothetical very large set of numbers from which this particular set was taken. For example, suppose we roll one of the icosohedral dice a very large number of times, and average the resulting numbers. It is easy to see that if all the numbers are really equally likely, the average should be exactly 4.50. But if we roll only 36 times, the resulting average may differ somewhat from 4.50, and if we roll only 9 times the average is not likely to be very close to 4.50. Thus, we distinguish between the sample distribution, the set of numbers obtained in a particular experiment, and the parent distribution, the hypo­ thetical very large set of numbers from which the sample is taken. Similarly, we distinguish the parent mean, which in this case is exactly 4.50, from the sample mean, which is in general somewhat different. In most cases we expect that the sample mean will be very nearly equal to the parent mean if the sample is very large. Similarly, we can introduce the terms sample variance and parent variance.

experiment

•

28

Construct a 360-digit random-number table, using the three icosohedral dice, and recording results in Table 8. In reading the numbers from the dice, always read in the same order (e.g., red, yellow, blue). Why is this important? Is it possible to distinguish a random-number table generated three digits at a time from one generated only one digit at a time, as with only one die? Enter in Table 9A the num ber of times each number (0 through 9) appears in the table, and from these counts calculate the probability of occurrence of each number. How do your results compare with the probabilities from the parent distribution? Select from Table 8 a subset of 36 digits. It is best to take a sequence of

probability distributions

75, the the ion, md

,me it is To !ers md lIed ion.

ave

40)

dar t of we age illy the nes

MS-4

TABLE 8

ODD ODD ODD ODD ODD ODD ODD DDD ODD ODD ODD ODD ODD ODD DDD

DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD

DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD

DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DOD

DDD DDD DDD DDD DDD DDD DDD DDD DDD ODD DDD DDD DDD DDD DDD

DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD DDD

DDD DDD DDD DDD DDD DOD DDD DDD DOD DCID DDD DDD DDD DDD DDD

DDD DDD DDD DDD DDD ODD DDD DDD ODD ODD DDD DDD ODD ODD DDD

leTS

po­

consecutive digits, to avoid bias in their selection. Enter their frequencies and computed probabilities in Table 9B. Compute the sample mean and variance

rly, the

lect

the

nee

nt

ice, lys sit ,t a

ars ~of

the

:of

TABLE 9A

Number Frequency 0 1 2 3 4 5 6 7 8 9

Probability

9B

Number Frequency

Probability

0 1 2 3 4 5 6 7 8 9

of these numbers. Compare your results with the parent mean and variance computed below. Also compute the mean and variance for the full Table 8 and compare with the parent values. The calculations are fairly simple if instead of working with individual numbers you use the frequency of occur­ 29

mathematics and statistics

rence of each number. For example, if the number ni occurs Fi times in the sample, then the sample mean is given by

n2F2 + ... L niFi 1+ n_ = -n1F... - - - - _. __.. = - Fl + F2 + ... L F;

(41)

in which the sums involve only 10 terms instead of 36 or 360. Similarly, the sample variance is given by (12

L

- ii)2

= --'--'---_ _

(42)

LFi

We also note that in each case L Fi is equal to the total number of digits N in the sample. In the present case we expect for the parent distribution Fi = N /10 for all i. Then Eq. (41) gives L ni

ii

lO

= 45 .

and Eq. (42) gives 2 (1

L (ni

ii)2

= --';'-'.1-0-··

2

=

L n. I
2

= 28.50

- 20.25 = 8.25

(43)

It would not be surprising if for a limited sample of 36 or 120 entries we find that the mean is not exactly 4.50. We should expect, however, that the agree­ ment improves with larger sample sizes. Try to make an intuitive judgment as to whether your results are reasonable, and note whether or not agreement improves with sample size. Just what constitutes a reasonable deviation will be discussed in Experiment 5.

PAIRS OF DIGITS

Now select 100 pairs of digits. This is done simply by taking the first two digits from the left in each of a series of three-digit groups. Now take the sum of each pair, to obtain 100 numbers in the range between 0 and 18. Note that the sums of digits are not equally probable because most of them can occur in several different ways. Table 10 shows in incomplete form the various ways in which each of the sums can occur, and the corresponding probabilities. Complete this table. Do the probabilities add up to unity? Enter in Table 11 the frequency with which each value between 0 and 18 occurs in your IOO-pair sample. Divide by the number of pairs to obtain the sample probability and compare with the parent values as given in Table 10. Compute the mean for your sample and compare with the parent value of 9.00. You may wish also to compute the variance for your sample and com­ pare it with the parent value of 16.50. Note that the computed variance for the sum of two digits is just twice the variance for a single digit. Why?

questions

3D

1

When a weather forecaster says that the chance of rain is 1 in 10, what does he mean? In what sense is this situation a statistical one?

2

Among 5 random digits (each between 0 and 9) what is the probability that all are 7's? That none is a 7? That exactly one is a 7?

probability distributions

TABLE 10

Sum

o

Probability 00

1/100 = 0.01

01

10

2

02

11

20

3

03

12

21

30

4

04

13

22

31

16

79

88

97

17

89

98

18

99

e

MS-4

2/100

= 0.02

3/100 = 0.03 4/100 40

0.04

5/100 = 0.05

5

11

6 7

8 8

i)

9

d 10

It It

11

II

12 13

o

14

n

it

15

Ir 8

8

e

3/100

= 0.Q3

2/100

= 0.02

1/100

= 0.01

l.

)f

l­

Ir

:8

It

3 Prove that for any set of N numbers, the average of the deviations from the mean is equal to zero. 4 What are some possible sources of bias in your random-number table which

would make it not truly random? 5 For any set of N numbers n;, prove that the variance is given by - (n)2 where
6

(12


Show that the variance of the sum of p numbers nj is just p times the variance of a single number n j • Note that whereas the mean is proportional to p, the standard deviation increases only as the square root of p. Thus, the more digits in the sum, the narrower the distribution is as compared with the mean. 31

mathematics and statistics

TABLE 11

Number

Frequency

Probability

Number

0

10

1

11

2

12

3

13

4

14

5

15

6

16

7

17

8

18

Frequency

Probability

9

7 In Table 10, show that the probabilities add up to unity, without actually

adding them. Hint: The sum of the first N integers is tN(N

+

I).

8 Derive the values given for the parent mean and variance for the distribution of sums of two random digits. 9 Suppose a table of random octal-base digits is constructed by rolling one or

more octahedral dice. Find the mean and variance for the distribution of digits, and for the distribution of sums of two digits.

32

experiment

MS-5

binomial distribution

introduction In this experiment you will use your random-number table and some other simple experimental equipment to study a particular probability distribution of wide usefulness, known as the binomial distribution. To introduce the binomial distribution, we begin with several coin-tossing problems. When a symmetric coin is tossed, the probabilities for it to land "heads" or "tails" are each t (i.e., 50%). We have no control over the way it lands, and each toss is independent of all previous tosses. Thus if we have just tossed a head, the probability of tossing another head is still the coin doesn't remember the previous toss. What is the probability of tossing two heads in a row? This is a different question. The success of this event depends on the successes of two inde­ pendent events, each one of which has a success probability of t; the proba­ bility for the composite event is the product of the separate probabilities, or l Another way to obtain this result is to note that when the coin is tossed twice, there are four equally probable results:

HH

HT

TH

TT

Only one of these four is the event we want, so its probability is;t. We would obtain the same result if we tossed two identical coins at the same time, instead oftossing one coin repeatedly. The time sequence is ofno consequence. In three tosses, each of the three has two possible outcomes, so the total number of equally probably events is 23 , or 8. They are HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

t.

Thus the probability for three heads in a row is In fact, the probability for each of the above events is (1)3, For N tosses, the probability of any specified arrangement of heads and tails is (1)N. We now ask a slightly different question. What is the probability for exactly two heads in three tosses. We see that there are three different arrangements which give this result-HHT, HTH, and THH. Each has probability t, so the total probability is i. Similarly, the probability for exactly one head in three tosses is also l The probability for some number of heads between zero and three is t + i + i + t = I, which is not very surprising. What is the probability for n heads in N tosses? First, the probability for any particular arrangement of n heads and N - n tails is (t)N. But how many different arrangements of n heads are possible? That is, what is the number of combinations of N things taken n at a time? To answer this question it is use­ ful to imagine we have N coins, each with its own landing place. In distribu­ ting the n heads we have a choice of N places for the first one. For each of these we have a choice of any of the N - I remaining places for the second, 33

mathematics and statistics

for each of these N - 2 for the third, and so on, until we place the nth head in any of the (N n + 1) remaining positions. Thus the total number of arrangements of n heads among N tosses would appear to be N(N

I)(N

2)(N

3) .. · (N - n

+

1)

(44)

This is not correct, and the reason is that we have counted as separate many arrangements that are really equivalent. We do not care which of the n heads are in the various positions; a head is a head is a head. Thus to get the correct number of arrangements we must divide by the number of ways of rearranging the n heads among themselves. To compute this, note that in a rearrangement of n objects we can choose anyone of the n first, any of the remaining n - 1 second, and so on, until at the end only one remains. Thus the number of arrangements of n objects, usually called the number of permutations of n objects, is simply n(n - l)(n - 2) ... (3)(2)(1)

= n!

(45)

The expression n! is read "n factorial" and is an abbreviation for the product of all the integers in decreasing order, starting with n. By definition, O! = 1. Finally, the number of different ways of arranging n heads among N trials (which is usually called the number of combinations of N things taken n at a time) is N(N - I)(N - 2) ... (N n + 1) (46)

n!

This expression, usually denoted by numerator and denominator by (N

(~) can be simplified by multiplying n)!, which gives

(~) = (N N~)! n!

(47)

Finally, the probability for n heads in N tosses, which we denote by P N(n), is (48)

To check, we may compute the probability for two heads in three tosses, which we already know is i. In this case n = 2 and N = 3. We find

3 8 It works! You may want to check some other possibilities.

We now consider a slight generalization. In the coin toss the probability for an individual head was t. Suppose we again have N independent events, but the probability for a win in each is not t but some other number p between zero and unity. What then is the probability for exactly n wins in N trials? The whole calculation goes through just as before, with one simple change. We have to find the probability for a specific particular arrangement of n wins, and multiply this by the number of ways of choosing n wins among N trials, which is again given by Eq. (47). The probability for a specific arrange­ n losses, in which each win has a probability p and ment of n wins and N 1 - p is each loss a probability q (49) 34

binomial distribution

MS-5

Thus the probability for exactly n wins in N trials, when each trial has a win probability of p, is

PN,p ( n)

n

=pq

N-n(N) n

=

p" N-n N! q (N - n)!n!

(50)

We note in the coin-tossing case that we have p = q t, and this expression reduces to the previous result. As a simple application of the generalized probability formula, consider rolling several icosohedral dice, For example, when rolling three dice, what is the probability of getting two 7's? Here the probability p for the individual event is lo, and we have N = 3, n = 2. Thus the probability is

P 3 ,lilO(2) =

C~YC~y-2 (3 ~!

0.027

The probability distribution given by Eq. (50) is called the binomial distribution because of its close relation to the binomial expansion coefficients. It is not difficult to show, in fact, that

(q + p)N

=

nt

(:)qN +

pnqN-n(:)

(~)qN-lp + (~)qN-2p2 + ... +

(~)pN

(51)

Thus, we see immediately that H

L

PH,p(n) = 1

(52)

n=O

as expected. (Why is this expected 1) The mean value of n for the binomial distribution is of interest. Returning for the moment to the problem of flipping N coins, it is obvious that the average or mean number of heads is the total number of trials N multiplied by the probability (t) of heads in each trial, or N12. It is plausible, although somewhat less obvious, that even when p is not equal to t, the mean value ii is equal to

ii = Np

(53)

This can be shown to be true, in fact. The proof involves some "acrobatics" and will not be given here. * By similar acrobatics* one can calculate the variance for the distribution, which measures the "width" of the distribution. This turns out to be given simply by (}2

POISSON DISTRIBUTION

*

= Npq

(54)

When N is large, the binomial distribution formula becomes quite unwieldy because of the presence of factorials of large numbers. Fortunately in this case, there are approximate representations which are much easier to use. We consider here the appropriate approximation when p grows very small as N grows large, so the mean ii Np remains finite, leading to the Poisson distribution. A different approximation, appropriate when p does not grow small as N grows large, will be considered in Experiment MS-6. This is the normal or Gaussian distribution. For details of the proof, see H. D. Young, "Statistical Treatment of Experimental Data," McGraw-Hill Book Company, New York, 1962.

35

mathematics and statistics

The approximation valid for very large N and very small p is as follows: In this limit the only values of n with appreciable probabilities are very small compared to N. Consider first the factor N! -=-=(N-=----:-:-! = N(N - 1) ... (N

n

+ 1)

(55)

This is a product of n factors, none of which is significantly different from N. We therefore replace Eq. (55) by N n • Second, we write the factor qN-n as qN-n = (1 _ p)N-n = (l - pt (1­

(56)

and note that the denominator is very nearly equal to unity because it is a number very close to unity raised to a not very large power. What remains is then () ~ p"(l _

P N,p

n

P

)N N" = (Np)n(l - pt n! n!

(57)

We now introduce the abbreviation a = Np, using it to eliminate N: (58)

All that remains now is to evaluate the limit lim (1

p)l/p

p .... O

This limit is discussed in all books on elementary calculus and is shown to have the value lie, where e is the base of natural logarithms. Finally, we obtain (59)

which is called the Poisson distribution. It contains a single constant a instead of the two constants Nand p which characterize the original binomial distribution. The reason for the difference is that we have taken the limit of the binomial distribution as N -+ 00 and p . --). 0 such that the product Np remains finite. In view of the way we have obtained the Poisson distribution, it should be clear that this distribution is the appropriate one when the number of individual events is extremely large and the probability p for a win in each Np remains finite. One of event is extremely small, so that the product a the most common applications of the Poisson distribution is the description of radioactivity. We may have 10 20 radioactive nuclei, each with a probability 10- 19 for decaying in a given time intervaL Then the total number of decays in this time interval is distributed according to the Poisson distribution, with N 10 20 , P 10- 19, and a = 10. We can also find immediately the mean ii and variance for the Poisson distribution directly from the corresponding quantities for the binomial distribution, given by Eqs. (53) and (54). Since q is very nearly equal to unity, we find, in terms of the parameter a, the very simple results ii=a 36

(60)

MS-5

binomial distribution

experiment The random-number table constructed for Experiment MS-4 provides a number of interesting illustrations of the binomial and Poisson distributions. Taking the three-digit random numbers as obtained from the dice, make a frequency count of the number of 7's in each group of three digits. That is, how many three-digit numbers contain no 7's, one 7, two 7's, or three 7's? Compare your results with the predictions ofthe parent binomial distribution. Compute the sample mean and variance and compare with those of the parent distribution. Now consider groups of nine digits in the random-number table. Make a

frequency count for the number of 7's in each square group of 9; there are 40

such groups in the table. Record your results in Table 12. Now repeat this

Number

Frequency

TABLE 12

2

3

4

5

6

7

8

9

o

o

2

3

4

5

6

7

8

9

count for each of the other digits and add the frequencies across each row to obtain the frequency count for all digits. Finally, to obtain the probabilities, divide by the total number of measurements. With 40 boxes and 10 digits, this is 400. 37

mathematics and statistics

Compare your results with values of the binomial distribution P lV , l/lO(n). In computing values of this distribution, some labor can be saved by using a recurrence relation which expresses the value of P( n + 1) in terms of the value of pen). For the binomial distribution the appropriate relation, which can be verified by direct substitution of Eq. (50), is peN - n) q(n

+

(61)

1) PlV,p(n)

Thus, it is necessary only to compute PlV,iO) and use this recurrence relation. In general, this procedure can be somewhat hazardous, since any errors in the early stages will propagate through repeated use of the recurrence relation. In this particular case, though, the values of P decrease so rapidly with increasing n that there is no difficulty. In fact, you should find that for n > 4, the values are less than 10- 5, so there is no point in going any further. Now consider the Poisson approximation for the distribution of digits in the groups of9 random digits. We haveN = 9andp = lo,soa = Np = 0.9. Recompute the probabilities for values of n from 0 to 4 using the Poisson distribution. Of course, we should not expect too precise agreement, because N 9 is a long way from N = 00, but the comparison is still interesting. Note in particular that the binomial distribution has the same value for n = 0 and n 1 (an accident resulting from the particular values of Nand p used), whereas the Poisson distribution gives values for P(O) and P(l) which are about 5% too large and too small, respectively. Does the accuracy of the approximation improve with larger n, or does it get worse?

questions 1

Derive the recurrence relation given by Eq. (61).

2

Derive the following recurrence relation for the Poisson distribution: PaCn

+

a

1) = - - I Pin)

n+

3

Among a large number of eggs, I% were found to be rotten. In a random sample of a dozen, what is the probability that none is rotten? One? More than one?

4

A man starts out for a Sunday afternoon walk, playing the following game. At each street corner, including the starting point, he tosses a coin. If it is heads he walks north one block, if tails he walks south one block. Find the probability distribution for the possible distances from the starting point after N tosses. If the man plays this game on several successive Sundays, find the average distance from the starting point after N tosses. How does this average vary with N?

5' Suppose that the number of babies in a delivery follows a Poisson distribu­

tion, and that the probability of twins is quintuplets.

38

150'

Find the probability for

experiment

M S -6

normal distribution

introduction The normal or Gaussian distribution is the limiting case of the binomial distribution when N grows very large and p remains finite (as distinguished from the Poisson distribution, in which N grows large and p very small so that the product Np remains finite). The normal distribution is important for a variety of reasons. It is a useful approximation for the binomial distribution when large numbers are involved and the binomial distribution becomes unwieldy. More important, it is often found experimentally that random errors associated with physical measurements are distributed according to the normal distribution; it is thus of central importance in the analysis of experimental errors. To derive the normal distribution from the binomial, we assume that the significant values of n are very large, since N is large, so P changes relatively little from one value of n to the next. In this case we can regard n as a con­ tinuous variable, rather than an integer. In addition we make use of the fact that the binomial distribution becomes more and more sharply peaked as N increases, as shown by the fact that n is proportional to N, while (1, which Thus only the measures the width of the distribution, increases only as values of n relatively close to ii have significant probabilities. We begin with the recurrence relation introduced in Experiment MS-5, Eq. (61). Regarding P as a continuous function of the variable n, we can pen), taking approximate the derivative of this function as pen + 1) tln = 1. That is,

ft.

P'(n)

~ pen + 1)

Pen)

=

[~~~ + ;; -

Because only large values of n are significant, we replace (n tor by n. Then rearranging and using the fact that p + q P'(n) pen)

Np

n

qn

1] pen) +

(62)

1) in denomina­ 1, we obtain (63)

We now make use of the fact that only values of n near n Np are important; we replace n in the denominator by the average Np. Note that it would not be a reasonable approximation to make this replacement in the numerator, since it is the difference of nand Np, and is thus much more sensitive to small changes in n than is the denominator. We ~hen find n

P'(n) pen)

Npq

(64) 39

mathematics and statistics

or, in terms of the mean ii (54)],

=

Np and variance a 2 P'(n) Pen)

Npq [cf. Eqs. (53) and

ii - n

(65)

=---;;z

We can now integrate both sides, writing the integration constant as In C: 1n Pen) = In C

(66)

or, taking antilogs of both sides, Pen) =

Ce-(n-ii)2i2a

2

(67)

The value of the constant C is determined by the requirement that the total probability of all possible values of n must be unity. Instead of summing over n, we now integrate; C must be chosen to satisfy the relation

f Pen) dn =

(68)

1

Evaluation of the integral requires some trickery, and we simply state the result that C must be equal to 1/(2na)1/2. Finally, the normal distribution function is Pen)

=

1

e-(n-Np)2/2Npq

(2n) 1/2(Npq) 1/2

=

1

e-(n-ii)2/ 2a 2

(2na)1/2

(69)

Ordinarily the second of these forms is used, but we write both to show the relationship to the binomial distribution with the same mean and variance. FIGURE 7

P(n)

n-3u n-2u n-U

n + U n + 2u

n + 3u

Figure 7 is a graph of Eq. (69), showing the significance of ii and a. The curve is symmetric about n = ii, and a as always characterizes the "width" of the distribution. At the values n = ii ± a, the curve has dropped to e- 1/ 2 of its maximum value at n = ii. It can be shown that the total probability for a value of n between these limits is about 0.683. This value is obtained by 40

normal distribution

MS-6

integrating Eq. (69) between the limits n = jj (J and n n + (J, which requires numerical approximations. Similarly, the probability for n to fall between ii _2(J and n + 2(J is 0.954, and for it ± 3(J it is 0.997. That is, there is a probability of only 0.003 for a value of n to differ from jj by more than

5)

±3(J.

6)

experiment 7)

We can now apply some tests to our random-number table to find out whether it is really random. Consider, for example, the digit frequency obtained in Experiment MS-4. With 360 digits, we expect on the average to obtain 36 of each number from 0 to 9. Suppose we find 39 sevens; does this mean the numbers are not really random, or is this number within the realm of reasonable probability? To answer this question, we note first that the probability for a certain number n of 7's in a 360-digit random-number table is given by the binomial distribution with N = 360, p = lo. The mean of this distribution is n = Np = 360 x = 36, as just observed. The standard deviation is given by

al er

8)

ne )0

9)

(J

= (Npq)1/2

(360

X 1~O)1/2

= 5.7

~

6

Approximating the binomial distribution by a normal distribution, we note that there is a 68% chance that any digit frequency will be within one standard deviation of the mean (i.e., between 30 and 42), and only a 5% chance that it will not be within two standard deviations (i.e., between 24 and 48). Note also that the probability for it to be exactly 36 is rather small. Compute this probability. Using the above criterion, check your digit frequencies from Experiment MS-4 for evidence of non-randomness. Another test using the same principle slightly disguised is to ask whether the random digits are more likely to be odd or even. The probability distribu­ tion for the number of even digits in a table of 360 is a binomial distribution with N = 360 and p = t (since an individual digit is equally likely to be odd or even). Find nand (J correspondingly to these values, and from these com­ pute the limits of the 65 and 95% probability brackets. Count the number of even digits in your table. Is there any indication of nonrandomness? Another application of this method is the determination of whether or no~ a coin really has equal probability for heads or tails. Suppose we flip a coin 100 times and find 52 heads and 48 tails. Is the coin lopsided? Using the normal approximation for the binomial distribution, we may again compute ranges of n (with a mean of 50) corresponding to various probabilities. If the observed value of n is outside the 95% range, we suspect something is amiss. Select a coin, flip it 100 times, record the results, and interpret them.

ne

:e.

1" /2

)r.

'y

ANALYSIS OF RANDOM IN MEASUREMENTS

One of the most important applications of the normal distribution function is concerned with the analysis of random errors in measurements. This is dangerous territory for the uninitiated; an unwary student can become so 41

mathematics and statistics

overawed by the machinery of statistical data analysis as to think that in it lies the entire secret of success in handling experimental errors. Such an attitude would be dangerously misguided. It is helpful to distinguish between systematic errors and random errors. Systematic errors, as the name suggests, are intrinsic in the experimental setup. Simple examples are a voltmeter which is not properly zeroed, a ruler which is inaccurate because of thermal expansion, a thermometer which reads correctly only when totally immersed, and so on. Systematic errors may change with time either because of some property of the system or because of variation of some external influence such as temperature of the room, line voltage, building vibration, or stray magnetic fields. Very often in experimental work systematic errors are more important than random errors. They are also much more difficult to guard against and deal with. There are no general principles for avoiding systematic errors; only an experimenter whose skill has come through long experience can design experiments to avoid systematic errors, and detect and correct them when they occur. Random errors are produced by a large number of unpredictable and unknown variations in the experimental situation. They can result from small errors in judgment on the part of the observer, such as in estimating tenths of the smallest scale division. Other causes include any unpredictable fluctu­ ations in the experimental conditions, provided these fluctuations are really random. It is found empirically that such random errors are frequently distributed according to the normal distribution function, and use of this fact can be useful in minimizing the effects of random errors. We give here a few useful results. Suppose we measure a certain physical quantity several times, obtaining several values each of which deviates somewhat from the true value of the quantity. If these deviations are due purely to random errors, and if the errors are distributed according to the normal distribution function, then it can be shown that the best estimate of the value of the quantity which can be obtained from the data is the arithmetic mean or average, which is a reason­ able enough result. Now, how reliable is this mean? One can calculate a standard deviation for the individual observations, and this gives an index of their reproduci­ bility. We expect the mean to be more reliable than the individual measure­ ments. In fact, if we take several sets of N observations each, compute the mean and standard deviation for each set, and then find the standard devi­ ation for the means, we can show that this is smaller than those of the indi­ vidual measurements in the set by a factor of approximately )"N. That is, (Jmean

and the standard deviation of the mean is usually taken as an index of the precision ofthe mean. Note, however, that this conclusion is valid only if one is absolutely certain that the errors are entirely random, and that there is no systematic error which could contribute an uncertainty greater than that indicated by the purely statistical considerations. Now suppose we want to compute a quantity Q which is determined from a series of measured quantities a, b, c, ... , by means of a known relation which we may write in general as

Q 42

f(a, b, c, ... )

normal distribution

MS-6

Suppose further that each of the quantities a, b, ... , has an associated standard deviation (1a' (1b' •••• What is the standard deviation of the resulting value of Q? We can show* that the standard deviation of Q, which we may call (1Q should be calculated from the prescription (1Q

2= (OQ)2 oa

2 (1a

(OQ)2 2 +

+ ob

(1b

This is the basic relation for propagation of errors, which is the analysis of the effect which errors in individual numbers have on numbers calculated from them.

questions 1

Considering the normal distribution as an approximation of the binomial, do you expect the approximation to be most accurate for values of n near ii, or for values far from ii, in the "tails" of the distribution? Explain.

2 The "probable error" of a distribution is defined as the error such that the

probability of occurrence of an error whose absolute value is less than this value is one-half. Using a table of probability integrals, find the probable error for the normal distribution and express it as a multiple of (1. Is this the most probable error? If not, what is? 3 Find the variance terms of (1.

<en -

ii)2)av for the normal distribution expressed in

4 Find the distance from the center of the normal distribution to either in­

flection point in terms of (1.

•

H. D. Young, "Statistical Treatment of Experimental Data," McGraw-Hill Book Company, New York, 1962.

43

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berkel

hys;

s

labo~

t ry,

alan m . por is, university ot caflfornia. berkeley hugh d . young, e n ,-'g/e-mellon university velocIty an

cce/erat/on

M-1

collisions

M-2

dis ipatlv

forces

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M-4

fore d oscillations

M-S

me raw-hili book company neW' y ork london

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mech

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mechanics Copyright f 1971 by McGra w-Hi ll. Inc.

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th

m -

123456n90 BAB A 798765432 10 The first edition of th Berkeley Physics Laboratory copyright 1963 , 1964, 1965 by Education Development Center \\<1 . supported by a grant from the 'ationa l Science Foundation to EDC. This material is available to publishers and auth ors o n a royalty-free basis by applying to the Education De velopment Ce nter.

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contents

berkeley physics lab o rato ry, 2d edition complete con ten ts of t h e 12 units

mathematics and statistics derivatives and Integrals trigonometric and exponential functions loa d ed dice p r obability distributions binomia l distrIbution norma l distribution

MS-' M5-2 MS-3 MS-4 M5 -5 M5 -6

mechanics

M-' M-2 M-3 M-4 M-5

v elocity and acceleration co llisions dissipative forces periodIc motion fo rced oscillations

electronic instrumentation v o ltage, current. and resistance measurements f/-2 m easurement of ac voltage and curren t f/-3 waveform measurements f/-4 comparison of variable voltages f/-5 transducers f/-'

acoustics and fluids AF-' A F-2 A F-3 AF-4 AF-5 AF-6

acous tic waves acoustic diffraction and Interference acous tic In terferometry fluid f low vis c o u s fl o w turbulent fl o w

microwav e o ptics MO -I MO-2 MO -3 MO -4

mic rowave production and reflection in t e r fe ren c e a n d diffraction the klystron mic rowave propagation

laser optics LO -I LO-2 LO-3 LO-4 LO -S

reflection and r e fraction of light p o lariz a tion of light diffr action o f lig ht Inte rfe re n ce of ligh t holograp hy

atomic physics

fields

A P -I

F-'

r adial fields F-2 im age charges F-3 fie ld /lnes and reciprOcity F-4 the magnetic field F-5 m agnetic coupling

AP-2

A P-3 AP-4

AP-5

atom iC s p e ctra photoelec tric effec t the photom ultip lie r and photon nOise Io nization by e l ectrons e lectron d if f ra c tion

nuclea r physics electrons and fields fF-' fF-2

fF-3 fF-4 fF-5

acceleration and deflection of e lec trons fo cusing and intensity control m g netic deflection of e le ctrons helical motion of ele ctrons va cuum diodes and the magnetron condition

N P-I NP- 2 NP-3 NP-4 NP-5

g e ig er-mueller t u be r a d io a ctive d ecay th e scintillation c o unter beta and g a mma absorption neutron a ctiva tion

semic onductor elect r o nics electric circuits f C-' resistance-c a p a cita n c e circuits f C-2 r eSistance -inductance circuits fC-3 LAC circuits and oscillations fC-4 c oupled oscillators fC-5 periodic structures and transmiSSion lines

SE-1 S E-2 SE-3

S E-4 SE-5

SE-6

s e mico n ductor d iodes tu nn e l diodes and relaxation oscillators th e tran sistor t r ansisto r a m plifiers pOSitive f eedback and o sc illation n e gative fee dba ck

mechanics

INTRODUCTION

In this series of experiment yo u will ob erve several basic phenomena of classical N ewto nia n mecha ni cs. A variety of experimental setups will give yo u experienc in applying the b' sic princip les of mechanics, such as ewton's laws o fm o tioo, m o me ntum and energy r lat ions a nd t he associated ki nematic de criptions of motion, t the anal ysis of phenomena whjch you can observ . a nd mea 'ure . Most of the experimental situations wil l be simple eno ugh so you can represen t the properties o f the sy tern under study by mea ns of sim ple mathematical rela tions. For example, if the amount a pri ng stretches is app roximately proportional to the fo rce making it stretch. we exp ress this beha vio r with the rela tion F = - kx. It mu t be understood , h owever, that such rela tionships a re seldom exact, but instead Cl re pa rl of an idealized model used to represent the p rop rties of the system. Many real pr ings do beha e very nearl y accord ing to the a bove relation, although if the p ring is stretched too far, the elo ogation may no I nger be di rectly propoTti nal to the force. In additio n , a perma n ot st ret h may result. T hus, am thema tical analysis of a physical y te rn almost always involves the u se of idealized models which prov ide an pproximate d scription of the pro perties of the system . It is im po r tant to bear tlus in mind wh n comparing your ana lytical pred ictions with actual o bservations of the behavior of the system ' the two will seldom agree exactly. The disagreement can be cau ed either by experiment al erro r (that is, errors in the measuremen ts) or by the lack f preci ion of the m odel. or bo th, a nd it is important to u nder ta nd the di sti nc tion. Al though much of your experimental work is concerned with making quantitative mea urements, the importance of qualitative observations shou ld not be overlooked. Often qu alitative o bservations, including t he effects of changing the vat;able quan tities in the experimen tal setup, will help you gain additio nal io ight and phy ical intuition for the physics of the sit uatio n. It is al ways u eful to record these quali tative observa tions a s well as the numbers resu ltin g from your quanti ta tive measurements, for later reference. T he experiments in this serie all ma ke use of a linear air tfll ck a simple but el gant d vice which perm its the o bservatio n of motion with al mo t com­ plete a bsence of friction. Gl iders or sleds are suppo rted above a straight track on a cushion of air the o rder o f 0.1 mm thick, p roduced by blowing air o ut of rows of small holes in the track. T he p rinciple is th ame as that of the s -called hover-craft · the only fric tio n is t ha t associat d with t he viscosity of t he layer of air on which the glider floats . T o this basic setu p we add t iming device, bumpers on the ends of the glider for various k inds of collisions, and equipment for appiying contro!Jed time-vary ing fo r es to glider , T h is auxiliary equ ipment is descri b d in the ind ivid ual experiment . T h e timing measu rements needed in studying motion can be made in a 1

variety of ways . T he two methods used in these experiments are an rdinary stopwatch, which measures a time interval, and a spark-reco rding etup, which us s a strip of waxed paper tape attached to the track a nd an electrode att ached to the glid r. A timing d vice generate voltage pu lses at regular interval , a nd the re ulting sp rk holes in th waxed paper provide a r ord of the po ition of the glider at each time. Other techniques which you may wish to explore include strobo copic pho tography, either wi th a multiple­ flash strobe li ght or a rotating-disk shutter, use of photocells Witll a light beam interru pted by the glider, and use of Doppler radar. [n a ll these schemes, the object is to observe the motion of the glider with ut having th o bservations affect the motion.

CAUTION

2

Several genera l precautions should be observed wh ile using the air tracks . Although tlle track appear qu ite rigid, their use ma kes them extremely sensitive to small changes in alignment. Be careful not to bum p, jar, or drop the gliders unnecessa rily. Drop­ ping the gliders from a height of even a few inches will probably ruin them. To avoid damaging surfaces. do not slide gliders on track when the air supply is turned off. W hen ad ditional weights are attached to gliders, be sure to add them symmetrically · if the glider is I psided, it will rub on the track . When the spark timer is u ed, be extremely careful not to toucb the parking wire; shocks fr m high- oltage sparks are not always lethal but they can burn mall, deep, and peculi arly painful holes in you r skin.

experiment

M -1

v eloc ity a nd acceleration

introduction According to Newton's first law, an object set in motion on a perfectl y smooth, level, frictionless surface continues to move in a straight line with constant velocity. By observing how closely the gliders on the air track obey this prediction, you can form some conclusions about how straight and level the track is and about how important the small friction caused by viscosity of the supporti ng air layer is. According to N ewton's second law, when a force is applied to an object, the object experiences an acceleration proportional in magnitude to that of the applied force . This relationship is usually expressed as

l: F= ma

(I)

,in which the symbol L , which means sum, indicates that if more than one force acts on the object, the vector sum of fore s must be used . In this experi­ ment the principal forces will be constant ; that is, they will not vary wit h time. A simple way of providing an accelerating force is to tilt the track by a meas­ ured amount; the acceleration may be predicted from the angle of tilt , and it may also be determined experimentally from measurements of the positions of the object at a succession of time intervals.

experiment 1

op eration of th e air tra ck F irst familiarize yourself with the operation of the air track by turning on the air supply and observing the motion of the gliders on the track, including collisions with spring bumpers on the ends of the track and with bumpers on the gliders. N ow carefully place a glider near the cen ter of the track and release it so as to give it no initial velocity. Adjust the leveling screw at the end of the track until the glider accelerates neither to the right nor to the left. Now move the glider to other positions on the track and check to see how strai ght the track is. If the track has been properly aligned , all points on the tra ck should be within 0.002 in. of a straight line. After the track is leveled, carefuUy avoid bumping it, since small irregularities in the tabletop can have a significant effect if the track is moved .

2

constan t veJoci.ty Attach a compression spring to one of the end bumpers with a piece of Scotch tape. Practice pressing a glider against it and releasing it, until you can 3

mechanics

launch the glider with a speed of about 10 to 20 em/sec. An alternative arrangement is to use a suitable mounting bracket to stretch a rubberband acro s the track near the end, and then press the glider against the side of the ru bberband and release it. Now insert a strip of waxed paper spark-recording tape, turn on the sparker, and launch the glider. Be sure to tum off the sparker as soon as the glider reaches the opposite end of the track to avoid confusing the spark record with holes for the return trip. Remove the tape and lay a meterstick on it, on edge so the markings are adjacent to the spark holes. Do not place the end of the meterstick at the first hole. (Why?) Without moving the stick, record the positions of the holes. Also determine the sparking rate. Plot a graph showing position of the glider as a function of time. What shape should this graph have? Determine the vel ocity of the glider during the first interval, and during the last interval you can measure, taking care not to use the interval during which the glider was launched. (Why?) Within the limits of experimental error, is the predic­ tion of Newton's first law borne out? If not, why? Now repeat this experiment, compressing the launching spring about half as much as in the first trial. How does the velocity compare with that in the first trial? What relationship does this suggest between spring compression and velocity ? Do the departures from Newton's first law seem more or less p ro nounced than in the first trial? Why might this be expected? Compute the change in velocity over a fixed length of track (say 50 em) for each trial, and compare the results. What does this comparison suggest? If the spark tinting arrangement is not available, you may time the glider with stopwatches for two successive 50-em intervals during the motion .

3 accelerat ed m otion When a body is released from rest near the surface of the earth, it accelerates downward with constant acceleration, provided air resistance is negligible and the body travels a distance which is small compared to the earth's radius. The magnitude of this acceleration varies by a few tenths of a percent, depending on location, but is approximately

g = 9.80 m/ sec 2 Ra ther than try to measure this acceleration directly, we can tilt the track a predetermined amount and measure the acceleration of the glider; from this we can compute g. If the angle of inclination of the track is (/., the acceleration of the glider down the track is given by a = g sin (/.

(2)

Derivation of this relation is left as an exercise. Many co mmercially made air tracks have a leveling screw with 20 threads per inch a nd a distance of exacUy 50 in . between the leveling screw and the feet on the opposite end. Thus one complete revolution of the leveling screw incunes the track at an angle whose tangent is 0.050 in./50 in. = 0.001. Since the angle is very small, the angle (in radians), its sine, and its tangent are all very nearly equal. Thus for each turn of the leveling screw the angle of track is increased by 0.001 rad (radian) or I mrad (milliradian). W ith a track inclination of 2 mrad, start the glider with zero initial velocity and make a spark record of its position as a function of time. Using a meter­ stick (as in Part 2 of this experiment), measure the position of each spark hole. 4

velocity and acceleration

M-1

The acceleration may be determined from the spark data in vario us way ' . The simplest way of doing this is to make use of the rela tion (3) for the object starting from rest a t position So and at time t = 0 a nd moving with constant acceleration a for ti me I. If s is plo tted as a fu nction not of t but of t 2, the result should be a st raight line whose slope is T he resul t for may be compared with the prediction of Eq. (2). Another method which prov id es more detailed in fo rmation is to com pu te differences between successive positions and from these calcul ate the a vera ge velocity in each time interval. Then by taking d ifferences between successive velocities, the average acceleration between each pair of successive in tervals can be obtained. This calculat ion can be arranged in tabular form, as in Experiment MS-l. It ,is easiest first to calculate the velocity in centimeters per interval, rather than centim eters per second, and 'i m il arly fo r the acceleration, and then convert into centimeters per second squared a t the end. Is the acceleration constant, within the lim its of experimental precision ? If not, why? Does the value of acceleration agree with the pred ictjon of Eq. (2), within experimental precision? If the spark-timing arrangement is not availa.ble, you may time the glid r with two stopwat0hes for two successive 50-em intervaJs during the m otio n. From these measurements you can calculate the average vel o ity in each mterval, and the average acceleration from the center of one interval to the center of the other. Can you think of a way of checking th e accuracy of your stopwatch times by timing a known interval repeatedly ? This experiment may be repeated with a larger track incl ination, say 10 mrad. Is the deviation from constant acceleration rela tively greater or less than in the first case? Why'?

tao

4

a

mass of glider The final measurement in t his experiment is to determine the m ass of a glider. This is accomplished by applyjng a known force to the glider a nd measuring its acceleration. T he track is returned to its level position and shou ld be carefully checked to see that it is level. W e a ttach a small weight (the order of 5 g) to a length of i-i n.-wide magnetic reco rd ing tape wh ich passes over the air bearing and is attached at its o ther end to the glider. Let the mass of the glider be M, and the mass of the end weight m. The tension T in the tape is less than t he weight of rn. because rn accelerates down­ ward as M accelerates. Thus the equation of m otion for m is rng -

T

=

rna

and that for Mis

T= Ma Combining these equations to eliminate T, and solving for M, we find (4) The acceleration a may be measured by any of the methods d iscussed a bove. In addition, you may wish to place additional weights on the glider' be care­ ful not to load it so hea ily that it no lo nger floats freely. 5

mechanics

You may now want to repeat orne of the earlier parts o f the experiment using a weighted glider. What systematic d ifferences would you expect?

questions ,

Approximately what is the percen t difference between sin If a = 100 ?

IX

and

CI..

if rJ. = IO?

2 How great is the experimental error in mea uring the posi tions of the spark

holes in the tape? Approximately wh t erro r in the value of accele ration might result from these errors ? 3 Should the friction due Lo the vi co ity of the air layer on whi ch the glider

floats be more signifi a nt at smaU velocities or at la rge velocities? W hy? 4 Make a rough estimate of tbe magnitude of th viscous frictio n force, using

your data for the level track . F ind the appro im ate accelerat ion for the level ru n, and find the fric tional force using F = rna. 5 How would you expect the viscou friction force to vary wi th the mas of the

glider ? Why? How migbttl'li prediction be check d experimenta lly ? 6 In Part 3 of this experiment, suppose the track is tilled at an angle a instead of being flat. Derive an exprcs ion fo r the accelerati on ; show that th e acceleration may be in eiLher direction, depending on the val ues of m, M , and ~ , and that there is a critical angle for wl'lich the acceleration is zero. Could M be determined from a measurement of this angle? How? What advantage would lh i scheme have?

6

e xp eriment

M -2

collisions

in troduction In this experiment you will investigate coi'lisions between gJiders on an air track. The principles used to analyze these collisions are Newton's second and third laws, which are used to develop the principle of conservation of momentum. In this experiment two gliders move on a horizontal track with no horizon­ tal forces except the forces they exert on each other during a collision. Let the masses be m l and m 2 , and their velocities VI and V 2 . Velocity is a vector qua ntity ; we can take VI to be positive if Inl moves to the right, and negativ if to the left, and the same for V 2 . Both VI and V 2 are functions of time, since they change d uring the coll ision. While the gliders are in contact during the collision, they exert forces on each other. Let the forces on m I and m 2 be F I and F}., respectively, with the same sign conventions as for the velocities. N ow, according to Newton's second law, and

(5)

Newton's third law states that the two interaction forces FI a nd F2 have equal magnitude but oppo ile direction, so w also have (6)

Combining this with Eq. (5)

which may be rearranged (7)

The quantity In l VI is defined to be the momentum of the fi rst glider, usua lly denoted by P I' and simila rly for the other glider. Thus, Eq. (7) states that the total momentum P I + P2 does not change during the collision, inasm uch as its time derivative is always zero at each instant during the collisio n. (8)

This is a very powerful resul l inasmuch as it makes no detailed assumptions about the fo rces which may vary in a co mplicated way during the collision. The result is often expressed by saying that the momentum is consen:ed in the colli. io n. Of COllfS , the principle is valid only when there are no hori­ zontal forces on the gliders other than their mutual forces. (W hy?) It is 7

mechanics

important to note that we have not proved that kinetic energy is conserved in the collision. As we shall see, kinetic energy mayor may not be conserved, depending on the kind of collision. It is customary to classify coJiisions according to the relative velocity of the two bodies before and after the collision. If the relative velocity has the same magnitude before and after, the collision is said to be perfectly elastic. If the relative velocity has smaller magnitude after than before, the collision is semielastic, and if it is zero after the collision (i.e., if the two bodies stick together), it is completely inelastic. The ratio of final to initial relative velocities is called the coefficient of restitution, denoted bye. For a perfectly elastic collision e = I, for a completely inelastic one e = 0, and for a semi­ elastic collision e is between 0 and I. In the setup used in this experiment, mass m 1 is given an initial velocity vo, and mass m 2 is initially at rest. Let the velocities after the collision be VI and v 2 . Then by definition the coefficient of restitution is v

VI

-

2 e = -=-_ ---.C.

(9)

Vo

If m 2 is larger than m l ' V I may be negative, but the relative velocity after the collision is stiU given by V 2 - V I . It is of interest to compare the initial and final kinetic energies. We shall derive the relationship o nly for the special case of equal masses, so m l = m 2 . In this case the conservation of momentum equation (10) takes the simpler form

(11 ) This equation can be combined with Eq. (9) to express

VI

and

V2

in terms of

Vo and e; the resulting expressions are VI =

i(l -

e)vo (12)

= t(1 + eko

V2

Now let R be the ratio of finall to initial kinetic energy; that is,

R

=

tl11 1 V l

2

+ ±m2 v/

I

( 13)

2

'imlvo

For the particular case of equal masses, this becomes V

2

+ v/

l R = ---"--.,........:'-

2

(14)

V0

Inserting Eqs. (12), we obtain the simple result

R

=

to + e

2

( 15)

)

This shows that the final kinetic energy is the same as the initial only if the collision is perfectly elastic (e = I); otherwise R is less than unity and the final kinetic energy is less than the initial. Note that the minimum value of R is t. A similar expression may be derived for the general case of unequal masses. The derivation is somewhat more involved; the result is

(16)

which shows again that kinetic energy is conserved only when e 8

= I.

collisions

M-2

experiment 1

equal masses

Place two identical gliders with spring bumpers on the track. Carefully place one glider at rest at about the center of the track, and direct the other toward it with a speed of about 20 cm/sec. In order to study the collision in detail, the velocities before and after the collision are needed; these can be obtained in various ways. One procedure is to time each glider for a fixed interval (say 50 cm) with a pair of stopwatches. As an alternative, an ordinary spark-timer arrangement can be used to measure the velocity of one glider. If a double­ wire spark setup is available, it may be used to find the velocities of both gliders. What conclusions can you draw concerning momentum transfer? Energy transfer? The coefficient of restitution? You may want to repeat the experi­ ment several times, especially if stopwatches are used, and compute the average value of the coefficient of restitution. For an inelastic collision, a small piece of Scotch tape with adhesive on both sides may be attached to each bumper spring. Be careful not to touch the adhesive surfaces, since fingerprints make them much Jess sticky. Again determine the initial and final velocities. Is momentum conserved? Energy? What happens to the energy that is lost? 2 unequal masses

Add weight to one glider, or use a heavier glider, to investigate collision with unequal masses. Make qualitat,ive observations for both m 1 > m 2 and m 1 < m 2 , and note the results. Choose one particular combination for quantitative observations. Note that in general it is necessary to measure three velocities. The initial and final velocities of m 1 can both be obtained from spark-timer data, and the final velocity of m 2 with a stopwatch or a second spark-recording wire. Investigate conservation of momentum and energy. If time permits, investigate an inelastic collision with unequal masses. 3

action at a distance

In the collision experiments using bumper springs on the gliders, the collision is practically instantaneous, compared with other times involved in the experiment. If we could observe the details of the collision (by using, for example, a very high-speed motion-picture camera), we would find that the bumper springs become distorted during the collision. As they flatten, the force between gliders grows to a maximum and then decreases again, becom­ ing zero as the bumpers move apart. By taping small ceramic magnets to the ends of the gliders you may set up a "soft," but perfectly elastic, collision. The two-sided Scotch tape may be used again to attach the magnets. Be sure the magnets are oriented so that they wjll repel each other, and do not let them strike each other; they are made from a brittle ceramic material and are easily broken. Plan and carry out a collision experiment using the ceramic magnets as bumpers. Work out the details for yourself. If ceramic magnets are not available or if you wish to try another experi­ ment in which the collision takes place over an extended time, you may use oversize bumper springs on the glider. Make a loop of "clock" spring about 12 cm in diameter and attach it in place of one of the regular springs, but 9

mechanics

orient d ·'backwards. · This will be possible since the spring diameter is greater than th e length of the glider. The backwards orientation keeps the spring from placing the glider off balance. Study the collision between two gliders equipped with these soft springs. 4 mag netic intera ction f orce T he magnetic interaction force may be studied in more detail. It is of in terest to observe how the force varies with distance be tween magne ts. With the gliders at one end of the track with the magnets "facing" each other, raise the opposite end of the track. The glider nearer the raised end will come to equilibrium at a position where the magnet force just balances the compo­ nent of gravitational force rng sin IX down the track. Measure the distance between magnets carefully· repeat for several track elevations, being sure to measure the elevation so a can be computed for each position. Compute the magnet force at each position , and from this data plot a graph showing magnet force as a function of distance. 5

m easuring potential en ergy The pot ntial energy due to the magnetic interaction can be measured directly. With the track inclined as before, move the upper glider up the track a meas­ ured distance and release it with zero initial velocity. It will move to wi th in some minimum separation between magnets and then bounce back up the track. At the point of closest a pproach the glider has zero velocity and, there­ fore, zero kinetic energy . T he increase in magnetic p otential energy must therefore equal the decrea e in gravitational potentia ~ energy , which is equal to mgs sin rJ. , where s is the distan ce down the track the glider moves. U se several values of s, and cha nge IX if necessary. By measuring the minimum eparation dlwing the bounce for each set up, it is possible to obtain the magnetic potential energy as a function of distance between magnets. Draw a graph of potential energy as a function of distance. F ind the slope of tbe curve at several points and use the rel ation dV

F =- ­

dx

to determine the force. Compare your results with those obtained by measuring F directly. Alternatively, you may wish to perform a numerical integration of the force function, perhaps by counting squares on the graph, to obtain V.

questio n s 1 Suppose we could place a small explosive charge on one of the bumpers so that it detonates at the time of co llision, pu hing the two gliders apart. Will momentum still be co nserved ? E plain. W ill kin etic energy be conserved? What value will the coefficient of re titution have? 2

10

Must the spring forces be proportional to displacement (that is, F = - kx) in o rder for the collision to be perfectly elastic? If not, what is the necessary condition which the force must satisfy?

collisions

M- 2

3 W hat effect will the visco us fr iction of the supporting air layer have on your co nclusion regarding conservation of momentum in collisions? 4 Derive Eq . (16) for the general ca e of uneq ual masses. 5 Show that Eg. (16) red uces to Eq . (15) in the limit when the two masses become eq ual. 6 Using your data o n the magnetic in teraction force, make a rough ord r-of­ magnitude estimate of the time of the collision with magnetic interactions. That is, during how long an in terval is there sign ificant magnetic interaction?

11

experiment

M -3

d issipative forc es

introduction In this experiment you will study a variety of dissipative forces, so called because they act to dissipate mechanical energy. These forces, which are also called dampingforces, include the various kinds of friction, magnetic damping forces, interaction forces in collisions which are not perfectly elastic, and others. During this experimen t, observe the feet of the air track to make sure it is not sliding on the table top. Why is this important? If necessary, prevent sliding by either clamping the track, placing double-sided Scotch tape under the feet, or reducing the initial velocity of the glider. You have already observed that gliders on an air track are not completely frictionless. The principal source of friction is the viscosity of the thin layer of air between the glider and the track. We can show that the total viscous force is directly proportional to the surface area A of the layer and to the relative velocity v of gJider and track, and inversely proportional to the thick­ ness d of the layer. Thus, the viscous friction force may be written as F = _ I'/Av d

(17)

where 1'/ is a constant characteristic of the fluid (air) called the viscosity. For our purposes, the most important feature of this force is its proportion­ ality to velocity, and we shall represent the force simply as

F = -bv

(18)

where b is a constant whose value depends on the dimensions of the setup and the properties of air, and the negative sign indicates that the direction of F is always opposite to that of the velocity. When a glider moves on a level track with no other force except viscous friction, its equation of motion is

F= ma

or

dv -bv = m­ dt

(19)

showing that the instantaneous rate of decrease of velocity is proportional to the velocity itself. With a given initial velocity Vo, the glider slows down rapidly at first, then less rapidly. It is easy to find the total distance the glider travels before stopping. We express dv/cit in terms of civ/dx using the chain rule for derivatives as follows:

dv dv dx dv - = - - =-v dt dx cit dx 13

mechanics

Inserting this result in ELj. (19) , dividing out the common factor v and re­ arranging. we obtain

dv dx

b m

This differentia l equation may be integrated immediately:

b v=--x+C m where C is an integration constant. If va is the initial velocity at the point x = 0, then C must have the value va' and we find v

=

b m

(20)

va - - x

This equation shows that the glider comes to rest (v = 0) after traveling a distance x equal to

mvo b

x = --

(21)

This result may also be derived by equa ting the impulse of the force to the total change of momentum:

f F dt = f - bv dt = f - b dx =

- bx = - mvo

Another example of a velocity-dependent frictional force (damping force) is the effect d ue to eddy currents in a conductor resulting from motion in a magnetic field . When an electrical conductor moves through a magnetic field . the changing magnetic flux in the conductor induces currents of magnitude direc Uy pro po rtional to the rate ofchange of flux, and thus to the velocity. T il se currents in turn experience a force which at each point is proportional to the field at that point. T he direction of the force on the con­ ductor is always such as to oppose the relative mot io n ; thus, this force can be represented the same way as the fo rce due to air viscosity, as F = -bv. In this case, the constant b is proportional to the electrical conductivity of the material, to the area of conductor over which the magnetic field extends, and to the square of the magnetic field intensity. (Why?) When both magnetic and viscou s damping forces are present then of course the total force is the sum of the individual contributions. In this experiment permanent magnets may be mounted on the gliders, and the magnetic damping is caused by eddy currents in the track as a result of its motion relative to the magnets. A third type of dissipative force in the air-track setup is associated with the behavior of the spring bumpers. W hen one glider collides with another glider or with the end of the track, the relative velocity after the collision is somewhat smaller in magn itude than before the collision; the ratio of these two reJati ve velocities i called the coefficient of restitution e, just as in Exp eriment M-2. Ex perience sh ows that for a given bumper arrangement e is very nea rly indepe nde nt of ini nial velocity. An interesting application of all these types of dissipative forces is provided by the air-track analog of a bouncing ball. The track is tilted to an angle !J. and a glider is released from rest at the to p end. It bounces at the bo ttom end, but does not quite regain its original height. A fter a series of bounces with 14

dissipative forces

M-3

succe sively decreasing heights, the olider eventually comes to [ st at the bottom of the tilted track, If the initial distance up the track is Xo, then after the first bounce the glider reaches a po int X I ' after the second X2, and so on. These distances can ea ily be measured experimenta ll y by using the scale on the air track. The appropriate equations of motion for this syste m, including vi COliS (or magnetic) friction, the bumpers, and the tilt of the track, can b solved exactly. However, the exa t solu tions are rather compl icat d and not very illuminating, and an approximate analysis i more u seful. Even though both viscous and bumper dissipation are present we can analyze the e two effect s separately. Later, it will be seen that in some cases one, or the other of these is the dominant influence on the glider motion. Considering first the viscous force, we note that the in iti al potential energy relative to the bottom of the track is n1.qxo si n ct, and after tlle fi rst bounce, it is m.qx 1 sin Ct. We let tlx be the decrease in height after this bounce, so 6.x = Xl - xo' The corre ponding loss of energy is mg sin

Ct

(22)

6.x

Si nce the gravitational force is conservati ve this loss of energy is due entirely to the work done against the frictional force. Thi work ca n be calcul ated approximately by assuming that friction is small compared to the gra vitational fo rce, so the motion is nearly the arne as it would be in the ab ence of all frictio n. The work done by fricti n during the first descent from initial position Xo is

w = rxo F dy = rxo- bv dy

Jo

(23)

Jo

where the var iable y is in troduced temporarily to r present the instantaneous distance of the glider F OIn its starling point Xo' The acceleration of the glider approximately a = g sin Ct so the speed vat any position y is given by

v2

=

2ay

=

2.q sin ay

Substituting this expression for y into Eq . (23) and integrating, w fin d

W

XO

=

I

o

-

b(2ay)1 /2 dy = _

2b(2a) 1/2·(X 0

3

)3/2

(24)

The work done on th return trip is approximately the same, so the total change in energy due to viscous friction is 2W. F inally, combini ng this result willi Eq. (22) and solving for tlx, we obtain

(25)

Thus, the change in height after the fir t bounce is proportional to the ~ power of the original h ight. Similarly, the change in height after the second bou nce is proportional to x 1 3 / 2 with the same propo rtio nality constant and so on. This relationslllp may be checked experimentally. Turning now to the bumpers, we recall fro m Exp riment M-2 that the coefficient of restitution e is defined as the ratio of relative velociti s after and before a collision. Since kinetic ener gy is proportional to 1,2, the r tio of kinetic energies just after and j ust before impact is e 2 . Neglecting the energy los through viscous friction this energy correspond s to the potential nergy 15

mechanics

at maximum distance up the track before and after the bounce. Since these potential energies are proportional to distance, we immediately find

and so forth, so that the difference in height after the first bounce is .1x

=

Xl -

Xo

= -(1 -

2 e )xo

(26)

and that after the second is .1x =

-(1 - e 2 )x I

and so on. Thus, we see that if the bumpers are the principal mechanism of en rgy loss, the decrease in height after each bounce is directly proportional to the height before that bounce, rather than to the J power of the height as when viscous fri ction is the dominant effect.

experim e nt 1 visco us dam ping After ca refu lly leveling the track, launch a glider, and measure its initial velocity and the total distance it travels before stopping. For this measure­ ment the bumpers may be considered perfectly elastic. From these data, determine the damping constant b, using Eg. (21). Add wei ghts to the glider to approximately double its mass, and repeat the above o bservations and calculations. How should the value of b in this case compare wilh the previous value? Why ? 2 magnetic dam ping To observe magnetic damping, attach four ceramic magnets symmetrically to the glider. Attach enough weight to another glider to give it the same total mass as the glider with magnets. Place the two on the track and push them together (wi th the magnetic glider in back) to give them the same initial veloci ty. No te that the magnetically damped glider lags increasingly behind the ot her. Determine the damping constant b for the magnetically damped glider by the same method described above. Note that you are measuring here the 10 lal h due to both magnetic and viscous damping. 3 bouncing ball For the "bouncing ball" experiment, tilt the track about 5 mrad (milliradians). Be sure to record the tilt. Release a glider from the top; record its initial po ition and its maximum height after each bounce. Note that the position at the bottom of the track may not be at the zero point of the scale in which case it must be subtracted from each reading. Record the positions first, then make the subtractions. (Why?) 4 modified bouncing ball Tf time permits, repeat the bouncing ball experiment with magnetic damping. You may also wi h to try different track slopes. The coefficient of restitution of the bumper may be varied by wrapping a rubber band around it several times. An interesting variation is to attach a small piece of Silly Putty to the 16

dissipative forces

M-3

bumper. Does the coefficient o f restitution appear to val)' with velocity? How does the behavior of a highly viscous fl uid such as Silly Putty differ fro m that of an elastic solid such as rubber? 5

analysis Detailed analysis of the bouncing ball data is facilitated by using a graphical technique in whlch a special graph paper called " log-log" pa per is used. This paper is ruled so that the distance along each scale is proportional not to the number on the scale, as with ordinary graph paper having uniform ruting but rather to tbe logarithm of that number. Thus wh n one plots a graph showing, say, a versus b, on log-log paper, the effect i actually to plot log a ersus log b. To understand the usefuJne s of this techn ique, consider again Eq. (25). Taking logs of both sides and rearranging, using fam il iar properties of logarithms, we find

3 log ( - ~) = 210g x

+

23i2 b log 3ma 1/ 2

(27)

Thus if the bouncing ball behaves according to this equation, the g raph of - Ax versu x on log-log paper [ which is in effect a plot of log ( - ~~) versus log x] should be a straight line with a slope of l However, if the behavior is represented ins tea by Eq. (26), then we have Jog ( - ax) = log x

+

log (I - e 2 )

(28 )

Again the graph of log ( - Llx) versus log x should be a straight line, but this time with a slope of unity. Hence, the slope of the graph at ach po int tells us which damping mechanism is dominant in that region. Before making log-log graphs of your data, you may wish to try to pred ict the shapes of the curves. It is helpful to note that the largest velocities occur in the first few bounce, so the damping forces are largest at fir t, bec ming successively Ie s important after several bounces. However, if the velocities become too large, the approximations used to derive Eq. (25) may not be vatid, and departures from the predicted slope oft may be observed. In which direction will the actual slope differ from the predicted value?

questions Why is the magnetic damping force propo rti oal to the square of the magnetic fie ld inten ity? 2 Show that the quantity blm has units of time. W hat is the significance of this

time in the experiment? For example, how is it related to the time rvquired for a glider on a level track to come to half its initial velocity when given an in itial velocity V o ? 3 D iscuss in detail the nature of the approximations u ed to derive Eq. (25) .

4 When Silly Putty is u ed on the bumpers, how does the coefficient of restitu­ tion depend on the r lative velocity in the collision? Is it greatest for high­ velocity collisions or for vel)' low-velocity coll isions ? Can you understand this behavior on the basis of the proper ties of Silly Putty itself? 77

dissipative forces

M-3

bumper. Does the coefficient of re titutjon appear to vary with velocity? How does the behavi r o f a highly viscous fluid such as Silly Putty differ from that of an ela tic solid such a rubber ? 5

analysis

Detailed analysis of the bouncing baHdata is facilitated by using a graphical techniq ue in which a special graph paper called " log-log' paper is used. This paper is ruled so that the di stance along each scale is proportional not to the number on the scale, as with ordjnary graph paper having unjform ruling, but rather to the logarilhm of that number. Thus when one plots a graph showing, say, a versus b, on log-log paper, the effect is actually to plot log a versus log b. To understand the usefulness of this technique, con ider agai n Eq . (25). Taking logs of both sides and rearranging, using familiar pro perties of logarithms we find 23 / 2 b

3

log ( - ilx)

=

2 log x + log 3ma! /2

(27)

Thus if the bouncing ball behaves according to this eq uation, the graph of - ilx versus x on Jog-log paper [which is in effect a plot of log ( - Ll:~) versus log x] 'hould be a straight line with a slope of However, if the behavior is represented instead by Eq. (26), tben we have

t.

log (- ilx)

=

log x

+ log (1

- e2 )

(28)

Again the graph of log ( - ilx) versus log x sho uld be a stra ight line, but this time with a slope of unity. Hence, the lop of the graph at each poin t rells us which damping mechanism is dominant in tha t region. Before making log-log graphs of your data, you may wish to try to predict the shapes of the curves. It is helpful to note that the la rgest velocities occur in the first few bounces, so the damping forces are largest at first, becoming successively less important after several bounces. However, if the velocities become too large, the approximation used to derive Eq . (25) may not be valid, and departures from t he pred icted slope of1 may be bserv d. In which direction will the actual slope differ from the predicted value ?

QuestIons 1 Why is the magnet ic damping force proportional to the square of the magnetic

fi eld inten ity? 2 Show that the quantity blm ha uni ts of time. What i the significance of this

time in the experiment? For example, how is it related to the time required for a gl ider on a level track to come to half its initial velocity wh n given an initial velocity o? 3 Discuss in delai l the nature of the approxima tions used to derive Eq. (25). 4 When Silly Putty is used on the bumpers, how does the coefficie nt of resti tu­

tion depend on the relatjve velocity in the colli: ion ? Is it greatest fo r high­ velocity collisions or for very low-velocity collisions? Can you under tand this behavior on the ba is of the properties of Silly Putty itself'. 17

mechanics

5 For a glider with only viscous air damping, how does the damping constant b vary with the mass of the glider? Why should this variation be expected? 6 When a glider on a tilted track is given an initial velocity va' show that if the track is sufficiently long, the glider will reach a final velocity ("terminal velocity") which is independent of Va. Derive an expression for the terminal velocity . 7 Is the effect of air surrounding the glider significant in comparison with the

effect of the air layer between glider and track, in determining the total frictional force? Explain.

18

experiment

M -4

periodic motion

introdu ction Examples of oscillatory or periodic motion are familiar to everyone. An especially simple kind of periodic motion is represented by the behavior of the harmonic oscillator, which serves as an idealized model to represent the most important features of other periodic motions. The essential character­ istics of the harmonic-oscillator model are the following: A mass is acted on by a force which is proportional in magnitude to the displacement of the mass from an equilibrium position, always in the direction toward the equilib­ rium position. Thus the acceleration of the mass is also proportional to its displacement from equilibrium. The motion of the mass is such that its displacement from equilibrium is a sinusoidal function of time. The frequency of oscillations is independent of amplitude.

Only the first of these is really essential; the second and third follow from it, as we shall now show. Let the displacement of the mass m from equilibrium be x. Then the force is given by

F= -kx

(29)

where k is a constant called the force constant for the system. Such a force can be produced by a spring obeying Hooke's law. According to Newton's second law ,

-kx

=

ma

d 2x dl 2

= m-

(30)

The displacement x must be given by a function of time which satisfies Eq. (30) , that is, which is a solution of this differential equation. It is easy to verify that the functions

x = Xo cos wt x

=

Xo sin wt

(31 )

are solutions, where Xo is a constant called the amplitude and w is an abbrevi­ ation for the quantity (k/m) 112. Here Xo is determined by the way the system is initially set in motion, whereas w depends only on the basic properties of the system, the constants k and m . Each time the quantity wI increases by 2n, the motion goes through one cycle. The time for one cycle is called the period, denoted by T. We see that Tis given by

T

= -2n = w

(m)1 /2

2n -

k

(32) 19

mechanics

Theredprocal of the period is the number of cycles per unit time, orfrequency, denoted by! We see that W

=

211: 211:f= ­ T

(33)

The quantity w represents the time rate of change of the quantity wt in Eq. (31). Since wI plays the role of an angle, w is often called the angular frequency . Often. however, the single word frequency is used to refer to w rather than! The system shown in Fig. I has approximately the properties described. FIGURE 1

Cord

A glider on a horizo ntal air track is attached at its ends to identical springs. Each spring ha a force constant k o ; that is, to stretch either spring a distance x requires a force F = kox. In the equilibrium position both springs are stretched the same amount, so the total force is zero . W hen the mass is displaced a distance x to the right of its equilibrium position, t he fo rce of the left spring increases by kox, whereas that of the right spring decreases by the same amount. The result is a net force to the left with magnitude 2kox, so the force constant to be used in the above equations is k = 2k o. The for ce on the mass is a conservative force; hence the total energy is constant. When the mass reaches the endpoints of its motion and stops, the energy is ent ir Iy potential energy (as in a stretched spring); when it passes the equilibrium position, the energy is entirely kinetic energy. During each cycle energy is transformed from kinetic to potential and back, but the total energy is constant. It is not difficult to show that the average potential energy is equal to the average kinetic energy, and that each is equal to half the total energy. Experience hows that when any real mechanical oscillating system is set into motion the oscillations eventually die out and the system comes to rest at its equ ili brium position. The position of the mass is given as a function of time not by a simple sinusoidal function but rather by a function with the gene ral shape of Fig. 2. This effect, not predicted by the simple model dis­ cussed above, is due to the presence of damping forces ,in addition to the elastic restoring force represented by the force constant k. Familiar examples of d amping are the vi cous and magnetic damping studied in Experiment M-3, a U kinds of fr ictional forces, air resistance, and many others . The effect of dam ping may be added to our model. Following the results of Experiment M -3, e assume that the damping force, whether due to viscous friction or magnetic damping, is proportional to velocity and can be repre­ sented by

F

dx

= -bv = -b ­

dt

(34)

where b is the damping constant, characterizing the strength of the damping force. Clearly, the rate at which oscillations die away depends on the magni­ tude of b; a large value of b means rapid decay, and the converse. The 20

periodic motion

M-4

FIGURE 2

additional force given by Eq. (34) must be included in the differential eq uation expressing Newton's se ond law, which now becomes

d 2x m dl 2

+

bx b dt

+

kx

=

0

(35)

The relation of dampi ng to the properties of the system (the constants k, and b) can be explored in mo re detail. T wo ap proaches are possible, oDe an approximate analysis using energy considerations, the other ma k ing use of the general solution of Eq. (35). The second is rather invo lved, and we consider here only the energy m ethod . We begin wi th t he fo llowing question: If the maximum displacement (amplitude) for a given cycle is x o, how much energy does the system lose during that cycle ? T he in stantaneous rate of loss of energy is the rate of doing work agai nst the damping foroe, which is simply the magn itu de of the force (bv) times the velocity v, or b! 2 . This quantity varies during the cycle but the total energy lost is still given approximately by the average rate of energy loss during the cycle (the average value of bv 2 ) multiplied by the time required for a cycle, which is In,

(36) To find the average va lue of v 2 , we note t hat the average kineti energy <~mv2 > .v for a harmon ic oscillator is equal to its average pote ntial energy <~kX2 >a\" so each of the e quan tities must equal half t he total energy E. T hus, we have 1-m< v2>.v = -tE. T he a verage rate of loss of energy is then

(-dE)' dl

av

bE = - .v = -m

(37)

and the energy loss during one cycle is b - 2n (km)1 /2 E

(38) 21

mechanics

Now dE/cit is not constant over a cycle, but is greatest when v is greatest, and zero when v is zero. If we ignore this variation and consider how the energy decreases on the average, we see that Eq. (37) is a differential equation for E and that its solution gives the energy as a function of time. The solution of Eq. (37), as we can verify easily by substitution, is (39) where Eo is the initial total energy, at time t = O. That is, the energy of the oscillator decreases exponentially. The time required for the energy to decrease to lie of its initial value, called the relaxation time, is given by (m/b). It is useful to introduce a constant called the quality factor Q, defined as 2n times the ratio of maximum energy stored in the system to the energy dissipated in one cycle. An expression for Q is easily obtained from Eq. (38): mw

2nE

Q = llE =

b =

(mk)1 /2 b

(40)

Having found how the energy of the system decreases with time, we may now ask how the amplitude, which is often more directly observable, decreases with time. Since E at any time is proportional to the square of the amplitude, the variation of Xo with time must be given by a function which is the square root of the function describing the time variation of E, that is, by a function of the form

Specifically, the amplitude must be given by (41) where Xo is the initial amplitude at time t = O. Thus, we see that the relaxation time for the amplitude of the oscillations (the time required for the amplitude to drop to li e of its original value) is

2m

(42)

r =-

b

We can also define the half-life TI / 2 during which the amplitude drops to half its original value; this is given by TI / 2

=

r In 2

=

2m In 2

b

I .386m

(43)

b

Combining either Eq. (42) or (43) with Eq. (40) gives the results

Q=

~ wr = ~ C;)(:I /~) = In\ T~2

(44)

Thus, Q can be obtained in terms of directly observable characteristics of the motion. Finally, we consider briefly the problem of a system containing two masses; the simplest example is shown in Fig. 3. If one of the masses is displaced and FIGURE 3

Cord

22

periodic motion

M-4

released, the resulting motion will not be sinusoidal. However, the system does have possible motions in which each mass moves sinusoidally. One possibility is for the two masses to move exactly in unison, so that the distance between them is constant. A little consideration shows that in this motion the center spring does not contribute to the restoring force on either mass, so the effective force constant is simply k o. Thus, we expect the frequency of this motion to be given by w

=

(:Y / 2

(45)

A second possibility is for the two masses to have exactly opposite motions. In this case the mid point of the middle spring does not move; the'motion of each mass is as though it were acted on on one side by a spring of force constant ko and on the other side by a spring halfas long. Halving the length doubles the force constant (Why?), so the total effective force constant for each mass is 3k o, and the corresponding frequency is (46)

These predictions can be checked experimentally. The damping character­ istics can also be investigated; a decay time and a quality factor can be determined for each motion. Any motion of a system of coupled oscillators such as this, in which all masses move sinusoidally with the same frequency , is called a normal mode of the system. Each normal mode has a characteristic frequency relationship between the motions of the various elements of the system.

experime nt

(

1

In order to compare the above theoretical predictions with the observed behavior of the system, the mass and spring constants must be known. The suggested procedure for measuring the spring constant is shown in Fig. 4. One end of the spring is attached to the binding screw at the end of the track opposite the air pulley ; the other end of the spring is attached to the glider with a binding screw. With a piece of masking tape, attach a piece of magnetic

)

e

spring constant

FIGURE 4

"

d Track

]

23

mechanics

recording tape to t he top of the glider. Pass the recording tape over the air pulley and attach a lotted weight hanger to the other end. Be sure there is sufficient air supply to the air pulley so the recording tape does not bind. Note the equilibrium positi on of the reference li ne on the glider. Now add weights in 1O-g increments, up to a total of 100 g, recording the position of the reference line for each value of total weight. Do not stretch the spring more than 20 cm' beyond this it will be permanently deformed and will not return to its original length when the weight is removed . Plot a graph ofextension ofspring as a function of applied fo rce, remem ber­ ing that the force on the spring is the weight (mg) of the total mass on the end of the tape . F rom this graph determ ine the onstant k Q •

2 simple harmonic motion To observe simple harmonic motion, remove the tape fro m the glider and attach a second spring, identical to the first. Attach a piece of cord to the end of this spring. Pull the cord enough to stretch each spring about 10 cm and tie to the end of tbe track. Now displace the glider about 5 cm from its equilibrium po ition and release it. Observe the motion, noting the tran sfer of energy between the glider and the springs. T ime 10 cycles of the motion. Find the period T and the frequency f Repeat the measurement with maller and larger vibration amplitudes, recording the amplitude for each trial. Do you find any significant variation in frequency? From the measured frequency and force con tanl, comp ute the rna s of the glider. Compare with the mass measuremen t in Experiment M- l. If this measurement is not available, or if you are usin g a different glider, measure the mass with a balance, compute the period from Eq. (32), and compare with the mea ured value of T. Is the difference within the range of experimental error ? You may wi. h to add slotted weights to th glider accessory rod to increa e its mass. Again compute the period from Eq. (32) and compare with the measured value of T.

3

damping Damping may be observed with the same experimental setup as above. C are­ [ully displace the glider 5 cm from equilibrium and release it with no ini tial velocity. Count the number of cycles fo r the amplitude to decrea e to half its original val ue. Compute the Q of the system and the relaxation time T. Also compute the damping constant b; compare with the value obtained in Experiment M - . Now add damping magn ts to the glider and again determi ne Q. Compare your result with Q for a glider of the same mass bu t only viscous damping. What would Q be with only magnetic damping ? Finally, you may wish to add mass to the glider and o bserve how Q changes with mass. You may fwd that as m is increased , Q increase , goes through a maximum, and til n drops off. Why does Q vary in this way?

4 coupled oscillators For the tudy of coupled oscill tors assemble the system shown in F ig. 3, with the cord again pulled tight enough to extend each spring about 10 m. Displace o ne mass, bolding the other fixed , and r lease both masses at once. Note the complex nature of the motion. Now try the following : Displace both masses toward the center by the same amount and release them. D o s the motion now appear to be sinusoidal ? This mode in which the masses move in opposi te directions is called the symmetric mode. Measure the ti me for 24

periodic motion

M -4

10 oscillations, compute the period or frequency, and compare with the pre­ d iction of Eq. (46) . N ow displace the two masses in the same direction by equal amounts and release them. Is the motion sinusoidal? Again determine the frequency, and compare with the theoretical prediction. This mode is called the antisymmetric mode.

ld

le re m

5

:rld

ld ld ld its er

f

.s,

modified osci llators An interesting modification of this system is to replace the middle spring with a spring of very small force constant. In this case the symmetric and anti­ symmetric modes will have very nearly the same frequency. (Why?) Such a system may be constructed by using a O.I-lb constant-force spring to couple the two oscillators. Assemble such a system and displace one mass h iding the other stationary, and release both masses together. What happens? H ow can this behavior be understood in terms of the normal modes of the system? You may wish to measure the number of cycles needed for the amplitude to decrease to t of its o riginal value, and compute Q. Note that these quantitie may be different for the two modes . Compare the two Q values with Q for a single mass. Now place a strand of Silly Putty across the center spring and repeat the determination of Q. What do you fi nd? Why should your results be expected?

~n

lte

nt nt 2), ge ler

.2)

questions 1

H ow is the motion of the system shown in F ig. I related to the motion of a simple pend ulum?

2 Why is it desirable to use two springs in the setup shown in Fig. I, rather than

a single spring?

re­

~al

3

alf he ·3. lre 19.

If a spring with spring constant ko is cut in half, what is the fo rce constant of the resulting spring?

4 If two identical springs, each with spring constant k o, are connected in series,

what is the resulting spri ng constant ? What if they are connected in parallel ? 5 If a harmonic oscillator with only viscous damping has a Q value given by

Q,,, and with only magnetic damping a value Qm, show that the value when both kinds o f d amping are present is given by

~es

la

-

I

QIOI

6 3, m.

ceo ,rh !be

I

1

Qv

Qm

=- +­

Q \OI

Is the quantity m appearing in the freq uency expressions the inertial mass or the gravitational mass of the glider? D oes it matter?

7 In the abo ve analysis of harmonic oscillators, the masses of the springs have been neglected. Will the effect of spring mass be to increase or decrease the freq uency? Explain. Make a rough estima te of the order of magnitude of the correc tion ; i.e., is it 0.1%, 1%, 100%, or what? 8 For the coupled oscillator system, is the effect of spring mass more important

for the symmet ric or the antisymmetric mode ? Explain. 25

experiment

M -5

forced oscillations

introduction In this experiment we study the behavior of a harmonic oscillator when a sinusoidally varying force is applied to the mass in addition to the spring and damping forces considered in Experiment M-4. This situation serves as a model for a wide variety of practical situations in which a vibrating mechani­ cal system experiences a periodically varying force. As we shall see, the system can be made to vibrate with the same frequency as that of the applied force, with an amplitude that depends both on the magnitude of the force and on its frequency. When the force frequency is close to the natural vibration fre­ quency of the system, the amplitude of this "forced oscillation" can become very large, a phenomenon known as resonance. The experimental setup for this experiment is very similar to the one used for Experiment M-4. The principal difference is that the cord which was previously tied between one spring and the end of the track is now tied between the spring and a device which gives the other end of the cord a sinusoidal motion with adjustable amplitude and frequency, as shown in Fig. 5. This motion produces a sinusoidal change in elongation of the spring, FIGURE 5

Il-JUk

O

ko

Sine drive Scotch yoke

II ~l'--I........o---------nij)'w

which, in turn, exerts an additional sinusoidally varying force on the mass . Let the motion of the cord be described by r cos wIt , where r is the.amplitude of t he motion, and 0/ its angular frequency. Then if the spring constant is ko , the additional sinusoidal force applied to the mass is

F = kox = kor cos w't

(47)

We note that w' is not necessarily equal to the natural frequency w = (k/ m)1/2 of the system, which is the frequency at which it would oscillate in the absence of the sinusoidal force. The effect of the sinusoidal driving force is to induce a sinusoidal motion of the mass, with the same frequency as that of the force. To see how this comes about, we first note that Newton 's second law now contains an additional force term. Neglecting damping for the moment, we find that the equation analogous to Eq . (30) is (48) 27

mechanics

We ask whether there is a sol uti on of this equa tion which has the form

x = xa cos w' t

(49)

where w' is the same frequency a in Eg. (47), and xa is a co nstant to be de­ tenuined. Substituting this trial sol ution into Eq. (48), dividing out the common factor cos w 't, and solving fo r x a, we find (50) where, as before, w is the natural frequency of the system and k = 2kQ' Equ ati on (50) shows that Eg. (49) is a solution of Eq. (48) provided the amplitude Xo of the oscillat ion is given by Eq. (50). When the driving fre quency w' is less than the natural frequency w , the amplitude is positive and the forced oscil lation i in phase with the driving force. When w' > OJ, the oscillation is 1800 (a half-cy Ie) out of phase with the driving force. When w' = w, Eq. (50) predicts tha t the amplitud becomes infinite. This results from the neglect of damping' a more detailed analysis including damping I ads to an amplitude function which has a maximum, but not an infinite di continuity, when w' is close to w, as shown in Fig. 6. As mentioned pre-

FIGURE 6

w viously, this peaking of the amplitude of the forced oscillation is known as resonance. In forced oscillations of a damped oscillator, energy is continuously dis­ sipated by the damping fo rce, but the driving force replaces this loss by doing work on the system. The ampl itude of the forced oscillation is determined , in fact:, by the requirement that the average rate of Joss of energy due to damping is equal to the average rate at which the d riving force does work. This relation may be used to calcula te the amplitude at resonance. We shall not discus the calculation in detail, but the principal features of the result can be obtained rather simply. First, as shown in Experiment M-4, Eg. (40), the average rate of energy 10 s is propo rtional to E/Q; in tu rn , tbe total energy E is pro portional to the sq uare of the amplitude X o 2. Thus tbe rate of energy loss is proportional to Xo 2/Q. The rate at which the driving force does work is propo rtional to the velocity which is proportional to x o, and is also proport ional to the ampli­ lude of the force fu nction, which is proportional to r. Thus, the rate of doing work is proportional to xo r. Putting the e pieces together to equate the rate of dissipation to the rate of doing work by the driving force, we find

-lF = (const)xaf X 2

Since Q is a dimensionless ratio, dimensional considerations show that the proportionality constant must be a pur (dimensionless) number. A more 28

forced oscillations

M-5

de tailed ca lculation shows that its value is !, and the correct relationship is simply

~)

(51)

e­ Ie

We must emphasize that Eq. (51) gives the amplitude only at resonance, si nce only then is the driving force in phase with the velocity of the mass. At other freq uencies, the two are not in phase, and the rate at which the force does wo rk is no longer simply proportional to Xo and to r . Tbe sin usoidal di pIa ment of the cord is achieved by a mechanica l device caIJed a " Scotch yoke"; the principle is shown in Fig. 7. An eccentric

0)

ile ~g

Cord

FIGURE 7

ve flJ,

:n ts ilg

te 'e­

pin rota tes at uniform speed, d riven by a dc motor. The pin moves up and down in a chan nel which is constrained to remain vertical. Thus, as shown in the figure, the horizontal displacement of the channel is given simply by x

= r cos w it

where r is the di tance of the pin from the axi of rotation, and W i is the a ngular velocity of rotation. By changing r we may change the amplitude of the displacement. T he frequency of the d isplacement is controlled by controUi ng the voltage supplied to the motor. For the particular motors used, the speed is nearly propo rtio nal to the applied voltage. T he reason for this simple rela tionship, briefly, is tha t the rotation of the moto r's a rmature in the stationary magnetic field induce a vo ltage called the "back emf" (electromotive force) in the armature. Th speed of the mo tor increases until the back emf exactly equals the supply voltage (0 glecting po tential d rops due to resistance in the wind­ in gs). Since the back em f is directly proportional to motor speed, the speed is proportiona l to voltage. This relationship may be checked experimentally. Once the relationship is establi shed, the easiest way to measure th freq uency of the d riving mechanism is t o measure the supply voltage and use the relation­ ship just discussed to compute the frequency.

as ~ s­

ng :d, to rk.

~ll an

'gy he to ihe

)li­ ng !lte

experiment 1

he ore

motor calibration With the motor and Sco tch yoke assembly mounted on t he support beam of the a ir track , connect the motor and a voltmeter to a low-voltage power 29

mechanics

FIGURE 8 +r7----------------~------------------_.

+ Low-voltage power supply

v

de motor

supply as shown in Fig. 8. Set the meter for a full-scale range of 30 or 50 V. If you are using a vacuum-tube voltmeter, it must be connected to a 110-V power outlet, turned on, and allowed to warm up for a few minutes, and then set to zero by turning the ZERO ADJUST control with the test lead wires con­ nected together. Then connect it to the power supply and motor, connect the power supply to a power outlet, turn on the power supply, and increase the supply voltage until the motor starts. The motor will not rotate stably below a certain critical voltage, and may stop. If it does, turn up the supply voltage until it starts again. For each of several voltages up to (but not exceeding) the maximum voltage for which the motor is designed, measure the time for the motor to make 10 revolutions (or a greater number if you think greater precision wil\! result). Compute the frequency, and plot a graph of frequency as a function of voltage. This will be your calibration curve in subsequent parts of the experiment. Is it a straight line? rfnot, can you devise a quantitative measure of the deviation from a straight line? Does it pass through the origin? If not, why not? 2 resonance Now set up the harmonic oscillator and the sine-drive unit as in Fig. 5. Adjust the Scotch yoke so that the amplitude r of the driving displacement is between I and 2 mm. Vary the motor speed and find the speed for maximum displacement or resonance. Using your motor calibration curve or timing the motor directly, determine the resonance frequency. Compare with the natural frequency of the system as determined in Experiment 4. If they do not agree (within limits of experimental error), can you suggest a reason for the discrepancy? Determine Q and compare your result with the value obtained in Experiment M-4. 3 m agnetic damping Add damping magnets to the glider and again determine the resonance frequency . Compare with your previous result. Again determine Q and compare your result with the vaLue obtained in Experiment M-4 . 4 normal modes In Experiment M-4 we studied the normal modes of oscillation of the system shown in Fig. 9. We found that there are two normal modes : for the symmetric mode, the frequency is OJ

30

s

=

Cf~O) 1/2

forced oscillations

M-5

FIGURE 9

) and for the antisymmetric mode it is Wa =

v. ·v

en -n­ he he

Jm to vill Ion the ure

(:Y

I 2

Adjust the motor speed to excite the antisymmetric mode. Determine the frequency of this mode and compare with the result obtained in Experiment M-4 from free oscillations. Make the same determination for the symmetric mode. 5

ad ded damping By placing a strip of Silly Putty across the center spring you can damp the symmetric mode but not the antisymmetric mode. Why? Find the maximum amplitudes of the two normal modes under this condition.

6

added mass Add a small additional mass to one glider but not to the other. Find the two normal modes. How do the normal-mode motions differ from the behavior when the masses are equal?

lot ,

questions . 5. ent um i ng the not the ned

Why is the graph of motor speed versus voltage not a straight line? 2

Why does the graph of motor speed versus voltage not pass through the origin?

3

When the motor is first turned on, the glider may appear to move at first with an irregular, nonsinusoidal motion , which eventually becomes sinusoidal. Why?

4

Show that the velocity of the glider is a sinusoidal function of time, and that (t cycle or n/2) out of phase with the displacement.

it is 90° mce a nd

the . the

± n/2,

5

Show that if the phase of the velocity relative to the driving force is then the average work done on the system by the force is zero.

6

How will the maximum amplitude of the forced oscillations at resonance change if the mass of the glider is decreased?

7

On the basis of results from Experiment M -4, would you expect the maximum amplitude for the coupled oscillators to be greater for the even mode or for the odd mode, assuming the driving amplitude is the same? Explain .

31

• electroniC instrumentation Copyright © 1971 by McGraw-Hill, Inc.

All rights reserved. Printed in the United States of America.

No part of this publication may be reproduced, stored in a

retrieval system, or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher.

Library of Congress Catalog Card Number 79-125108

07-050483-0

1234567890 BABA 79876543210

The first edition of the Berkeley Physics Laboratory

copyright © 1963, 1964, 1965 by Education Development

Center was supported by a grant from the National Science

Foundation to EDC. This material is available to publishers

and authors on a royalty-free basis by applying to the

Education Development Center.

This book was set in Times New Roman, printed on

permanent paper, and bound by George Banta Com­

pany, Inc. The drawings were done by Felix Cooper;

the designer was Elliot Epstein. The editors were Bradford

Bayne and Joan A. DeMattia. Sally Ellyson

supervised production.

contents

berkeley physics laboratory. 2d edition complete contents of the 72 units

and statistics

v'

derIvatives and integrals trigonometric and exponential functions loaded dice probability distributions bInomial distribution normal distribution

acoustics and fluids AF- t AF-2 AF-3 AF-4 AF-5 AF-6

acoustic waves acoustiC diffraction and Interference acoustic interferometry fluid flow viscous flow turbulent flow

microwave optics velocity and acceleration

collisions

dISSipative forces

periodic motion

forced oscillations

instrumentation voltage. current. and resistance measurements measurement of ac voltage and current waveform measurements comparison of variable voltages transducers

MO- t MO-2 MO-3 MO·4

microwave production and reflection interference and diffraction the klystron microwave propagatIon

laser optiCS LO-t LO-2 LO-3 LO-4 LO-5

reflectIon and refractIon of light polarization of light diffractIon of light interference of light holography

atomic physics radial fields Image charges fIeld lines and reciprocity the magnetic field magnetic coupling

AP- t AP-2 AP-3 AP-4 AP-5

atomic spectra photoelectric effect the photomultiplier and photon noise ionization by electrons electron diffraction

nuclear physics acceleration and deflection of electrons focusing and intensity control magnetic deflection of electrons helical motion of electrons vacuum diodes and the magnetron condition

NP· t NP-2 NP-3 NP-4 NP-5

geiger-mueller tube radioactive decay the scintillation counter beta and gamma absorption neutron activation

semiconductor electronics resistance-capacitance circuIts

resistance-Inductance cIrcuits

LRC CircuIts and oscillations

coupled osclllators

periodic structures and transmissIon /rnes

SE- t SE-2 SE-3 SE-4 SE-5 SE·6

semiconductor diodes tunnel diodes and relaxation oscillators the transistor transistor amplifiers positive feedback and oscillation negative feedback

...

electronic instrumentation

)

rk tal its lnt

'ns

,Ie rre

nole

rem

INTRODUCTION

Electronic instruments playa central and indispensible role in present-day science and technology. Nearly every measurement made today in physics, chemistry, or biology requires sophisticated electronic instrumentation. Design of electronic instruments has become a highly specialized field in itself, but every scientist and engineer needs some working knowledge of electronics to understand the applications of electronic instruments to his particular problem. In this series of experiments we shall study the operation of several basic instruments, including meters, power supplies, signal generators, and oscilloscopes. Each of these has individual characteristics and limitations which make it suitable for certain applications, and it is important to understand these characteristics. Equally important is the need to understand the interactions between an instrument and the system it is observing. No instrument can observe a system without interacting with it in some way, and the behavior of the system is always changed to some extent by the presence of the instrument. Thus, the general objectives of these experiments are to learn the character­ istics of a few basic instruments and their limitations in terms of stability, sensitivity, and accuracy, and to study their interactions with the systems with which they are used.

experiment

E'-I

voltage, current, and resistance measurements

introduction The immediate purpose ofthis experiment is familiarization with an ordinary vacuum-tube voltmeter (often called a VTVM) and its uses in measuring voltage, current, and resistance. An important indirect purpose is to show, using this instrument as a simple example, how one can adopt a critical attitude toward the functions of measuring instruments, including their sensitivity and precision and the ways in which they interact with the system being measured. Most ordinary electrical meters used for voltage and current measure­ ments use a device called a d'Arsonval movement. This consists of a pivoted coil which can rotate in a magnetic field, attached to a spiral spring which tends to return the coil to a certain equilibrium position. A typical arrange­ ment is shown in Fig 1. When a current passes through the coil, the magnetic field exerts a torque directly proportional to the current, giving the coil an angular displacement until this torque is just balanced by the restoring torque of the spring, which is proportional to the angular displacement. FIGURE 7

Amperes

Shunt 3

electronic instrumentation

Thus the angle through which the coil turns is directly proportional to the current through it; by adding a pointer and a scale, we have a current­ measuring device. Typical d'Arsonval movements in portable YOM's require a current of 200 ~ for full-scale deflection and have an internal resistance (the resistance of the moving coil) of the order of 750 n. In the simplest version of a current meter one simply permits the circuit current to flow through the meter. Of course, the current must be limited to the value corresponding to full-scale deflection; larger currents could cause mechanical damage to the coil or pointer or could bum out the coil. To measure larger currents a shunt resistance is placed in parallel with the meter, as shown in Fig. 2, so that only a fraction of the total current flows through FIGURE 2

1 J/Yib ,.

~M ~ l'<:5t~ ~ ~ ?o\i) ~r r:r11I ,:.:12Jt5f-sr" c(.;tr~nr ~

:tot)" J)A

0;;

".L~...L 7 <6 "':

f
:e"t-''''~~

...,

7~11s;o

()

0

~

1>­

Shunt

~

e..,. =

7tA..

t:p

the meter itself. For example, if a full-scale reading of 2000 IlA or 2 rnA is required, with the typical meter movement described above, we use a shunt resistance of 750 n/9; when 200 ~ flows through the meter, 9 x 200 IlA or 1800ilA flows through the shunt, for a total of 2000 1lA. The combined resistance is then 75 n. Can you Qrove thi§.? ... Because of the internal resistance of an ammeter, there is always a voltage drop between the terminals when a voltage is being measured. One some­ times considers an idealized ammeter which can be inserted in a circuit to measure current without causing any voltage drop. Such an idealized meter would have zero internal resistance; it is important to understand that real meters never attain this ideal, although in some cases the internal resistance is small enough to be negligible. ~J'ji~ rnti~ 4i?r Antp. cVVYen-fc.";J1f) To use a d'Arsonval meter to measure potential difference (voltage) ~ .;:!R 4'1'lL series resistor is therefore 250 kn 750 n. Note that the voltmeter always must draw current from the circuit it is measuring. We sometimes speak of an S0 i)\j/~ i?~'i>il)Jonce (l'nk",q{ il""SoiSJ ::7(0.4 ~dealized v~ltmeter which d.raws n? cu~rent and therefore has ~n infini~e Internal reSIstance, but as With the IdealIzed ammeter such a deVice can, III ond Mt :z..«J1m. practice, only be approximated. I. r.r . f, . (i I I(f vpta Electronic amplification, with either vacuum tubes or transistors, is often it> Ste a M·ue-Iin (ffl 1 rnt-r J used to extend the usefulness of meters. For example, vacuum-tube volt­ sV V in Fu fl Sea ft. meters typically have an internal resistance of 1 J Mn, and voltage ranges as low as 0.01 V (full-scale). With amplification it is also possible to construct

-to

4

1

K:: \\ "'Id' ..a.. \I:. o· <1\ V

,

..

voltage, current, and resistance measurements

EI-1

FIGURE 3

e

~-------------v--------------~

l----(~r:·

1---­

an ammeter with much greater sensitivity than would be possible with a d' Arsonval meter alone; in some cases, the voltage drop across the instru­ ment may be reduced compared to that for the unamplified meter, although this depends on the design of the particular instrument. An additional instrument to be used in these experiments is a low-voltage de power supply, which converts the llO-V-ac line power to dc, with adjust­ able voltage up to a maximum of about 35 V, and maximum current of about 200 mA. An idealized power supply would supply a fixed potential difference independent of the current drawn from the supply, just like a battery with zero internal resistance. Such a device can be approximately but not exactly realized in practice. It is customary to represent the behavior of a power supply as an idealized supply in series with a resistance, analogous to the internal resistance of a battery, and called either internal resistance or output resistance. For some power supplies the output resistance varies with the setting of the output level control. To measure resistance of a circuit element, the most straightforward procedure is to Pass a current through the resistor, measure the current and the resulting voltage drop, and compute the resistance from Ohm's law, R = VII. To simplify resistance measurements, VTVM multimeters com­ monly contain an internal voltage source, usually a small dry cell, connected as shown in Fig. 4. The value of the resistor Ro depends on the setting of the scale selector switch on the meter. Note that when the external resistance across the meter terminals is equal to Ro, the measured voltage drops to half its peak value. Thus, the center-scale resistance reading for each range is equal to the corresponding value of Ro. Note also that the device llibeled

+

v VD:J!. Qv 1.

--

Unknown resistance

;:: 1

'R·l

+ ~\IfIY.' 1

\J1't -:;. ~otl?lIl'\\(

®: "1

..," 9..~ t.\

5

electronic instrumentation

simply Vin the diagram may be a voltmeter using electronic amplification, as described above.

experiment

10 'aVOid shacK 1 ~nd e kvatecI

1.41

50

ill.

pill- tk.

thaI-

{Cl,)
iH not

rl>fmfiaL.

1

V£; \.}~(. f''>~ suV (YlClt:&

i, ,Pc-t'

Ordinary vacuum-tube voltmeters require a source of power, commonly 110 V ac, 60 Hz. The ac receptacle should also provide a "ground" terminal, to match the three-contact plug on the meter power cord. This ensures that the meter case is at ground potential (Le., the same potential as the building plumbing, steel framework, and so on) and prevents a shock hazard which would occur if the meter case were at an elevated voltage with respect to ground. After the instrument has been plugged in, turn on the power switch and wait a minute for the vacuum tubes or transistors to come to operating temperature and to stabilize. voltage measurements With the voltmeter function switch set to dc+, and the voltage range switch set at 50 V, touch the probe contact to the "common" or "ground-" meter lead and adjust the meter to zero using the ZERO-ADJUST control (not the screw on the meter itself, which should be adjusted only with the power oft), If the probe has a switch, be sure it is set to dc.

FIGURE 5

de Probe

+~------------------~------------------~

Low-voltage power supply

looQ lOW

Ground

Assemble the circuit shown in Fig. 5. Plug in the low-voltage power supply, set the voltage control to minimum, and turn it on. Read and record the voltage for each position of t~ voltage control knob from I (MIN) to 9 (MAX). For 'l'lIIn 15' V. f:·S ~ ._­ voltages below 15 V you may wish to repeat the readings on the 15-V meter scale. Any discrepancy between these readings and the previous ones is probably due to improper zero-adjust; when the scale switch is changed, the . zero-adjust should be rechecked. Any remaining discrepancy probably 1 1,Paa," ~ .;..1ft;'!1 results from the nonlinear behavior of the meter circuits. Before disconnect­ ~jjof f It) .' ing the circuit, turn the voltage control to MIN and .turn off the power. V --:1: 1 I} .•~ With the voltage contrpLa.t an intermediate range, replace the 150-0 l l wc,.tI- ~ ~ ;;:: ::tI.2.tl.lresistor with a 1500-0, I-W resistor. Does the voltagechange? How much? ~ Can you determine the internal resistance of the....J?Q.wer sllppl):: ~ this observation ? -~-. ~----.-. --~ 6

-----

voltage. current. and resistance measurements

2

EI-1

cu rrent measu rements

Assemble the circuit shown in Fig. 6, using your VTYM or a separate milliammeter. If the VTVM is used, set the function switch.1..o....MA. and the range switch to 500 rnA full scale, and tum it on. Check the zero~adjust, then tum on the power supply. Read and record the current for each position of the power-supply control from 1 (MIN) to 9 (MAX). For currents below 150 rnA FIGURE 6

Low-voltage

power supply

V::

v;::

you may wish to repeat measurements on a lower scale. At the MIN setting you may observe a substantial discrepancy in the current reading between the 0.5-rnA scale and the 1.5-mA scale. Can you understand this discrepancy in eter? This resistance is usually larger terms of the in ut re i on the lower scales. h? : 1 ~ Plot a graph ofcurrents at the various settings as a function of the voltages at the corresponding points. Do your data points lie on a straight line? What does this indicate? To what do you ascribe any deviation from the line? How might you improve the precision of your J1leasurements? From the best VII, straight line through the data points, determine the resistance R which is the reciprocal of the slope of the line. Compare your measured value with the nominal value of the resistance. Do the values agree within the precision specified by the manufacturer?

'R 1

1~~\tf

("U-'2.-I-

~).r l­

~

v

~ IiJOO A. • 1. .t

~o(J.n. x Ix u{,

\/:: '·J.x Iv ~I VolJs' .

. ~rak(M --:)0

3

resistance measurements

To use your VTVM to measure resistance, disconnect the probes from the circuit to which they were previously connected and switch the function ..~1. 'If iYtvit sw.itch to OHMS. Be ~ure the switch in the ~~obe is away from the dc position . WIth the range sWItch on the x 10 pOSItlon, "short" the probes (connect -\tIr~" SM I: . tv'Qte s.tlYQ iVr tpIO 12f9.s11wt· them together) and adjust the OHMS AD] control for zero on the resistance scale. Disconnect the probes and adjust the ZERO AD] control for zero on the voltage scales; recheck the OHMS AD] and repeat if necessary. To determine the value of an unknown resistor, simply connect the resistor between the probes. Compare the result with that determined from current and voltage measurements. Attempt to read the resistance on the x 1 and x 100 scales. Be sure that the top and bottom of the scale are in adjustment for each range. How do the values read on the various ranges compare? How do you account for any discrepancy?

4

resistance of a tungsten filament

To determine the resistance ofa tungsten filament, assemble the circuit shown in Fig. 7. Instead of using the VTYM separately as a voltmeter and an ammeter, we can make the two voltage measurements shown, and compute 7

electronic instrumentation

FIGURE 7

Low-voltage

power supply

the current I from the voltage drop VI - V2 across the resistor Ra. using Ohm's law: (1)

In principle, one could measure the voltage across Rl directly; the problem that often arises in practice is that with some instruments the negative sides of both the power supply and the VTVM are permanently grounded, so that one side of any voltage being measured must be at ground potential. There are many advantages in maintaining common grounds for instruments, one of which is minimizing shock hazards. Measure current versus voltage for a type 47 tungsten-filament light bulb. If V2 exceeds 8 V, the bulb will bum out. The value RI is a 250 Q lO-W resistor. Plot a graph of current through the bulb as a function/of voltage. Do the data points lie on a straight line? From your data, compute the filament resistance for each setting of the voltage control. What is the ratio of the resistance of the filament at the MAX position to that at the MIN position? The type 47 lamp is rated at 6.3 V and 0.15 A. The nomi­ nal operating resistance is therefore 42 Q. Compare this value with the maxi­ mum computed resistance. The nominal operating temperature of an incandescent filament is 2575 K (kelvins), which is over eight times room temperature (nominally 293 K). Do your measurements suggest that the resistance of the tungsten filament increases faster than linearly with temperature or slower than linearly? With the function switch set to OHMS determine the resistance ofthe tungsten filament. How does the value which you obtain compare with the values previously obtained? What does this suggest about the magnitude of the voltage Vo as shown in Fig. 4? By using a second meter as a voltmeter you may wish to measure the voltage at the terminals of the ohmmeter to check your conclusion.

v-=-G.·~V c>"d )-:.a·I~Pr

"

''-..,.

",./

.../

5

semiconductor diode

Replace the light bulb in Fig. 7 by a general-purpose germanium diode. Obtain the voltage-current curve for this device using the same procedure as for the light bulb.

CAUTION

Do not permit the current to exceed the diode's rated value, or the device will be overheated and its characteristics irreversibly changed.

Reverse the polarity of the device (interchange the connections to its two terminals) and again obtain voltage-current data. Plot the data for the two polarities on the same graph to form a continuous curve passing through the origin. What do you conclude about the properties of this device? 8

voltage, current, and resistance measurements

EI-1

questions

v==-

12· t

Y~ :::

fl.

-b

'2GO)(/O

1

For an ordinary voltmeter consisting of a microammeter and a series resistance, show ~hat the total series resistance needed is proportional to the full-scale voltage reading desired. Specifically, show that for .a 200-I.lA full-scale meter, the series resistance must be 5000 II for each volt of full­ scale reading desired. Such a meter is often called a 5000-ohms-per-vol} meter. '-.>/ ~c>c>OA./Y'" It- ~ I'V 2 In using the ~TVM to measure voltage across a 150-ll resistor, what is the order of magnitude of the current flowing through the meter? Is this signifi­ cant compared to the current through the resistor? Would the situation be different if the meter were an inexpensive 100-ohms-per-volt multimeter?

A

3 Suppose an ohmmeter is to be built using a 1.5-V battery and a voltmeter with input resistance of 11 Mll. What value of Ro (in Fig. 4) should be used for a half-scale reading of 50 ll? What voltage range on the meter should be used? In ~sing the VTVM>tQ W@sure current through a 150:9 resistor, how does the voltage drop across the meter compare with that across the resistor? 5 Does a light bulb obey Ohm's law? 6 Is the resistance of a light bulb, as measured using the resistance scales on the VTVM, related to the slope of the curve of current as a function of voltage? Explain. The term "nonlinear device" is used to refer to any device for which the relation between voltage and current is not a straight-line graph. Which of the devices studied in this experiment are nonlinear? Can you think of other nonlinear devices?

1'svoJ k

3<~-y

1(" "" sc.Jl. va I+"'IUO :J:\'cfOS5 p.~ tN. lnoice 0/; 12" ~1l 1:1 tfiI. seQ.. -r is VJal-F I'ts f.s. \Jq/.,u;. ~ 1'(,110 Ifs, ':j(j'1.i 1417(7..A..., 111" +- Y6U.. Y Urll07lSLOY1 12. is f< ()_ '

rot

s

\/-;.

~o I -t !Z.t€iJ.1:

:L -:. lZol

'0

~

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'0

le

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.......

I

..c..4.

1\ lilA.

Ro

experiment

E1-2

measurement of ac voltage and current

Introduction In Experiment EI-I we measured voltages and currents whose magnitudes and polarities did not change with time. In this experiment we consider alternating (or ac) voltages and currents, which change polarity and magni­ tude in a regular periodic (cyclic) pattern. The most common ac voltage or current is a sinusoidal function of time; this is the sort of current usually produced by ac power generators oand distributed for general household and commercial use. In addition, many (although not all) electronic oscillators produce ac voltages that are sinusoidal. A sinusoidal voltage may be described by an equation of the form V(t)

=

Vo sin 2nft

(2)

In this expression Vo is the maximum value of V (since the sine function never has a magnitude greater than unity) and fis the frequency of alternation equal to the reciprocal of the period T, or the time for one complete cycle. When time t increases from zero to a value equal to Tor llf, the quantity 2nft increases by 2n, corresponding to one complete cycle. These relations are shown in Fig. 8, which is a graph of Eq. (2) and shows Vas a function of time.

•

FIGURES

V

o~-------,~----------~--------~+----------.~

2T

-Yo - - - - - - - - - - ­

The magnitude of an ac voltage may be characterized by the amplitude V o, but a more common practice is to use the root-mean-square (nns)

voltage, which is defined as the square root of the average value of V 2 , averaged over a complete cycle or a large number of cycles. The reason for this seemingly arbitrary choice can be appreciated by considering the power dissipated in a resistor. An instantaneous voltage V across a resistance R 11

r

I

electronic instrumentation

produces an instantaneous current I and an instantaneous power dissipation P given by

P

VI=

=

V2

(3)

R

If the voltage varies with time, the average power Pay is still given by IIR times the average value of V 2 , which is V;ms. Formally, Vrms is defined as follows: 1 ( -T

fT

V 2 dt

)1/2

(4)

0

F or a sinusoidal voltage this expression can easily be evaluated, using an integral table if necessary. The details are left as an exercise; the result is Vrms

~ ........",. Vt-.yl'H

~

= 1.20 v. .o:..¥r%-. y r;

1

= (T

IT V 0

2 0

2

sin 2nft dt

)1/2

(5)

When we say that the ac line voltage is 120 V, this is the rms voltage. For example, a 100-W light bulb has an average power dissipation Pay of 100 W; this is related to Vrms by (6)

R

where R is the bulb resistance at operating temperature. For this bulb, we find

R = (120 V)2 lOOW

144 n

How can ac voltages and currents be measured? Clearly, the meters discussed in Experiment EI -I cannot be used directly. An ordinary d'Arsonval meter would tend to deflect in one direction during one part of the cycle, and in the other direction during the opposite-polarity part. In fact the meter movement with its mechanical inertia, of course, cannot follow the variation of voltage if the frequency is more than a few hertz (cycles per second), and instead reads the average current through it. For a sinusoidal current, the average value is zero, since it is positive half the time and negative the other half. The simplest way to measure ac current is to convert it into pulsating dc current by means of a device called a rectifier, incorporating one or more circuit elements called diodes. An ideal diode has the property that it conducts current in one direction with no resistance and completely blocks current (i.e., has infinite resistance) in the other direction. Actual diodes do not quite realize this idealized behavior (cf. the diode studied in Experiment EI -1), but practical diodes have a resistance in the "reverse" direction several orders of magnitude larger than in the forward direction. The usual symbol for a diode is shown in Fig. 9a; this symbol always shows the direction of low resistance, the "forward" direction. If a diode is placed in series with a meter and a source of ac voltage, as in Fig. 9b, the resulting voltage across the meter will vary with time as shown in Fig. 9c. The average value of this voltage is not zero, since the negative half of the cycle has been removed. The average value for the positive half is given by 1 TI2 12

fTI2 0

Vo sin 2nft dt =

2 Vo n ~

(7)

measurement of ac voltage and current

£1-2

FIGURE 9

+

C\::F' -.-.tll--­

v

11'\ __ (.\ 16 (:k

voe

(b)

(a)

/~

v

'\IO-Vf~(t\ .

L..--~-'~

r---------~,~---------J----------~,-------------t

T

2T

(c)

The average value for the negative half is zero, so the average for the whole cycle is half the average fo~J:t~~p()~i!i~~}1!lf, or -

~-"

.. ,---~-~,--

I Vav = - Vo = O.31SVo n

.'11ft

l

)

. f.,4'rl'L) d t -:: \Iv ~d\ ~l-t t T/~ C: ~\f ~yQ) j

1{{(Q)-t) \N\lo"(

:;;:::..

m.(

*_

A more common scheme, and one that increases the possible meter sensitivity, is to use a combination of diodes in a full-wave rectifier, which inverts the negative half of the cycle instead ofjust blocking it, producing the waveform shown in Fig. lOa. One commonly used full-wave rectifier circuit uses four diodes arranged as in Fig. lOb. The direction of current flow through the meter is independent of the polarity of the alternating voltage. Can you verify this? When a full-wave rectifier is used, the average voltage across the meter is given by

. ~v\\ 'tC\\~(P.\\CiY\

~ 6.~~

Va.

=~o = O. 637Vo

n

Thus for a meter with a full-wave rectifier, the rms value of a sinusoidal voltage is greater than the rectified average by a factor of 2J2/n or about ~ V",'I 1.111. This correction could be made by including a different set of scales on i i ;;J. ~(\IIII"4t the meter for ac, but it is more common to include a I-Mn resistor in the meter probe, which is in series wi th the 10-Mn internal resistance of the meter y _ 71 ,I r} for the dc ranges but is shorted out by a switch for the ac ranges. This gives (1115 .-!.--- v'V'1'/Ilw;..v a conversion factor of 1.100, which is within 1% of the correct value. , t).. fi It is important to understand that the relation between rms and rectified-. .n -.J fIA I ;;JJaverage voltages derived above is valid only for sinusoidal voltages. For an ~. V.,.dI·$ -;: ;. ()Iy~ ac voltage with more complicated wave shape, such as a square wave or a sawtooth wave, the rms and rectified-average voltages are defined in the same ,\II •. t . .

\I

.rr.

I

~

--.

13

electronic instrumentation

FIGURE 10

O~·-------------7r~T--------------~T~--------------t

2

(a)

(b)

\k{\t~ ) V(f(.ay~,

Vr.tfI.~ ~

way, but the relation between the two is different. In all cases, the meter will read a value which is 1.100 times the rectified average. Why? The above discussion of the operation of an ac voltmeter is somewhat oversimplified. First, diodes used in the rectifier circuits never have the idealized behavior described. The result is that the meter reading is pro­ portional to the rectified average voltage only at relatively high voltage levels; at lower levels the relationship is more complicated. Consequently, meters have to be calibrated empirically, and each ac scale is somewhat different from the dc scale with the same full-scale reading, especially near the zero point. Second, many rectifier circuits contain capacitance at their output. The effect of this capacitance is to produce an output voltage closer to the peak voltage (or peak-to-peak voltage) than to the rectified average. At very low frequencies the voltage will sti11 approach the rectified average. In the examples of ac voltages considered above, the average value of the voltage (as distinguished from the ~ctified average or th~e) is zero, as a result of the symmetry of the waveform. A"Sinusoidal wave, for example, has a negative half-cycle of the same shape as the positive half-cycle, but with the sign reversed. Not all ac voltages have this property; Fig. 10 shows a simple example ofa voltage whose average value is different from zero. More generally, any voltage which is asymmetric, that is, in which the shapes of the positive and negative portions of the cycle are different, may be expected to have an average value different from zero. Nevertheless, anyac voltage, symmetric or not, can always be represented as the sum of two voltages, one a constant voltage, the other an ac voltage whose average value is zero. That is, any ac voltage Vet) can be represented in the form [V(t)

=

Vo

+

VI(t)]

where Vo is constant and VI has the property (V1 )av 14

(8)

= O. To prove this

measurement of ac voltage and current

EI-2

latter property, we transpose Vo in Eq. (8) and take the average of both sides over one cycle: Vt (t) -I

IT VI(t) dt

=

Vo

Vet) -

1 IT Vet) dt - 1fT Vo dt

ToT

T

0

0

or (9)

We have replaced (VO>av by Yo, since Vo is constant. Equation (9) shows that if we choose Vo to be equal to (V)av, this guarantees that (Vt>av is always zero, as stated above. This representation of an asymmetric ac voltage in terms of a constant term plus a time-varying term with zero average value is often very useful, and one refers to the "de" (Vo) and "ac" (VI) components of a time-varying voltage, with the actual voltage Vet) always equal at each instant to the sum of dc and ac components. The averages are related in the manner discussed, and similar relations can be derived for thte rms values, as follows. First, from Eq. (8), ~

=

(Vo

+

V I )2

=

VOl

+

2VOVl

+

V/

We integrate both sides of this expression from 0 to Tand divide by T:

T r T Jo V I

III

2

T r dt = T Jo V I

2 0

2V

dt

+ /

rJo VI dt + TI JorTV/ dt T

The term on the left is equal to V;;"s' The first term on the right is simply Vo 2 , the second is zero because it is proportional to the average of VI' and the last is equal to vI rms' Thus, we have the simple result

s; rs nt ro

It. lIe ry he

:0, Ie, lth

;a )re :he to ted 1ge ted (8)

:his

/2

Vrms = [Vo: + VI rmsr \ --.---__P....!.L_ -----Pr.f!:.." -..

In circuit analysis involving transistors, vacuum tubes, and similar devices, it is often very useful to think in this way of time-varying voltages and currents in terms of their dc and ac components. It is customary in some circles to express voltage ratios in terms of decibels, abbreviated db. One takes the logarithm to the base 10 of the ratio and gives the result (which is a dimensionless number) the unit bel, named for Alex­ ander Graham BelL A decibel is a ~n!!u~J l!-J'.e.t Thus a voltage ratio of 2 is expressed in decibels as follows:

10 loglo 2 = 3.0103 Suppose the manufacturer of a VTVM states in his specifications that the response drops 2 db at 100kHz from the midfrequency response. Then

-­

-,-

-2.

10 loglo ---.:.:::.::..::::= = -2

VIOOkHz

= 0.631

Vmid

so that at 100 kHz the meter reads only about two-thirds the correct value. In addition to studying ac current and voltage measuring instruments, you may also study in this experiment the properties of a signal generator used to produce a sinusoidal voltage with variable amplitude and frequency. The behavior of the signal generator when connected in a circuit can be repre­ sented as an ideal voltage source with zero resistance, in series with a resist­ ance as in Fig. 11. This quantity, called the internal resistance or more 15

electronic instrumentation

FIGURE 11

Signal generator

commonly internal impedance, is important because it determines how the output of the generator varies when the load resistance (the effective resist­ ance of the external circuit) changes. If the internal impedance is large com­ pared to the load impedance, then changing the load has very little effect on the current but changes the voltage significantly. Conversely, if the internal impedance is very small compared to the load, then changing the load has little effect on the voltage but a significant effect on the current.

experiment 1

4(>

voltage measurements

Connect the VTVM and the signal generator to the ac power. Set the function switch on the VTVM on ac and check the probe switch to be sure it is in the proper position. Set to the 15-V scale. to IDV, I "10 'tV Set the frequency of the generator to 400 Hz by setting the dial to 40 and the frequency multiplier to 10. Set the amplitude control to MAX and the range to ('; 10 V. Measure and record the output voltages with the range switch in the .Arvlf 10 V, I V, and 0.1 V positions, using an appropriate meter scale for each. Which scale on the meter face should be used? How reliable are the indicated ('~ .qcel-+t:­ rms voltages on the generator range switch?

6 0 ~y--J ((Yt(,

IOV)l V )O-11I 2

V'c) .

-'

'-vi

~~

.---;:;::-­

output control

To check whether the output level increases linearly with the position of the output control, measure the output voltage at each mark on the output dial and plot the measured voltage as a function of control position. What can you conClude about the accuracy of this dial? Does it depend on the position of the voltage range swi~

\12­

3

rectifier characteristics

If the rectifier had the ideal characteristics described in the introduction, then at each value the rectified-average voltage would be directly proportional to the amplitude of the ac voltage. In that case the dc and ac scales on the meter would differ by a constant conversion factor, as discussed earlier. In fact, however, rectifiers never behave precisely in this idealized way. In the direction of easy current flow, the current is often nearly proportional to 16

measurement of ac voltage and current

EI-2

voltage at sufficiently high values, but at small values it may be more nearly proportional to the square of the voltage. In this case the ac scales on the meter must be modified accordingly, since the meter deflection will correspond to V/ rather than Vo itself. The actual scales on the meter are determined empirically, and are correct only for the particular rectifier used in the instrument. To study the rectifier characteristics, read from the scales several dc values and the corresponding ac values. Plot the dc value as a function of ac value, and note the regions in which the two are proportional. 4 log-log plot of characteristics

To examine this relationship in more detail, replot these values on log-log paper. On a log-log plot a straight line with unit slope corresponds to a direct proportionality, whereas a line with a slope of one-half corresponds to a "square-law" dependence. At what approximate voltage does the transition from "square-law" to linear behavior occur? This transition voltage depends on the characteristics of the rectifier. 5 frequency response The above measurements have all been taken at a frequency of 400 Hz. Around this frequency the measured voltage should not be sensitive to frequency changes. However, for frequencies very much higher or lower than this central frequency the voltmeter reading may drop off. With the sine wave output set at MAX and 10 V, measure the apparent voltage as a function of frequency, taking rather large steps so as to cover a broad range without an excessive number of readings, e.g., 100Hz, 200, 500, 1000, 2000, and so forth. Pay particular attention to frequencies above 40 kHz and take additional measurements if indicated. Using log-log paper, plot-the apparent voltage as a function of frequency. At what upper and lower frequencies is the apparent voltage down 2 db from the midrange value? How does this result compare with the manufacturer's specifications? 2hJfl I04L-'!:>v ztt:

6 internal impedance To measure output current and determine the internal impedance of the generator, return the generator to 400 Hz, and set the amplitude controls to MAX and 10 V. Set the function switch on the VTVM to MA and the range switch to 50 rnA. Note that the current is determined by the "open-circuit" vortage, as meaS'iired by the voltmeter with its very high internal resistance, and by the total circuit resistance, including the internal resistance of the generator and the very small internal resistance of the aml;l1eter. That is, I

Thus, if the voltmeter can be considered to have infinite input resistance and the ammeter zero input resistance, the internal resistance of the generator is simply the ratio of the "open-circuit voltage" (with only the voltmeter con­ nected) to the "short-circuit current" with no external resistance except the small ammeter resistance. Read the short-circuit current, and then repeat with the range switch in the I and 0.1 V positions. Using this data and the open-circuit voltages measured previously, compute the internal resistance for each range. You may also wish to investigate whether the internal resistance depends on the setting of the output level control. 17

electronic instrumentation

What is the relation between rms, rectified average, and peak voltage for the "square wave" shown in Fig. I2? For the sawtooth wave?

v ----

V(~(~.K~c\ i71-kA\f

-

"TIl.

-:>'

t;;,. tv dt~Vc>

0

IT 2

T

§.T 2

2T

hilI -:; ~ (vdt"" ~ (a)

~~ V(~dih-e~"" V~t.\"-) v

VCi-),:

Vu

'T'

t·

(b) 2 If a meter is calibrated to read rms voltage for a sine wave, what conversion factor must be applied to read rms voltage of a square wave? In studying the apparent variation of output voltage of the signal generator with frequency, how can you be sure the observed behavior is due to fall-off in sensitivity of the meter and not to actual decrease in level of the output voltage? 4 Suppose a certain voltage changes when the frequency is varied, so as to be

proportional to 1/[ How many decibels per octave is this? An octave repre­ sents a factor of 2 in frequency; the term is derived from the diatonic scale in music, in which the frequency ratio ofthe first and eighth notes is a factor of2. 5 Using the setup for obtaining the internal impedance of the signal generator,

is there any way to measure that quantity and the input resistance of the voltmeter separately? Explain. 6 An idealized current generator is a device which produces a current in an external circuit whose magnitude is completely independent of the circuit resistance. Show that such an idealized device would have an infinite internal impedance. 7 An idealized voltage generator is a device which produces a terminal voltage whose magnitude is completely independent of the resistance of the circuit to which it is connected. Show that such an idealized device would have zero impedance. 18

experiment

E1-3

lNaveform measurements

introduction

I '

1

• ,~'E1X

1 rY'i

We have seen in Experiments EI-I and EI-2 how, with the use ofa volt-ohm­ milliammeter, we can measure the average (dc) component of a voltage or current and also measure the average value of the rectified ac component. Even if we know that the waveform is sinusoidal, this information is not sufficient to establish the frequency of the wave. And for waveforms which are other than sinusoidal the information provided by the meter readings offers very little help in establishing the shape of the wave. If the voltage changed very slowly (over seconds or minutes) we could, in fact, use a dc voltmeter and measure the voltage as a function of time. The chart recorder is an automated version of such a device. But if the voltage changes are rapid compared with a fraction of a second, neither the meter movement nor the chart recorder can follow the voltage changes. If we are to record the waveform of rapidly changing voltages, we shall require a device with a much more rapid response. The cathode-ray oscilloscope is just such a device. Oscilloscopes are enormously useful, both in the physical sciences and in the life sciences. In all kinds of electronic circuitry, especially that involving pulses or nonlinear behavior, the oscilloscope provides information not readily obtainable in any other way. Two typical applications from the life sciences are in cardiography and neurophysiology. During contraction and relaxation of the heart muscle, cell membranes create electrical potentials which can be measured using oscilloscopes. In neurophysiology, oscilloscopes are often used to observe muscle response; muscle reactions generate small voltages that can be taken directly from the muscle with probes and imposed on the vertical input of the scope. The indicating device in the oscilloscope has no mechanical motion, but uses a beam of high-speed (the 0!de~~LLQ.7_m/sec) electrons in a special vacuum tube called a cathode-ray tube, usually abbreviated CRT. The principal components of a typical CRT are shown in Fig. 13. Electrons are boiled off from a hot cathode, and accelerated and focused by a series of electrodes, the details of which need not concern us here, * to form a narrow, well-defined beam. Electrons in the beam move freely in the evacuated glass envelope ofthe tube until they strike the screen; this is coated with a material called a phosphor, which glows when struck by the beam. The result is a small bright spot of light at the center of the screen, visible outside the tube. The beam can be deflected by two pairs of electrodes called deflection plates, shown in Fig. 13. A potential difference imposed between the plates of either pair produces an electric field transverse to the direction of the beam, and the resulting force on the electrons deflects them, so the beam strikes the screen off center. In fact, we can show'" that if the electrodes are properly The detaiJs of CRT operation are investigated in Experiments EF·J through EF·4. 19

electronic instrumentation

FIGURE 13

Top view Bulb Neck

Base

.Drfllj. .u UJ

L.......L_...L. •

Aquadag coating

===!!!!!!!!!!!l__________________~ \

Electron gun

f '.

20

Fluorescent screen

(on inside face of tube)

I

Deflection plates

designed, the deflection is very nearly proportional to the voltage applied to the deflection plates. Two pairs ofplates, oriented at right angles, are provided, so the spot may be deflected to any point on the screen. The x and y co­ ordinates of the spot at any instant are then proportional to the voltages on the respective deflection plates. Typically, a voltage of the order of 200 V is required to deflect the beam to the edge of the screen. The electron path between each pair of plates is usually of the order of2cm, j/.,;c"J~ flJf il'''''>f>frrl' and the total beam path is the order of 30 cm. Thus an electron spends the order of 2 x 10 - 9 sec between a pair of deflection plates, and has a total time of flight of the order of 30 x 10- 9 sec, or 30 nsec (nanoseconds). Thus this device is capable of extremely rapid response to varying voltages, and we expect it to be useful up to a frequency of the order of 100 MHz, some seven or eight orders of magnitude faster than ty"plcarmeter movements. Thus the spot on the screen becomes a voltage indicator capable of very rapid response to a change in deflection potential. Even so, the human eye , ).I

~. f; -/ ;

cannot follow such extremely rapid motions. To circumvent this difficulty, ',C' x /5·'~ "r'~" ".~" 1 we use both pairs of deflection plates at once. The voltage Vet) to be observed . ' .~ \"" is applied to the vertical deflection plates, either directly or through electronic amplification, and a voltage is applied to the horizontal plates which increases unifer.ly with ti.e. Thus the vertical deflection of the beam is proportional to Vet), the horizontal deflection to time, and the spot traces out a graph of Vas a function of t. Even if this trace occurs in a very short time interval, the image persists on the screen for a time, just as a fluorescent lamp continues to glow for a fraction of a second after the power is turned off. The trace on the screen can be viewed with the eye or photographed for more detailed study. If a periodic waveform is to be observed, the horizontal deflecting voltage can be made to vary with the same frequency as the voltage being observed, sweeping uniformly across the screen during one cycle, then quickly jumping back to start the sweep for the next cycle. In this way one cycle of the voltage is traced out over and over again. The horizontal deflecting voltage ideally should have the behavior shown in Fig. 14; its appearance gives such a voltage the name sawtooth voltage. Applied to the CRT, it is also called a linear sweep voltage or linear time base, since its function is to sweep the beam horizontally at a constant rate. This voltage is usually generated by electronic circuitry built into the oscilloscope, called the sweep generator. The frequency of the sawtooth voltage must be variable, of course, and the circuit is always

waveform measurements

EI-3

FIGURE 14

lhe

designed so as to permit synchronizing the sweep frequency exactly with the frequency of the vertical deflecting voltage. Both the vertical and horizontal deflections are achieved by amplifying the respective voltages using variable-gain amplifiers to produce sufficiently large deflections and to permit adjusting the size of the picture on the screen. Additional controls permit moving the entire picture up and down or side­ ways. Thus, ifdesired, a small portion of the picture can be magnified on the screen for more precise measurements.

to

d,

D­

m is

n, he ne :tis

we

SUMMARY

To summarize, the essential functional units of a cathode-ray oscilloscope are as follows:

Cathode-ray tube

This is the indicating device, consisting of an electron gun, a deflection system, and a screen for visible display of the electron beam.

Power supply

The power supply must supply suitable potentials for the electrodes of the electron gun as well as current to heat the cathode. Typical accelerating voltages are of the order of 2000 V, although 10,000 V is not uncommon. Television picture tubes frequently use accelerating voltages of 15,000 to 20,000 V.

Sweep generator

The sweep generator must provide the "sawtooth" sweep voltage as shown in Fig. 14 with a variable frequency, and it must be able to synchronize this with a repetitive input voltage.

Signal amplifiers

Full-screen deflection requires about 200 V. To display small signals, as small as 0.1 V, each deflection voltage must be amplified; these amplifiers must provide a voltage gain (multiplication factor) of up to several thousand.

en

:ry lye

ty, 'ed tlie ses tlal .of the

A functional block diagram of the oscilloscope is shown in Fig. 15.

lies on led

FIGURE 15 0-----\

il.ge

ed, ing

Amplifier

Signal input Synchronizing

voltage

il.ge

lily

Sawtooth

na da :am .nie ney ays

Cathode-ray tubc .

21

electronic instrumentation

T he operating panel of a typical cathode-ray oscilloscope is shown in Fig. 16; each control is labeled and its function is discussed in the next paragraph. FIGURE 16

@

JNTJ!!N

e

ACOFF

FOCUS

\'ERTPOS

ROB P
@

....

~

HORI Fa 5XL ON

\"l:.Jll I P

HOB GAIN

l'1UtQ VER

Eft

+

+

EXT SYNC

GND

The controls shown in the upper-right corner of Fig. 16 all vary the potentials applied to the electrodes of the cathode-ray tube. The intensity control (INTEN) varies the voltage on GI, the first grid of the cathode-ray tube. Advancing this control makes the grid less negative with respect to cathode, permitting an intense electron beam and a brighter trace. The focus (FOCUS) control adjusts the potential of AI, the focusing electrode. The vertical position (VERT pos) and horizontal position (HOR pos) controls vary the de voltage applied to the vertical and horizontal deflection plates. These controls may be used to adjust the position of the trace on the screen. The frequency of the sawtooth generator is controlled by the horizontal 22

waveform measurements

v _ \v

'R-::: -=l

--t;r" ",. Icl·ft ""IM..Q..

FIGURE 17

EI-3

frequency selector and frequency vernier controls. The SYNC SELECTOR switch may be used to choose a synchronizing signal. Because of the proportionality of voltage and deflection, the oscilloscope can be used to measure voltages. This requires a calibration procedure to determine what deflection distance on the screen corresponds to I-V input, and so on. The calibration factor will change with the setting of the vertical amplifier gain control, so care must be taken not to change this control after the calibration is completed. In this experiment you will gain experience in using the oscilloscope to examine a variety of time-varying voltages, both sinusoidal and nonsinusoid­ ai, and in the proper use of the various controls on the instrument. In addition, you will investigate three characteristics of the instrument which are important for its proper use. Two of these, input impedance and frequency response, are closely related to corresponding properties of ordinary volt­ meters, studied in Experiment EI-2. Like a voltmeter, the oscilloscope draws current from the voltage source it is measuring; this property is described by assigning to the scope input terminals an input resistance, usually called input impedance. For example, if a voltage of 1 V (rms) causes a current of I !!A (rms) to flow in the leads connected to the scope, its input resistance is 1 V per I !!A or 1 MO. At very high frequencies the input no longer behaves as a pure resistance and is more accurately described by a resistance in parallel with a small capacitor, typically the order of a few picofarads. Because of internal circuit capacitances, the response of the scope to a given input voltage always drops off at very high frequencies. As with the voltmeters studied in Experiment EI-2, it is customary to describe this effect by stating the frequency at which the response has dropped 2 db (a factor of 0.63) from its midfrequency value. Depending on the uses for which the scope is intended, this "cutoff' frequency may be anywhere from 100 kHz to about 100 MHz. Another important characteristic, and one unique to the oscilloscope, is the sweep linearity, or lack of it. Ideally, the sweep generator should produce the voltage shown in Fig. 14, so the resulting horizontal deflection changes at a perfectly uniform rate until, at the end of the sweep, it flies back instanta­ neously to begin the next sweep. In practice, this ideal behavior can only be approximated. Typically, sweep generators found in the real world behave as shown (exaggerated) in Fig. 17. Toward the end of each sweep, the voltage changes less rapidly than at the beginning, as shown by the decreasing slope of the curve. The reason for this behavior is to be found in the generator V

r-~----~---------.~--~~--------~-----------t

T

2T

23

electronic instrumentation

circuit, which usually em ploys a capacitor discharging through a resistor, a situ tion to be investi gated .in detail in Experiment EC- 1. Clearly, if time measurem nts on variou pa rts are to be obtained by measu ring hori zontal distances on tb screen, knowledge of the extent of sweep nonlinearity is es entia!. FIGURE 18

,(J)

r~lurn

trac

(11 )

Similarly, the " Ay back" is never instantaneous but req uires a finite time (as in Fig. l8a) . Most scopes have a provision which turns off the beam during the small in terval when the Ryback occurs and the co rresp nding portion of the cycle is then not seen at all on the screen (as in Fig. 18b). The flybac k time be omes more significant as the frequency increases. Wby?

exper iment 1

24

oscilloscope operation To gain fa miliarity with the operation of the oscilloscope and the signal generator, con nect the sine-wave output of the gen ra tor to the vertical input terminals of the scope, maki.ng sure that the two "ground" termi nals are conn cted together. (Why?) T um on the scope and the generator, and after aIJowing them to wann up a few minutes, adjust the scope to obtain each of the pictu res in Fig. 19. Try changing the ampl ilude and frequency of the generator and observe what corresponding changes are needed in the scope controls. Repeat, u fig the square-wave output of the generator.

FIGUf

waveform measurements

EI-3

FIGURE 19

(b)

(a)

CAUTION

2

(c)

If the vertical and horizontal amplifier controls are turned all the way down, the spot will not move. If the intensity is too high, the local heating caused by electron impact will destroy the phosphor, leading to a " dead spot" in the screen. It is a good practice never to permit the spot to sit at one point on the screen.

osci ll oscope cali bration

To calibrate the scope for voltage measurements, introduce a I V peak-to­ peak voltage into the vertical in put. Some scopes have a calibration voltage terminal, in which case it should be connected to the nongrounded side of the vertical input. If your scope does not have a calibration voltage available, a simple calibration voltage source can be made using a 6.3-V filament supply tran former and two resistors, as in Fig. 20. Using the center-tap of the trans­ for mer gives a voltage from the transformer of ·~(6.3) or 3.15 V rms. This 470 r.!

FIGURE 20

6.3 "rms

calibrating voltage may be checked with an ac voltmeter. What rms voltage corresponds to I V peak-to~p eak? Wi th the calibration voltage connected to the vertical input, adjust the vertical gain until the peak-to-peak (p-p) deflection in inches is equal to the p-p Voltage. Now the amplitude of any other voltage can be determined by measuring the deflection on the screen. For voltages substantially larger or smaller than ) V you will want to change the calibration appropriately. 25

electronic instrumentation

Once set, the vertical gain should not be changed; otherwise the calibration will be lost. 3 voltage measurement

Change the 62-0 resistor in Fig. 20 to various other val ues (e.g., 22,47, 100 0); measure the resulting voltage with the scope and with an ordinary voltmeter, and compare results. 4 rectification

An interesting example ofan asymmetric, non sinusoidal voltage is the output of a half-wave rectifier, shown in its simplest form in Fig. 21. Calibrate the FIGURE 21

Semiconductor diode

15kH

--

vertical deflection as before; then connect the I-V p-p voltage to this circuit, and connect the output terminals to the scope input. Note the resulting wave­ form, and make any measurements on the screen that seem appropriate. Reverse the diode. What changes do you observe? We have already made the acquaintance of diodes in Experiment EI-2. An ideal diode conducts with no resistance in the "forward" direction but has an infinite resistance in the back direction. Thus, a positive voltage at the input terminals in Fig. 21 causes current to flow almost as though the diode were a short circuit, and the voltage drop across the IS-kO resistor is nearly equal to the source voltage, with only a very small voltage drop across the diode. But for a negative voltage the diode acts as a very high resistance, so most of the voltage drop appears across it, and very little across the IS-ld1 resistor. 5 rectified-average voltage

With the vertical deflection calibrated to I Vjin., determine the peak voltage in the forward direction. Connect a voltmeter across the scope input and measure the average dc voltage. Since this circuit acts as a half-wave rectifier (cf. Experiment EI-2), the average dc voltage should be given by (V)av =

1 11:

Vo = 0. 318Vo

where Vo is the amplitude of the sinusoidal input voltage. How do your experimental results compare with the expected ratio? Can you think of reasons for any observed discrepancy? 6 input impedance

To determine the input impedance, first calibrate the vertical deflection to I Vjin. Then connect the circuit shown in Fig. 22. If the values of the resistors are much larger than the output impedance of the calibration source, the 26

waveform measurements

EI-3

FIGURE 22

Calibration

source

Oscilloscope

voltage Vs at the vertical input terminals is related to the calibration voltage Vc by R in (10) Vs = R. + R Vc In

where Rin is the input resistance of the scope. Try various values of R in the circuit until the deflection drops to half the original value, and compute the input resistance of the scope. If you wish, increase the vertical gain to obtain a larger picture, and repeat the measure­ ments. How does the input resistance of the scope compare with that of the voltmeter used in Experiment EI-2? 7

frequency response

To study the frequency response of the oscilloscope, we would need a signal generator with a frequency range considerably higher than the expected cutoff frequency. An instructive (and simpler) alternative is to use the scope to measure the frequency response of the sine-wave output of the generators available. We shall assume that the vertical deflection circuits of the scope have flat response to welJ above I MHz, so any observed dropoff below I MHz is due to the generator and not the scope. If the vertical scope response falls off below 1 MHz, the measurements described below can still be made, but you will then be measuring the frequency characteristics of the scope rather than of the signal generator. Introduce a sine-wave voltage of about I V POp (peak-to-peak) into the vertical input of the scope. Measure the pop deflection at 50 Hz, 500 Hz, 5 kHz, 50 kHz, 500 kHz, and I MHz, and at whatever additional frequencies seem useful. What is the observed reduction in deflection (in decibels) at I MHz compared with a midrange frequency (say, 500 Hz)? Find the frequency at which the output has dropped 2 db from the midrange value. Compare your result with the manufacturer's specifications for the scope and the signal generator. 8

rise time

Another property closely related to frequency response is the rise time for a square wave for both the scope and the generator. A circuit whose frequency response drops off at I MHz cannot be expected to respond to sudden step changes occurring in a time less than the corresponding period, in this case I J.Lsec. For similar reasons, oscilloscopes usually have a lower-frequency limit which makes them incapable ofresponding to very slow voltage changes, say less than 10 Hz. Scopes referred to as direct-coupled (dc) scopes are an exception; they have no lower-frequency limit, and even respond to dc voltages. Introduce a square wave of amplitUde 1 V into the vertical input. Closely examine the waveform at a frequency of 50 Hz. If your oscilloscope is ac coupled you will observe some "droop" in the peak voltage. This results 27

electronic instrumentation

from the charging of a capacitor in the input circuit and is closely related to the origin of sweep nonlinearity. Increase the frequency of the signal genera­ tor and the sweep frequency of the oscilloscope by a factor of 10. The square wave should now be reasonably flat. Increase the two frequencies by another factor of 10. You may now begin to see the rising and falling edges of the square wave, which were not apparent at the lower frequency. What does this suggest about the ratio of the rise time to the dwell time? Increase the two frequencies by another factor of 10. Finally, at 500 kHz and at 1 MHz you will observe a more rounded waveform. With a square-wave frequency of 1 MHz, adjust the horizontal gain so that the horizontal scale corresponds to 0.5 Ilsec/in. The rise time is conveniently defined as the time for the voltage to rise from 10% of its peak value to 90% of its peak value. Determine the square-wave rise time and compare with the manufacturer's specifications.

9

high-frequency cutoff The high-frequency cutoff of the scope can be modified artificially by connecting the generator to the scope through the circuit shown in Fig. 23

FIGURE 23

100kQ

1

'V\N-v Signal generator

.0002 j.tf

0

I>---------i-O

(5) eRO

rather than connecting the two directly. For sufficiently low frequencies the capacitor acts as an open circuit, but at sufficiently high frequencies it be­ comes a short circuit and most of the voltage appears across the 100-ldl resistor rather than the scope terminals. The behavior of resistance-capaci­ tance combinations such as this is studied in detail in Experiment EC-I. We can measure the frequency response of this setup, pretending that the resistor and capacitor are inside the scope. Repeat the above measurements on frequency response, and find the frequency at which the response has dropped 2 db from its midrange value. Also measure the rise time for a square­ ~ave input. Compare your results with the previous ones when the generator and scope were connected directly.

10

28

sweep linearity Finally, we may look briefly at the matter of sweep linearity. The horizontal sweep is produced by discharging a capacitor through a resistor. Initially, the rate of discharge is nearly constant and the sweep is quite linear. However, as the capacitor discharges the rate of discharge decreases proportionally, resulting in a sweep which has some nonlinearity. Adjust the sweep frequency and vernier until about five cycles of your test signal appear on the screen, as shown in Fig. 24. For an input frequency of 60 Hz what is the sweep fre­ quency? You may notice that the display has a slight flicker. What causes this flicker? Why is it not so apparent when two cycles are represented? With the sinusoidal signal adjusted symmetrically, record the IJV"HIVll)

waveform measurements

EI-3

FIGURE 24

y 3

corresponding to zero vertical deflection. Note that two adjacent positions of zero deflection a re separated by I iosec. Compute the average sweep velocity belween zeros at the cen ter of the screen and at the right-hand side of the scr en. If you wish, you may plo t the velocity as a fu nction of time on two­ cycle semilog paper. You should find that th curve is a straight line. Repeat these measurements using a frequency of 10 kHz ; what differences do you Dote ? Is the degree of nonlinearity related to the total duration of th weep? Ideally the sweep velocity should be of the for m (11)

!le le-

where Vo is the initial sweep velocity a t the left side of the scr en, and! is the characteristic relaxation time. The derivation of this equatio n will be post­ poned to Experiment EC-l, where you will study transient phenomena in detail. F rom your plot, or alternatively, from a measurement of velocity at the left- and right-hand sides of the screen, compute! and co mpare with the sweep duration. Because of the nonlinearity of the sweep, in making quantita­ tive measurements one generally uses only the initial portion of the sweep.

I~

:1­ Ve he Its las retor

questions 1 What is the rms value of a I-V POp sine wave ? Of a I-V Pop squ are wave?

2 Show that the circuit of Fig. 20 gives the correct calibration voltage.

3 As an instrument for measuring voltage, what advantages doe the oscillo­ ~al

Iy, er, ily, cy as 're­ his

Dns

scope have over a conventio na l voltmeter such as those discussed in Experi­ ment EI-2 ? What disadvantages? 4 Suppose a sinusoidal voltage is introduced into the vert ical scope input, the

horizontal sweep fr q uency is set at 120 Hz, and three complete cycles of the sine wa ve appear on the screen. What is the freq uency of the input voltage? Is this a good way to measur frequencies? Explain. 5 Suppose that instead of using a sawtooth sweep voltage for the horizontal

deflection we lise a sinusoidal vol tage of the same frequency as a sinusoidal voltage applied simultaneously to the vertical channel. Sketch one possible pict ure which might appear. W hat is the importance of the relative phase of the two ignals in determining the shape of the picture? 29

electronic instrumentation

6 Derive Eq. (10) concerning the input impedance. From this, derive an

explicit expression for R in in terms of R , Ve, and Vs. 7

In the measurement of rise time of a square wave, are you measuring characteristics of the generator or of the scope ? How do you know?

8

If a sinusoidal voltage is applied to the vertical input of a scope, and the picture is as shown in Fig. 25, what would you expect to be the cause?

FIGURE 25

9

If a sinusoidal voltage is observed using a sweep-generator frequency which is double that of the sinusoidal voltage, how will the picture look?

30

experiment

EI- 4

comparison of variable voltages

Introduction In Experiment EI-3 we used the oscilloscope to observe the waveform of a variable voltage, that is, to plot voltage as a function of time. The horizontal deflection was produced by a sawtooth sweep generator, also called a linear time base. It is often useful to use a voltage other than a sawtooth for the horizontal

deflection. In this experiment we consider three different possibilities for the vertical and horizontal inputs: two sinusoidal voltages with different frequencies, sinusoidal voltages with the same frequency but different phases, and nonsinusoidal voltages. Lissajous figures This term refers to any superposition of two sinusoidal (simple harmonic) motions at right angles to each other. The simplest example is a superposition of two motions with the same frequency and amplitude. For example, if the same sinusoidal voltage is connected to both horizontal and vertical inputs, and the gain controls are adjusted so the maximum horizontal and vertical deflections are equal, then at any instant the x coordinate of the spot is equal to its y coordinate, and the resulting trace is a straight line oriented at 45° to !he axes, with total length equal to 2J2 times the amplitude of each separate displacement (horizontal and vertical). Why? Next, suppose again we have two sinusoidal signals with the same frequency and amplitude, but this time with a phase difference of a quarter cycle (n/2 or 90°). For example, suppose the vertical displacement is a quarter cycle ahead of the horizontal. This state of affairs could be described by the pair of equations x

=

Xo

cos rot

y = Yo cos (rot

+ ~) =

-Yo sin wt

(12)

Because the amplitudes are equal, Xo = Yo' In this case it is easy to show that the distance of the spot from the center of the screen, given by x o, is con­ stant, so that the trace is a circle; it is also easy to show that the spot traces out the circle in a clockwise sense with constant angular velocity equal to ro. When the phase difference is other than n/2, we can show that the trace is always an ellipse, with its axes tilted with respect to the horizontal and vertical axes. In fact, the orientation and shape of this ellipse provides a way to measure the relative phase of the two signals, as we will now show. Suppose that two sinusoidal signals have different frequencies. If the frequency ratio is a rational number (i.e., a ratio of two integers), then the trace must always be a closed curve which repeats itself over and over. For example, if the frequency ratio is 5 to 13, then 5 cycles of the lower frequency 31

electronic instrumentation

represen t the same ti me interval as 13 of the higher, so that after this time interval both signals have gone th ro ugh an integer number of cycles-and are back at the same point as at the beginning of the interval. If the frequency ratio is irrational (i .e., not ex pres ible as a ratib of integers), then the trace does not clos on itself, but fiUs up the en tire rectangle d etermined by the maximum horizontal and vertical d isplacement . F igure 26 sho ws a fe w FIGURE 26

:2

.'5

:;

5

3

simple exa mples of Lis ajo us figu r s. Note that in each case the freq uency ratio wi wx is eq ua l to the ra tio o f the total n umber of maxima in t he vertical direction (either up or dow n) to the total number of max ima in the horizontal di rection (either left or right). Is this true for every rational frequ ency ratio ? Li ssajous figures provide a convenient way to compare two freque ncies. T h us they can be used to measure the freq uency of an unkno wn signal by co mparing it with a signal of kn wn frequency. T he simplest case is of co urse a 1: I ratio , but there are many other po sibilities. phase measurements An im portant and useful applicati on of the oscilloscope is for measuri ng the relative phase of two sinuso idal signals of the same frequency. There are several ways to p roceed ; the simplest makes use of measu rements of the Lissajous figu re (always a line, circle, or ell ipse) formed when one voltage is fed to each scope axis input. Let the oscilloscope deflections be described by the equations x = Xo cos

wt

y = Yo cos (wt

+

¢)

(13)

T hat is , y has the same fr q uency as x, but y is ahead of it in phase by a phase angle ¢ , wh ich may be posi tive or negative. If ¢ = O. the trace is a stra ight line; if ¢ = n/2, it i.' a circle. F igure 27 shows the case in which ¢ is between 32

comparison of variable voltages

EI-4

o and n12, assuming the positive directions on the screen are up and right. To obtain the.phase angle, we note that when the trace crosses the horizontal FIGURE 27

----A----~

axis, y

±n12

=

O. This occurs whenever cos (OJt +
=

Xo cos ( ±;


=

±xosin
=

(14)

where we have used the identities for the cosine of a sum and difference. Thus, the distance B in Fig. 27 is given by 2xo sin
Another technique for phase measurement uses the scope's internal sweep generator with externally supplied synchronizing signals. We connect one of the two signals to the vertical input, switch the horizontal to the sawtooth sweep and the synchronization selector to external, and apply the vertical voltage also to the external "sync" input. We note the position of the trace on the screen. Then, keeping the original signal on the vertical input, we con­ nect the other signal to the sync input. Because this is out of phase with the first signal, the sweep will start at a different point in the cycle of the vertical input signal, and the picture will appear shifted to the right or left by some fraction of a cycle, depending on the phase difference of the two signals. Some oscilloscopes have a provision for varying the intensity of the beam in response to an external signal. This provides, in effect, a third coordinate for the display, and it is often called the "z axis" or, more descriptively, intensity modulation. If the scope you are using has this provision, the follow­ ing method can be used for phase measurement. Connect one voltage to the vertical input, the other to the z-axis input. If the two signals are in phase, the positive peaks will be the brightest portions of the trace, and the negative peaks the dimmest, as shown in Fig. 28a. Contrast can be improved by adjusting the beam intensity and the amplitude of the z-axis voltage. If the two signals are out of phase, the bright portions will be shifted by a corres­ ponding fraction of a cycle, as shown in Fig. 28b. This method is most useful when a square-wave signal is available which 33

electronic instrumentation

FIGURE 28

(al

(b)

(c)

is synchronized with one of the ine-wave inputs, as in the case in which the sin e-square wave generator is used for one input. In this case the trace is cut off sharply at the points in t he cycle correspond ing to the steps in the square wave. The trace then appear a in Fig. 28c, which shows phase differences of 0, n/4, and n12, respectively.

34

voltag e-cu rrent cha r acteristics Among the many other applications o f the oscill oscope one which is often useful is observing the voltage-cu rrent characteristics of d evices. The sim plest example is a resistor obeying Ohm's law, V = IR, for which the currelll is directly proportional to the Vol tage. T hus a gra ph of I as a functio n of V is a straight line. This is conveniently dis played on a scope by applyi ng a si nu­ soidal voltage to the resistor, applying this voltage also to the horizontal input and applying to the vertical input a voltage proportional to the currelll through th resistor. Applied to the resistor, this techn iq ue is trivial ; but there are many devices that do not o bey Ohm's law, and fo r wh ich current fl ow de pends on the sign of the applied voltage as well as its magnit ude. T he semiconductor diode discussed in Experimen t EI-2 is a simple exam ple. Any such device which does no t obey Oh m's law is called a nonlinear deL;ice. A voltage-current curve for a semicond uctor diode typically has the sha pe sh own in F ig. 29. The technique just described can bv used to di splay th is curve on the cop screen.

FIG ,

comparison of variable voltages

EI-4

FIGURE 29

----------------~L----------------v

experiment LI SSAJ OUS FIGURES To o bserve Lissajous figures, connect the horizontal input to a 60-Hz signal

and the vertical to a sine-wave generator. Most scopes have a LINE sett ing on the sweep selector switch. This applies a 60-Hz signal from th ac li ne to the horizontal input. If your scope does not have this provision, use a low­ voltage (6.3- V) signal obtained from the secondary of a small filament-supply transformer powered by the 110-V line. 1

Adjust the generator frequency carefully until you obtain each of the patterns of Fig. 30. How accurately is the generator frequency scale calibrated ? Tha t is, what dial setting corresponds to an actual frequency of 60 Hz? Record this value. Now shift the frequency slightly off the value which gives a stable pattern, and note that the pattern now shifts slowly through the whole range of patterns in Fig. 30.

lich the :e is cut : squa re ~n es of 2

is often ;im plesl lrrent is )f V is a

equal frequencies

multiple frequencies

Set the generator frequency near 120 Hz and adjust for a stationary pattern. Again read the frequency scale and record the result. Similarly, find the dial FIGURE 30

, a SInU­ rizontal

: current

,devices the ign lr diode ichd oes rYe for a chnique

I{J = ­

rr

'P

=0 35

electronic instrumentation

positions for 20, 30, 60, 120,240, 300, 360,420,480, 540, and 600 Hz, and construct a calibration curve (actual frequency as a function of dial setting) for this range of frequencies. 3

non multiple frequencies

Look for other Lissajous patterns in the vicinity of40 and 90 Hz, and at other frequencies that seem interesting. How do these differ from the previous patterns in which one frequency was always an integer mUltiple of the other?

PHASE MEASUREMENTS

To study techniques for measuring relative phase, two signals with an adjust­ able phase difference are needed. Figure 31 shows a simple way to obtain

FIGURE 37

A 100kQ

Signal generator

B

.01 ~f

-these voltages. The details of operation of such circuits will be studied in Experiment EF-I ; for the present we simply remark that the current through the resistance-capacitance circuit leads the voltage in phase by an amount which varies from nearly Tel2 at low frequencies to nearly zero at high frequencies. 4

lissajous figures

To check that the circuit does have this behavior, connect A to the horizontal and B to the vertical input as shown, vary the frequency, and observe the resulting Lissajous patterns. It will be necessary to vary the vertical gain to keep a constant vertical amplitude. Now set the frequency for a phase shift in the vicinity of 45°, and use the method described above (measuring the Lissajous figure) to measure the phase shift. Repeat these observations for at least two other frequencies, one higher and one lower than the first. Recheck the equality of the two amplitudes each time. If you wish, take several frequencies and plot a graph of phase-shift as a function of frequency. It may be useful to use semilog paper, with the frequency on the logarithmic scale. 5

external synchronization

Return the frequency to the same value as used for the first measurement above, and use the "external synchronization" method described above. Using the same circuit, connect the terminals labeled A in Fig. 31 to the vertical input, and also to the external sync input. Adjust the sweep frequency controls to obtain one cycle, with the sync amplitude control all the way down, and then turn up the sync amplitude just enough to stabilize the picture. It may be necessary to adjust the sweep frequency vernier slightly. Note the horizontal position of a recognizable point on the screen. A convenient 36

comparison of variable voltages

EI-4

procedure is to adjust the vertical position control so the trace is centered vertically, and then observe the horizontal position of the point where the trace crosses the horizontal axis. Also measure the horizontal distance corresponding to a half-cycle. (Why not a full cycle ?) Now disconnect the A output from the circuit of Fig. 31 and connect the B output instead. Again observe the position of the point where the trace crosses the x axis. From these observations, determine the magnitude and the sign of the relative phase of the two signals. Repeat, using the other two frequencies used above, and compare your results with the previous ones.

6

intensity modulation

To use the intensity modulation (z-axis) technique, return the sweep controls to the linear time base with internal sync. Connect output A to the vertical input and the square-wave output from the generator to the z-axis input. With the frequency set to the same value as for the first of each of the previous sets of measurements, check to see that the sine and square waves are in phase. Now disconnect output A and connect output B to the z axis. By measuring horizontal distances on the screen, determine the relative phase (magnitude and sign) of the two signals A and B. Repeat with the other two frequencies used above, and compare results.

VOLTAGE-CURRENT CHARACTERISTICS

As mentioned above, the voltage-current characteristics of nonlinear devices can be displayed conveniently on the scope face. We consider a semiconductor diode as an example in the circuit shown in Fig. 32. The voltage across the

FIGURE 32

Semiconductor diode

15kQ

IS-ill resistor is proportional to the current through the circuit. If we con­ nect this voltage to the vertical input and the generator voltage to the horizontal input, we obtain a plot of current through the diode as a function of voltage across diode and resistor in series. 7

diode characteristics

Set the generator frequency to a low value, say 100 Hz, and obtain the V-I curve. How does the pattern chan,ge with generator voltage? Reverse the diode and repeat the observation. What is the relationship between this picture and the previous one? 37

electronic instrumentation

8 high-frequency response Now increase the frequency of the signal generator until you see some opening of the pattern. This opening is associated with capacitance across the rectifying junction. Note the frequency at which appreciable opening develops. For frequencies in excess of this value the diode is no longer a very effective rectifier.

questions 1 . In Fig. 26a, if the horizontal frequency is ] 000 Hz, what is the vertical

frequency? 2

In the Lissajous figure of Fig. 27, what determines the direction (i.e., clock­ wise or counterclockwise) in which the spot traces the pattern?

3 Verify the statements following Eqs. (12). 4 Verify the statement made in the text about the relation of the numbers of

horizontal and vertical maxima in a Lissajous pattern to the frequency ratio. S Can you determine from these experiments whether a positive vertical voltage

causes an upward or downward deflection? Whether a positive horizontal voltage causes a deflection to the left or right? 6 In Fig. 27, derive a relation between the phase shift and the angle which the

major axis of the ellipse makes with the horizontal axis. 7 When the sweep synchronization method of measuring phase is used, does

a positive value of cjJ shift the pattern to the left or the right? Explain. 8 You may have noticed that the Lissajous figure formed with frequencies of 60 and 120 Hz is similar to that for 60 and 30 Hz. Discuss the similarities and differences. 9 Which of the three methods for measuring phase do you think is best? Why?

38

experiment

E1-5

transducers

introduction The tenn transducer, as used here, refers to any device which converts electrical information to some other form, or the reverse. Familiar examples of transducers include microphones and loudspeakers. A microphone con­ verts the pressure variations of an acoustic (sound) wave into corresponding voltage variations at its terminals, and a loudspeaker does the reverse, con­ verting a varying electrical signal into a sound wave. A record-player pickup cartridge has a similar function, converting mechanical vibrations of the stylus to electrical impulses. Transducers of various kinds are of central importance in most of the significant applications of electronic circuits. Additional examples can be grouped into various categories. Electro-optical transducers include the various types of photocells such as photo transistors, vacuum phototubes, cadmium-sulfide cells, and others. These either emit an electrical signal dependent on the light falling on them or use light input to control the flow of current in a circuit. Temperature-sensitive devices, such as thennocouples and thermistors, translate temperature variations into electrical signals. Mechanical defonnations, such as stretching of a material, can be converted into electrical signals using devices called strain gauges. A wide variety of particle detectors, including Geiger tubes, proportional counters, and various solid state detectors, report the passage of a charged particle through the counter by emitting a corresponding pulse; in some cases, the pulse amplitude is directly related to the particle energy. In this experiment we confine our attention to two commonly used trans­ ducers; one is acoustic, the other thermal.

ACOUSTIC TRANSDUCERS

The acoustic transducer is a type used to produce ultrasonic acoustic waves, that is, waves with frequencies above the range ofhuman hearing. For example, a wave with wavelength ;. 1 cm = 0.01 m corresponds to a frequency of f determined by the usual relation f = ciA, where c is the speed of sound. In dry air at 20°C, c = 344 m/sec, and the frequency corresponding to a wavelength of 1 cm is

f=

344 m/sec 0.01 m

34.4 kHz

A standard fonn of ultrasonic transduCt'r is shown in Fig. 33. This device is made from a cylinder of barium titanate, a material which can have a permanent electric dipole moment, below a certain critical temperature. The barium titanate is given a radial polarization by heating it, applying a potential difference between electrodes on the inner and outer surfaces, and 39

electronic instrumentation

FIGURE 33

then cooling it through the polarizing transition temperature, about 120 C. The polarization is then "frozen in" and persists even after the potential difference is removed. The remarkable property of barium titanate which makes it so useful in transducers is that if a potential difference is later applied to a cylinder which has been polarized by a potential difference of the same sign, the cylinder will shorten. Conversely, if the applied potential is opposite to the polarizing potential, the cylinder will lengthen. Thus by applying an alternating potential between inner and outer surfaces of the cylinder and adding a plate to the end, one may use this device as an acoustic transmitter, that is, as a source of sound waves. Conversely, a longitudinal compression on the cylinder produces a potential between the electrodes in the same direction as the polarizing potential; and a longitudinal stretching produces a reverse voltage. Thus this same device may also be used as an acoustic receiver. Such devices have been used successfully at frequencies up to 100 kHz. A very simple and inexpensive ultrasonic disk transducer has been de­ veloped and is used commonly for remote control of television receivers. Such a device is shown in Fig. 34a. The transducer is constructed from a disk of barium titanate to which an aluminum disk is bonded. A pair ofelectrodes, one in the center and one around the edge, are evaporated onto the barium titanate. The device is cooled through its polarizing transition at 120°C with C

FIGURE 34

Barium titanate

Aluminum

(a)

(b) 40

transducers

EI-5

a potential between the electrodes. Now if a potential is applied between the electrodes, the disk will warp, the direction of warp depending on the sign of the potential. Conversely, if we warp the disk we generate a corresponding voltage. Another interesting feature of this transducer is that it has an extremely sharp resonance frequency, or natural frequency of vibration, corresponding to a vibrational motion in which the disk warps one way and then the other, with the center moving in and out in opposition to the rim, as shown in Fig. 34b. We can show that the frequency of this motion is given approximately by

f=~2 2a

(16)

where u is the speed of sound in aluminum, approximately 6260 m/sec, I is the thickness of the disk, about 0.1 em, and a is the radius, about 1 cm. Note that this vibration frequency is inversely proportional to the square of the radius. The resonance of the disk gives this device interesting electrical properties, and it is of interest to observe how the electrical impedance varies with frequency. It turns out that the electrical behavior is quite similar to that which would be obtained from the circuit of Fig. 35, and so this is called an FIGURE 35

L

c equivalent circuit, for the transducer. At very low frequencies the inductance L is essentially a short circuit, and the dominant effect is that of the series capacitor c. This corresponds to the capacitance between the electrodes on

the barium titanate surface. Because of the very high dielectric constant of this material (about 3000), we obtain an unusually high capacitance, about 850 pF. At very high frequencies both capacitors are essentially short circuits and the behavior is dominated by the series resistor r, which represents dielectric losses in the barium titanate. Near the resonant frequency of the parallel L-C combination, which is given by I

f = 2n(LC) 1/2

(17)

the impedance would become infinitely large if it were not for the resistor R. The inductor L and the capacitor C correspond to the mass and elasticity, respectively, of the disk, and R corresponds to a damping force arising chiefly from the radiation of acoustic energy. Because of the sharp resonance in the mechanical vibration of the disk, we expect a corresponding sharp peak in the electrical impedance of the device at the resonant frequency. Furthermore, if the device can exhibit other normal modes of vibrational 41

electronic instrumentation

motion at higher frequencies, each of these should be accompanied by a corresponding peak in the impedance curve. These higher modes are not represented in the simple equivalent circuit of Fig. 35, but they could be included by adding more parallel L-C combinations. By measuring the impedance as a function of frequency, one can investigate the characteristics of this device.

THE THERMISTOR The operation of the thermistor is quite simple. The thermistor is made ofa

semiconductor material whose resistance decreases rapidly with increasing temperature, for reasons to be studied in the experiments on semiconductor electronics. To use the thermistor we need only construct a simple circuit to observe the changing resistance, such as the circuit shown in Fig. 36. One FIGURE 36

Thermistor

+.

-=- 3-6

V

50MA

A

typical thermistor has a resistance at 25°C of 135 kn with a temperature coefficient at this temperature of -4.6% per Celsius degree (CO), a much larger temperature coefficient than for ordinary resistors. The thermistor may be used as the basis of a temperature scale, or it may be calibrated at the ice and steam points and then compared with an ordinary mercury thermo­ meter at intermediate temperatures.

experiment 1 fundamental disk resonance

To study the electrical properties of the barium titanate transducer, assemble the circuit shown in Fig. 37. The voltage across the l-kn resistor is propor­ tional to the current through the transducer, which is inversely proportional to its impedance, assuming the total circuit impedance is large compared with 1 kn. Under what conditions is this assumption valid? Vary the fre­ quency of the signal generator to find the lowest-frequency resonance, and observe this frequency. Compare with the value predicted by Eq. (16). 2 higher resonances

You may find a second resonance at about four times the frequency of the first. This is the second harmonic; in this mode there are two radial nodes. In addition, you may be able to detect as many as a dozen weak higher resonances 42

transducers

EI-5

I

FIGURE 37

Transducer

lkn

in the frequency range up to 1 MHz. Measure as many resonance frequencies as possible. 3 thermistor characteristics

To observe the characteristics of the thermistor, assemble the circuit shown in Fig. 36. Measure the current flow at several easily reproducible tempera­ tures, such as the ice point, the steam point, and human body temperature. From this data, plot a calibration curve to use the circuit current as a temperature measurement. 4 linearity of temperature dependence

It is of interest to determine whether the thermistor resistance is a linear function of temperature, or some more complicated function. Immerse the thermistor and a conventional mercury thermometer in a water bath, and then vary the water temperature in about lOCO steps by heating it or adding ice, and record current as a function of temperature as measured by the mercury thermometer. Plot a graph showing resistance as a function of temperature. Is the relation linear? 5 voltage-current linearity

Many semiconductor devices are nonlinear, in the sense that current is not directly proportional to voltage. To investigate this aspect of thermistor behavior, immerse the device in a constant-temperature bath, such as ice­ water, and measure current as a function of voltage, using voltages up to 15 V. Two 7t-V "c" batteries in series provide a convenient variable voltage source. Do the results of this observation give any additional insight into the values of resistance determined in previous parts of the experiment?

questions 1 If a large sheet, comparable to a loudspeaker cone, were attached to the

barium titanate transducer, how would you expect the electrical properties to change? 2 Would you expect the behavior of the barium titanate transducer to be linear? That is, should the displacements be directly proportional to the applied voltage? Can you think of a way to check this linearity experimentally, at least in principle? 43

electronic instrumentation

44

3

Suppose that in a certain experiment a disk transducer is needed which is similar to that used in this experiment but has a fundamental resonant frequency of 20 kHz. If its thickness is 0.1 cm, what should be the radius of the disk?

4

To describe the state of a thermistor under given conditions, three variables are required: voltage, current, and temperature. The characteristics can then be represented in several ways: One way is to hold the temperature constant and plot current as a function of voltage, repeating this procedure for several different temperatures to obtain a set of curves. Sketch the shape that such a set would have.

S

With reference to Question 4, another way to represent thermistor character­ istics would be to hold the terminal voltage constant and plot current as a function of temperature, repeating for several values of voltage. Sketch the general appearance of such a set of curves. Can you think of a still different way to represent thermistor characteristics?

6

What advantages does a thermistor have over a conventional mercury thermometer as a temperature-measuring instrument? What disadvantages? Consider factors such as size, temperature range, accuracy, and any others you can think of.

2

3

4

5

6

7

B 910

2

3

4

5

6

7

B 910

2

3

4

5

ber eley phys·cs I

orator ,

dition

alan m. portis, unIversity of c lifo r , hugh d. young, carnegIe rn e l l o

berkf ley

('ltv

radia l fields ima g e charge fIeld lines and ree; the

a

I

oc ity

e tic flel

magnetic coupling

mcgra new york london

mexico

F-' F-2 F-3

F· 4 F-5

0

any

san franc i eco

d uss /dort

-hill b o k st louiS

~ rs;ty

penama

sydl t:y

toronto

fields Copyright 1971 by McGraw-Hill, Inc.

All rights reserved. Printed in the United States of America.

No part of this publication may be reproduced, stored in a

retrieval system, or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher.

Library of Congress Catalog Card Number 79-125108

07-{)50484-9

1234567890 BABA 79876543210

The first edition of the Berkeley Physics Laboratory

-copyright © 1963, 1964, 1965 by Education Development

Center was supported by a grant from the National Science

Foundation to EDC. This material is available to publishers

and authors on a royalty-free basis by applying to the

Education Development Center.

This book was set in Times New Roman, printed on

permanent paper, and bound by George Banta Com­

pany, Inc. The drawings were done by Felix Cooper;

the designer was Elliot Epstein. The editors were Bradford

Bayne and Joan A. DeMattia. Sally Ellyson

supervised production.

contents berkeley physics laboratory, 2d edition complete contents of the 12 units

mathematics and statistics MS-' MS-2 MS-3 MS-4 MS-5 MS-6

derivatives and Integrals trigonometric and exponential functions loaded dice probability distributions binomial distribution normal distribution

acoustics and fluids AF- t AF-2 AF-3 AF-4 AF-5 AF-6

acoustic waves acoustic diffraction and Interference acoustic interferometry fluid flow viscous flow turbulent flow

mechanics

microwave optics

M- t

MO- t MO-2 MO-3 MO-4

M-2 M-3

M·4 M·5

velocity and acceleration collisions dissipative forces periodic motion forced oscillations

electronic instrumentation voltage. current. and resistance measurements measurement of ac voltage and current waveform measurements comparison of variable voltages transducers

microwave production and reflection interference and diffraction the klystron microwave propagation

laser optics LO· t LO-2 LO-3 LO-4 LO-5

reflection and refractIon of light polarizatIon of light diffraction of light Interference of light holography

atomic physics radial fields

Image charges

field lines and reciprocity

the magnetic field

magnetic coupling

AP- t AP·2 AP-3 AP-4 AP-5

atomic spectra photoelectric effect the photomultiplier and photon nOise ionization by electrons electron diffraction

nuclear physics electrons and fields fF· t fF·2 fF·3 fF-4 fF-5

acceleration and deflection of electrons focusing and Intensity control magnetic deflection of electrons helical motion of electrons vacuum diodes and the magnetron condition

NP- t NP-2 NP·3 NP-4 Np·5

geiger-mueller tube radioactive decay the SCintillation counter beta and gamma absorption neutron activation

semiconductor electronics electric circuits resistance·capacitance circuits resistance-inductance circuits LRC circuits and oscillations coupled OSCillators periodic structures and transmission lines

SE- t SE·2 SE·3 SE-4 SE-5 SE-6

semiconductor diodes tunnel diodes and relaxation oscillators the transistor transistor amplifiers positive feedback and OSCillation negative feedback

fields

INTRODUCTION

The field concept is very useful in describing interactions between particles or objects. The gravitational interaction of two bodies can be described in tenns of forces, but often it is easier to speak of gravitational fields. Any massive object is said to establish a gravitational field in the space surrounding it, and any second mass present in this field experiences a force proportional to the field. If the field is produced by a single spherically symmetric mass, it is directed radially outward from the center, falling off as l/r2. If the field source is more complex, the field configuration is correspondingly more complex. With the interaction a potential energy can be associated, and also a potential energy per unit mass, called the gravitational potential. These experiments are concerned with the electric and magnetic fields associated with the interaction of charged particles. Interactions between charges at rest are described in tenns of electric field and electric potential. In vacuum these interactions are completely analogous to the gravitational interaction; when dielectric media or conductors are present, the charge configurations in the materials must also be taken into consideration. Forces between moving charges are traditionally described in tenns of magnetic fields; any charge in motion relative to a given frame of reference, or any current, produces a magnetic field. A charge moving through that field experiences a velocity-dependent magnetic field force. Denoting the electric and magnetic fields by E and D, respectively, and the particle charge by q, we represent the total force F on the charge when it is moving with velocity v by F

=q(E + v x D)

(1)

which is called the Lorentz force law. A magnetic field which varies with time produces an electric field; this phenomenon is called electromagnetic induction, and is described by Faraday's law of induction. Similarly, a time-varying electric field produces a magnetic field. All these time-dependent interactions could be described directly in terms of forces between moving charges, but the description in terms of fields is much simpler. In the first three experiments we study electrostatic fields and potentials. In the remaining experiments we investigate magnetic fields and electro­ magnetic induction. Some of these anticipate the Electrons and Fields unit in which you may investigate the trajectories of electrons in electric and magnetic fields.

1

experiment

F-1

radial fields

introduction In the first three experiments we study field and potential patterns in the vicinity of electrodes. Each conducting electrode forms an equipotential surface, and if a potential difference is imposed on two electrodes, an electric field pattern is established in the region between them. Ideally, we would like to be able to measure fields in vacuum in the vicinity of the deflecting elec­ trodes. Such measurements are possible, but they are difficult and not very illuminating to someone who is encountering the study of electrostatics for the first time. Instead, we shall study a much simpler problem, the potential pattern on a conducting sheet for various electrode configurations. The relation between the potential and field configurations on a two-dimensional conducting sheet and the configurations in the three-dimensional vacuum problem will be developed in the course of this set of experiments. In these experiments you may use as a conducting sheet a material called Teledeltos paper. This paper was developed by the Western Electric Company to be used with electric recording instruments. The paper is impregnated with graphite and aluminized on one side. Electrodes can be applied to the Teledeltos paper with silver paint. This paint, which is used for printed circuits, dries in air to a good electrical conductor.

experiment Potentials will be measured with a vacuum-tube voltmeter. One possible concern is that by placing the voltmeter probe in contact with the Teledeltos paper, we may alter the potential. This is a problem with typical meter move­ ments, which require substantial currents in order to obtain a deflection. With the vacuum-tube voltmeters, however, the current drawn from the Teledeltos paper is extremely small so that for most applications the change in potential will be negligible. We begin with a few preliminary measurements to acquaint you with the operation of the vacuum-tube voltmeter. Ifyou have not already performed Experiment EI-I, you will want to get some familiarity with the vacuum-tube voltmeter before beginning these experiments. 1

circular electrodes

Our first measurements will be performed with the electrode configuration shown in Fig. 1. Place a piece of Teledeltos paper, with this electrode con­ figuration, on a smooth, hard surface. With the probes provided, connect the positive terminal of a 7t-V "C" battery to the outer ring and the negative 3

fields

FIGURE 1

Teledeltos aper

1 Probe

Conducting circles

terminal of the battery to the central electrode. Place a centi meter scale along a radius of the ring. Measure and record th e potential as a functi on of radius at intervals fO.5 cm. What wo uld you expect for the variation of potential V with radius r for a different rad ial direction ? Measure a few poi nts as a check on your prediction . Based on your measurements of potential, what must be 1h direction of the electric field ? Using the measurements of potential we sha ll now compute the electric field . We know that the electric potential is the potential energy U per unit charge U

V= -

q

(2)

and the electric fiel d is the force per unit charge

E= ~ q

(3)

From the definition of potential energy, for a small displacement ell"

dV

=

- Edr

(4)

dV dr

(5)

Or, solving for E,

E=

Since we have measured the potential a t d iscrete point s, we shall be able to compute only the average field over successive intervals. Using the relation Eav

AV =

-

Ilr

(6)

compute the average fie ld for each interval tha t you have measured . Plot your computed values of the average field as a fu nction of I/r. What relation do you find between E and r? The significance of this o b 'erved relationship between E and r may not be immediately apparent. In o rder to see the connection with a physical field in vacuum, we shall need to discuss the beha vior of the system which we have been investigating. 4

radial fields

2

F-1

line charge

There are two ways in which we may discuss this problem, either in terms of charges or in terms of currents. Let us first look at the problem from the point of view of charges. We may not assume, as in the usual electrostatic problem, that the region between the central and outer electrodes is entirely charge free. What may we assume? We may assume that the electric field within the Teledeltos paper is everywhere parallel to the plane of the paper. If the field were not parallel, charge would build up on the surface until the field did become paralleL Thus, we are not studying a true three-dimensional field configuration, but rather a two-dimensional field configuration where the field is constrained to lie within the paper. This is not to say that there are no field lines outside the paper. The surface charge which constrains the field lines produces an external field also. Now, for what kind of an electrode configuration are the field lines all constrained to lie in a plane? Let us consider the field around a line of electrical charge, as shown in Fig. 2. We may find the field around such a line

FIGURE 2

lE ,I I, r' I I I

K E=­ r

I I

I

+++++++++++++++++++

charge either by direct integration or (better) by the Gauss theorem; we find that the field falls off as l/r, E

K

(7)

r

rather than as l/r2, which would be the result for a point charge. The constant K depends on the strength of the line charge and the units, and we need not be concerned with this aspect of the problem just now. If we establish the

potential at some radius a, then we can write the potential at any other radius as rK r V - KIn- -dr (8) a a a r

I

How may we establish the potential to be Va at all poiPts for which r = a? This may be done most easily by placing a metallic cylinder along the equi­ potential surface at r = a and setting this conductor at Va' What will be the potential at a radius r = b, where we shall assume that b is larger than a? From Eq. (8) we have b V - KIn- a

a

(9) 5

'Ids

or solving for K K

(10)

Eliminating Kin Eq. (8), we have VCr)

Vb In (ria) - Va In (rib) In (bla)

(11)

In particular, if Va is our reference potential, we may set Va

to obtain

V( ) r

= v: In (ria)

° In (bla)

(12)

Now we may interpret Eq. (12) in the following way: If we have an equi­ potential surface of radius a with potential zero and an equipotential surface at r = b with potential V o, then the potential at any intermediate radius will be given by Eq. (12). We may, for example, establish the potential Vo with a battery and forget about the line charge altogether. From our discussion of the electric fields within the Teledeltos paper and the fields between a pair of coaxial cylinders, we may now understand why Eq. (12) describes the potential between a pair of concentric rings painted onto the Teledeltos paper. It is easiest to look at this problem in terms of currents. We know that when a potential is applied to the electrodes, electric current will flow through the Teledeltos paper. Since the total current flowing out through any concentric ring must be the same, the current density (amperes per square meter) must fall off as l/r. Now, for typical conducting media, as we shall see later, the current density is proportional to the field, leading to the expectation that the field must fall off as llr in agreement with the result for the field around a wire. Of the two arguments, the one which is presented in terms of current flow is the simpler. It permits us to answer directly questions concerning more complex electrode configurations where it is not always easy to think of the charge configurations that must be built up. For this reason, in the experiments that follow we shall sometimes find it more convenient to talk in terms of current flow even though our principal interest may be in the potential for the related vacuum problem.

questions 1 Why does not current flow along equipotential lines ?

•

2

The concentric ring electrode arrangement on Teledeltos paper is analogous to two coaxial cylinders in vacuum in three-dimensional space. Can you think ofan arrangement which would be analogous to two concentric spheres in vacuum in three-dimensional space?

3 Suppose the Teledeltos paper were cut with a razor blade along part of one of the radial lines joining the two circles. Would the field configuration change? Explain.

radial fields

F-1

4 Suppose a third ring were painted with conducting paint on the Teledeltos

paper, concentric with the original two and between them in radius. Would the field configuration change? Explain. 5 From the answers for Questions 3 and 4, what can you conclude, in general, about the relation of cuts to field lines and the relation of conducting lines to equipotential lines ? 6 If the battery voltage in the experiment were doubled, how would the field

pattern change? The potentials? 7 Suppose the Teledeltos pattern were cut in half along a diameter. How would

the field configuration in the remaining half change?

"

7

experiment

F -2

image charges

introduction In this experiment we use electrodes on Teledeltos paper to study the field and potential patterns in the vicinity of two parallel line charges. Figure 3a shows two line charges with the same sign, and Fig. 3b shows two line charges with opposite signs. z

FIGURE 3

+\ +\

+ \

------'!'--'7fL-----'1c:---.::\------~\

/ /

\

-

x

/

' /2

+ + + + +

y

/

\

-._--- '[ \ // \ / ----------- "'.­

-~- ---

//

p

(a) z

+ + +

12.

2""'\

-\

+

0 - I ---~-~-~-=-\71 -----/~---y \

x

/

\ '0

1z

+ --g-\" / + -_fl \ // + +--- -----­ --- p/ + + +

///

(b) 9

fields

At points very close to either line charge, the equipotential surfaces are cylinders. Thus, we can simulate the situation shown in Fig. 3b by a pair of cylinders of radius a separated by a distance b and held at a potential differ­ ence 2 Yo. We assume that b is much larger than a. The potential due to the positively charged cylinder is then given by V - - V 1 -

0

~j~!Lll2

(13)

In (b/a)

and that of the negatively charged cylinder by V - V In (r 2 2 -

0

/'1

(14)

In (b/a)

The difference in signs of the two expressions reflects the fact that for the positive charge, V is negative whenever r > a, whereas for the negative charge, V is positive for r > a. For each, V = 0 at r = a. Thus the total potential is given by the sum of Eqs. (13) and (14): V = V

~(r~/!!2

(IS)

o In (b/a)

For points far away compared with the distance b between the line charges, this expression can be simplified further, as suggested by Fig. 4. From this figure, we obtain the approximate relations

r2

=

r+

tb sin e

(16)

Combining these expressions with Eq. (15) V

=inC%7a) [In (1 + !:!~~~) - In ( I ~~;!')]

(17) p

FIGURE 4

~ sin (J ----------~--------~~~--~--------~------------y

a

10

x

image charges

F-2

We now use the power series expansion In (I

+

1X2

IX) =

IX

+ "2 + ... +

(18)

noting that in this case IX is identified with (b sin e)/2r and by assumption is very small. We finally obtain

V=

Vob sine In (bla) r

(19)

We note that both the approximate expression Eq. (19) and the more general Eq. (15) predict that along the midplane between the charge lines, where = 0 and r 1 = r 2, the potential is zero. That is, this plane is an equipotential surface. This result can also be obtained from the fact that a test charge could be brought in from infinity along this plane without re­ quiring any work. The equivalent two-dimensional situation with Teledeltos paper is shown in F ig. 5a. The equipotential midplane is represented by a line, the perpen-

e

Teledel tos paper

FIG URE 5

1 2V,

II

TL-_-:---_-__·-~- , t-­

-- -- -- - -I i V o

~

I

TL-_-;-___ I

1/, In

'~- a "-- {I

(a)

(b)

dicular bisector of the line joining the two circular electrodes. Since this is an equipotential line, the situation would not change if it were painted with conducting paint. Furthermore, if o ne entire half were removed and the battery con nected to this line, as in Fig. 5b, the field and potential configura­ tions in the remaining half would not change! Thus, if the original problem had been to find the field and potential due to a conducting cylinder and a conducting plane, with their accompanying distributions of surface charge, we could have solved this problem by solving instead the problem with the two h ne-charge distributions. The negative charge line is in mirror-image position, relative to the positive line, and it is called an image charge . The concept of image charge is useful in simplifying a variety of problems in electrostatics ; the present situation is a simple example.

experime nt 1 circu lar el ect rodes The same experimental setup can be used as for Experiment F-l. Using the Teledeltos paper with two circular electrodes as in Fig. 5a, measure the 11

fields

potential along a radius directed at 45° to the line joining the electrodes. Plot Vas a function of l/r and compare your results with Eq. (19). Plot several equipotential lines in one half of the plane.

2 line-induced image Using the configuration of Fig. 5b, plot equipotential lines for the same values of potential used in step I, and compare results. Note that near the conducting line, the equipotentials are nearly parallel to this line. Why? Using the fact that at any intersection of a field line with an equipotential, the two have perpendicular tangents, sketch in a few field lines. The field lines always approach the conducting electrodes perpendicular to the lines. Why? 3 cut-induced image Considering the line joining the two centers, there is no current flow perpen­ dicular to this line. Hence, it should be possible to cut the entire setup in half along this line without changing the field pattern in the remaining half. Try it!

questions 1 Suppose in Fig. 3b the magnitudes of the two charge distributions (charge per.

unit length) are different. Is the midplane still an equipotential surface? Explain. 2 For the situation of Fig. 3a, with two charge distributions of the same sign

and magnitude, what is the equivalent two-dimensional situation with Teledeltos paper? 3 Suppose that each of the equipotential lines obtained in this experiment were painted with a thin line of conducting paint. Would the field pattern change? Explain. 4 Consider two perpendicular conducting planes, with a line-charge distribu­ tion parallel to their line of intersection. What is the equivalent two-dimen­ sional situation? Can you think of an equivalent image-charge situation? Hint: Three image-charge lines are required.

12

experiment

F -3

field lines and reciprocity

Introduction There are many experimental situations for which the electric field and potential cannot readily be obtained analytically and where it may be cumbersome to compute the field numerically. In cases in which the field is two-dimensional, it is often convenient to use conducting paper or an electrOlytic tank to obtain the field pattern. This is particularly useful in designing electron-beam devices with which one may want to investigate the effects of small changes in electrode configuration. In this experiment we shall study an electrode configuration used in the cathode-ray tube. This configuration is used for deflecting the electron beam and is a pair of flared parallel plates. First, we find the potential lines and then construct from these potential lines the field lines. Finally, we make use of a remarkable reciprocity relation to develop a new structure which permits us to construct the field lines directly. In a cathode-ray tube the electron beam is deflected transversely by passing the beam between a pair of parallel plates which are flared at the exit end as shown in Fig. 6a. An electron passing between the plates experiences a trans­ verse force F

(20)

-eEy

FIGURE 6

Real plates

Idealized plates

(a)

(b)

where - e is the charge of the electron and Ey is the vertical field. The total transverse impulse given to the electron may be written as (21)

where v is the velocity of the electron. Thus, the integral of the transverse field along the electron path is a measure of the deflection sensitivity. If the plates were very large compared to the distance a between them, so 13

fields

that edge effects could be neglected, the electric field between the plates would be uniform and given simply by

(22) where Vo is the potential difference between plates. In this case the total deflection would be proportional to I/a and also to the total distance b the electrons travel between plates. In the actual situation, the edge effects are not negligible, a nd it is useful to know how E varies along the electron path. From this we can numerically compute the quantity ('

=

J Ey dz

(23)

which is proportional to the electron deflection. For ideal parallel plates we would have

b Vo a

= -

exper iment 1

the zero potential line Using the experimental arrangement shown in Fig. 7, locate the zero potential line. Measure the potential as a function of z along a line 0.5 cm above the zero potential line. C ompute E as a function of z and numerically perform the integral indicated in Eq. (23) .

FIGURE 7

Cuts

I I

v

Probe

+

-=-7~V

Compare your result with that for ideal plates. What would be the reduc­ tion in deflection if the flared portion were omitted? Writing = Vo

(~) a err

what value do you find for (b/ a)err? Finally draw in the potential lines every half-volt. 14

(24)

field lines and reciprocity

F-3

2 the electric field The electric field, being force per unit charge, is a vector quantity, specified by its magnitude and direction. What is the direction of the field? It is clear that the component of the field along an equipotential line must be zero since no work is done in moving a charge along a line of constant potential energy. Then the only nonvanishing component of the field is that component normal (perpendicular) to the equipotential lines. Thus, the electric field is everywhere normal to the equipotential lines. The magnitude of the field is given by

E=

dV

(25)

where dVlds is the gradient of the potential along a direction normal to the equipotential lines. Where we have drawn only a discrete number of equi­ potential lines, we may compute the average field from the expression 8V &

(26)

where & is the perpendicular distance between two equipotential lines differing in potential by an amount 8 V. Starting with the top of the left-hand electrode in Fig. 7, draw a line emerg­ ing from the edge at 45°, and extend it so that at each crossing of an equi­ potential line the two are perpendicular. Similarly, extend the lines emerging from the edge at 30 and 60°. These are electric field lines. Similarly, draw field lines emerging from the bottom edges of the electrodes. Note that the equi­ potential lines and the field lines form a mutually orthogonal set of curved lines. There is an additional useful property of the field lines. To illustrate this property, compute the ratio of the separation between equipotential lines to the separation offield lines (8sIM) at a position near an electrode, as in Fig. 8a. Now compute the same ratio at a distance well removed from the elec­ trodes, as in Fig. 8b. Note that these two ratios are nearly equal. In fact, if we FIGURE 8

-l\~l ,

I

I

I .. I

\

~ I Lls I (a)

Lls

I I Lll I I

I

I .. I

(b) - - - - Field lines -------- Equipotential lines

were to draw more closely spaced equipotential lines and field lines, we would find that the ratio dsldl is an invariant. Now since the magnitude of the electric field is proportional to 1/&, it must also be proportional to 11M. That is, we may also represent the magnitude of the electric field by the density of field lines. We shall not pursue this connection further except to mention that the potential lines and field (or current) lines represent a kind of coordinate 15

I

fields

system. The transformation from the usual orthogonal coordinates into this system is called conformal mapping and is treated in the theory of functions of a complex variable. 3 reciprocity relations Is there any way in which we can plot the field lines directly? Note that the field lines (which are also current lines) are parallel to edges or cuts (because current cannot cross an edge or cut) and perpendicular to conducting lines. This situation is just the opposite of that for the potential lines. Thus, if everywhere we have an edge (or a field line), we replace it by a conducting surface and everywhere we have a conducting surface, we replace it by an edge, then our new equipotential lines would look something like the field lines for the original problem. Would the correspondence be exact? Although the proof is a bit involved, we can show that the correspondence is, in fact, exact. You may want to try this experimentally with the deflecting plates. Use Teledeltos paper as shown in Fig. 7, and with a razor blade cut lines on the right half of the paper which just correspond to the conducting lines on the left side. With conducting paint, paint in, at the top and bottom, the lines which have the same configuration as the field lines found on the left. Finally, connect the batteries between the two conducting lines and find the new potential lines, which are the field lines for the situation on the left.

questions 1 Why are the cathode-ray tube deflection plates flared instead of being simply parallel planes? 2 Can the reciprocity relation be extended to three-dimensional problems, oris

it valid only for two-dimensional situations? Explain. 3 Along a line joining the two flared edges of the deflection plates, how much does the field differ from the value Vola corresponding to large parallel planes separated by a constant distance a? 4 With respect to the reciprocity relation, what arrangement corresponds to the situation with two conducting circles, studied in Experiment F-2?

16

experiment

F-4

the magnetic field

introduction In this experiment we examine the magnetic field produced by a current in a solenoid. This study will aid in understanding the general relations between currents and magnetic fields. In addition, later experiments, particularly the Electrons and Fields unit, are concerned with the motion of electrons in magnetic fields produced by solenoids. This preliminary study will help establish the nature of the magnetic fields used in those experiments. In measuring magnetic fields, we shall be concerned with two questions: the variation offield along the axis of a solenoid, as in Fig. 9 and the variation FIGURE 9

p B

transverse to the axis in a region between two solenoids. These two cases correspond to two of the Electrons and Fields experiments, in which an electron beam is directed either along the solenoid axis or transverse to the axis, respectively. In later experiments we shall produce a constant magnetic field by imposing a steady (de) voltage across the solenoids. In this experiment it is more convenient to use an ac voltage source, resulting in an alternating magnetic field. This arrangement has the advantage that we can detect the alternating field easily by placing a small coil, called a search coil, in the field. According to Faraday's law of induction, a voltage is induced in the coil which is directly proportional to the rate of change of flux through the coil, and hence to the amplitude of the time-varying field. Thus, a measurement of this induced voltage provides a relative measure of field strengths at various points. To obtain the proportionality factor between field and voltage, we need to make one absolute field determination to calibrate the voltmeter. This can be done 17

fields

in various ways. The simplest is to make use of the expression for the field along the axis of a circular coil with N turns and radius a. The expression is B

2

= lioNI a

(27)

2

where r is the distance from the coil to the point where the field is measured, as shown in Fig. 9. By measuring the current in the coil, we can compute the maximum B and thus calibrate the induced voltage in the search coil with a known field. There are several ways to measure steady magnetic fields. A coil can be flipped over (causing a changing flux which induces a voltage pulse), or rotated (causing an ac voltage). Other methods rely on magneto resistance (the change of resistance of a material in a magnetic field), or the Hall effect (a voltage transverse to the direction of current flow, induced by a magnetic field).

experiment 1 field measurements Connect one of the solenoids to the sine-wave generator and connect the search coil to the oscilloscope as shown in Fig. 10. A 25-mH choke will FIGURE 10 x--~

Search coil Solenoid

18

serve nicely as the search coil. The coil should be mounted so that it can be moved easily in the vicinity of the solenoid. Center the pickup coil in the solenoid, set the sine-wave generator to maximum output at a frequency of about 2 kHz. You should obtain about a I-V (peak-to-peak) deflection of the oscilloscope. Rotate the pickup coil through 90° about a vertical axis and notice that the signal disappears. With the axis of the pickup coil perpendicular to the magnetic field, no magnetic flux links the pickup coil and no voltage is induced. With the axis parallel to the magnetic field, there is maximum flux linkage and maximum induced signal. Thus from the orientation for maximum (or zero) induced signal, we may determine the direction of the magnetic field. Place a sheet of graph paper under one-quarter of the solenoid as shown in Fig. 11. Determine the direction of the magnetic field at various positions on

the magnetic field

: field Ion is

F-4

FIGURE 11

(27) I----;-~-:-:--:-:---:-:-:-:--;-:-----,--- -

-- - ­

iu red, te the ~ith a an be e),

r

,tance effect gnetic

Graph paper

the paper. Mark a short line segment on the paper at each position to indicate the direction of the field. (A magnetic flux line is a line drawn in such a V, :.lY that it is everywhere parallel to the magnetic field.) In addition to representing the direction of the magnetic field, the number oflines per unit area may repre­ sent magnetic field intensity injust the same way that the electric field intensity may be represented by tbe density of electric lines.

e nt 2

field along axis Using a meterstick as a guide for the pickup coil and to measure di tance. obtain the magnetic field profile (or relative field strength) along the axi ', Record the pickup voltage every t em to a distance of about 20 em from the center of the coil. (Y ou may find it convenient to detennine the pi ckup voltage with an ac voltmeter rather than with the oscilloscope.) If you a re unable to locate the center of the solenoid accurately, you may measure the field a few centimeters beyond the center and locate the center from the field measuremen ts. Plot the magnetic field intensity as a function of position on a sheet of centimeter graph paper. Note that the magnetic field intensity has an inflec­ tion point (that is, a position at which the second derivative of the intensity with respect to position vanishes). Where is this inflection point with respect to the end of the solenoid? What is the ratio of the magnetic field at the end of the solenoid to its peak value? What is this ratio for a long solenoid?

3

field of two solenoids Using your data for the field along the axis of a single solenoid , how might you determine the field profile for a pair of solenoids, coaxial but separat d by a distance 2d as shown in Fig. 12? You may wish to make the computation and then measure the field directly for a pair of solenoids. Connect the two solenoids in parallel as shown so that the current through each solenoid will be approximately the same as for the single solenoid.

4

opposed f ields Can you predict the field profile when the fields produced by the solenoid are in opposition? You may wish to try this configuration by reversing the electrical connections to one of the solenoids.

5

dependence on distance from axis In order to compute the deflection of a transverse electron beam, it is necessary to know the magnetic field along a line perpendicular to the longi­

ct the e wil l

an be

or to lhout J coil With ~ne t ic

lei to lueed ii, we

wn In ns on

19

fields

FIGURE 12

x

tudinal axis of the solenoids, as shown by the y axis in Fig. 12. Place the solenoids on a common axis, separated by a distance of about 4 in. Place a meterstick on the table along the y axis and mount the pickup coil on the meterstick oriented so as to pick up the maximum signal. Record the pickup voltage every t em out to 10 em or more. Plot the field profile on a sheet of centimeter graph paper. How does the transverse field profile compare with the longitudinal field profile? 6

dependence on solenoid positions You may wish to repeat this series of measurements with the solenoids further separated or brought closer together. What differences do you note?

7

absolute field measurements In the above measurements we have been able to determine only the relative magnitudes of magnetic field, but not the actual value of magnetic field in Teslas at a given point. One way of determining the value of the magnetic field is to calibrate the pickup coil with the known ac field of a circular coil, as discussed in the Introduction, using Eq. (27). You should find it possible to detect the field at the center ofa 10-turn coil at a frequency of 2 kHz and in this way calibrate your pickup coil. The current will be essentially the short circuit current of the signal generator and may be determined either directly with an ac milliammeter or from a determination of the open circuit voltage and the output impedance as in Experiment EI-2.

8

field of a thick solenoid An alternate way of making an absolute determination of the field is to sum

FIGURE 13

Y

B

20

0

the magnetic field

F-4

Eq. (27) over all current loops in the solenoid for some convenient position of the pickup coil. With reference to Fig. 13, the incremental contribution AB to the field from the turns included in the cross section Ax Ay will be given by B = P-o NI Ax Ay 2 2(b - a)e

(28)

We may formally perform the integration to obtain B

P- 0 NI 4(b - a)e

I" y

r

Xl

2

dy

a

~.-~~~

(29)

",Xl

The integration over x may be performed first to obtain B

P-oNI 4(b - a)e

r" [(x/ +

X2

Ja

Finally, by making use of the substitution y = x tan

f

dy

y2)1/2 ­

(J

(30)

and using the result

!

(J d(J = 1 I + sin (J (31) sec 2 n I _ sin we may obtain a solution in closed form for B at any position on the axis. In order to simplify the discussion, we limit ourselves to the center of the solenoid for which Xl = e and X2 = + e. Integrating Eq. (31), we obtain

B=

4(b - a)

{[l + (e/b)2]1/2 + I} {[l + (e/a)2]1/2 - I} {[l + (e/b)2]1/2 - I} {[I + (e/a)2]1/2 + I}

~

0~

In the limit that e goes to infinity, Eq. (32) should approach the value for an infinitely long solenoid: P-oNI 2e

B=--

(33)

You may wish to check that Eq. (32) does, in fact, have the proper limits. F or a standard solenoid with N = 3400 turns and the dimensions a

Ii in.

b =

2t in.

(34)

we obtain B I

0.031

T/A

(35)

The centimeter-gram-second (cgs) unit of magnetic induction is the gauss. Using the conversion factor 1 T 1 Wb/m 2 = 104 gauss (l tesla equals 4 I weber per square meter equals 10 gauss), we may alternatively express a typical value of the induction as 310 G/A (gauss per ampere). Measure the dimensions of your solenoid and substitute into Eq. (32) to determine the induction at the center in teslas per ampere. (Be sure to express the distance b - a in meters.)

questions 1 If the frequency of the sine-wave generator is doubled without changing its amplitude, how does the search coil voltage change at a given point? Is it important to use the same frequency for all the measurements? 21

fields

2 At a frequency of 2000 Hz, is the impedance of the solenoid principally resistive or reactive? That is, is the coil winding resistance much larger than the inductive reactance, or vice versa? 3 How is the precision of your field measurements affected by the fact that the search coil has a finite size? Does the field vary appreciably over the dimen­ sions of the search coil? 4 Consider two coaxial thin coils, each with N turns and radius a, and separated

by a distance b. Derive an expression for the total field at points along the axis. Show that if b is greater than a certain critical value, the field has two maxima along the axis, but ifless, there is only one maximum, at the midpoint between the coils. Find the critical value of b. 5 For a very long, thin solenoid, show that the field on the axis at one end is exactly half the field at the center of the solenoid. 6 Discuss the relative advantages of the oscilloscope and the vacuum-tube voltmeter for measuring the induced voltage in the search coil.

22

experiment

F -5

magnetic coupling

introduction In this experiment we shall examine some remarkable properties of the mag­ netic coupling between a pair of coils. In Experiment F4 we passed an ac current through a large solenoid and measured the voltage induced in a small pickup coil as shown in Fig. 14a. What might we expect if instead of sending FIGURE 14

-

] Pick-up coil

Solenoid

--

Oscilloscope

-­

(a)

J

--

(b)

the current through the solenoid, we sent it through the pickup coil and measured the voltage induced in the solenoid as shown in Fig. 14b? We shall find that the field pattern generated with Fig. 14b is identical to that generated with Fig. 14a, and the re-lative signal intensities are also identical. Before proceeding to examine this problem experimentally we shall present a brief discussion of the reciprocity relation implied by Fig. 14. We begin with a discussion of the energy of interaction between currents. To simplify matters we restrict our discussion to two current-carrying coils 23

J

I'

fields

as illustrated in Fig. 15. We designate by <))ij the magnetic flux linking the ith circuit and produced by the jth circuit. Thus, the flux linking circuit I is the sum of <))11 (the flux produced by the current through circuit I) and <1>12 (the flux produced by circuit 2). Similarly, the flux through circuit 2 is the sum of <)) 22 and <))21' We define the self-inductance of circuits I and 2 as the number of flux linkages per ampere of current: (36)

where only the flux produced by the circuit itself is considered. Similarly we define the mutual inductances as follows: M 12

_ N 1<))12 -

(37)

12

In computing the magnetic energy, we have the self-energy terms which may be written as (38) In addition, we have an interaction energy which may be written either in terms of the current through circuit I times the flux produced by circuit 2 or the current through circuit 2 times the flux produced by circuit I (39)

From the second equality of Eq. (39), we may demonstrate by dividing both sides by 1112 the equality (40) Thus, if we are concerned with the interaction between a pair of circuits, we need consider only a single mutual inductance. Next, we show that the mutual inductance between a pair of coils can be expressed in terms of the geometric mean of the self-inductance times the geometric mean of the two flux coupling coefficients. Taking the product of the two expressions in Eq. (36), we obtain N 1N 2<))11<))22 =~-~~~~

L1 L

1112

2

(41)

Similarly, from Eq. (37),

M

2.

N N

<))12$21

1 2 = M 12M21 = -----T)-;~~-

(42)

Comparing Eqs. (41) and (42), we obtain M =

k(L1L2)1/2

(43)

where k

= (<))12~2Jc)1!2 <))22<))11

With reference to the experimental situation shown in Fig. 14, we designate the solenoid to be Ll and the pickup coil to be L 2 • In the situation shown in Fig. I4a, the current flowing through L2 is negligible so that <))12 and <))22 are' zero. The voltage induced in L2 is given by V21 24

=

-N

d<))21

2dt

(44)

magnetic coupling

F-5

FIGURE 15

In an appropriate frequency range the voltage across Ll is predominantly the back emf induced in the coil, so that we may write N tiW l l 1 dt

(45)

Since the fluxes are changing at the same frequency, we may write

V ll

N 2 <1>21 Nl 11

(46)

Now if we reverse the situation as shown in Fig. 14b, we have

V 12

tiW

12 = -N 1 - -

dt

V22 = -N2 V 12 V 22

dt

N 1 <1>12 N 2<1>22

-=--

(47) (48) (49)

Multiplying Eqs. (46) and (49), we see that Nl and N2 cancel and we may express the coupling coefficient in terms of the induced voltages k

V 12 • V21)1/2 ( VZ2 V l l

(50)

experiment 1 field observations Assemble the circuit shown in Fig. 14b with the sine-wave generator set to 2 kHz and maximum output. Place the field plot produced in Experiment F-4 and shown in Fig. 11 under the solenoid and move the pickup coil (now the signal-producing coil) in the vicinity of the solenoid. Observe that the orientation which produces maximum signal in the solenoid is exactly the same as that which produced maximum signal in the pickup coil in the previous experiment. This comparison suggests, qualitatively at least, the equivalence of M12 and M 21 • 25

fields

2 voltage ratio U sing the pickup coil at the center of the solenoid and the hookup shown in Fig. 14a, measure with the oscilloscope or with an ac-voltmeter the voltages VII and V 2I . Determine the ratio of V 2t1V 11 over a wide range offrequencies. You should find that the ratio is constant from approximately 200 Hz up to 20 kHz. Below 200 Hz the dc-resistance of the solenoid limits the current flow so that V2I drops with decreasing frequency. Above 20 kHz the capaci­ tance across the pickup coil produces a resonance (to be studied in Experi­ ment EC-4) which results in an increased signal. However, if we stay well within these limits, the assumptions of the previous section are well realized.

3 coupling coefficient Returning to a frequency of 2 kHz, change the experimental arrangement to that shown in Fig. 14b and determine V 22 and V 12 and their ratio. Finally, compute k using the expression given in Eq. (50). 4 dependence on position U sing the experimental arrangement of Fig. 14b, measure the voltage induced in the solenoid for the pickup coil along the axis of the solenoid. Compare your results with those obtained in Experiment F-4.

questions 1 Discuss qualitatively how the coefficient k defined by Eq. (43) is determined

by the geometry of the situation. For example, if the two coils are wound compactly on the same core so that all the flux which links one coil also links the other, what value does k have? 2 Voltage drops due to coil winding resistance are not included in the discussion of mutual inductance. Discuss some possible effects which coil resistance might have on your measurements. 3 Could a vacuum-tube voltmeter be used instead ofthe oscilloscope to measure induced voltages? What advantages and disadvantages would it have? 4 Can the value of k in this experiment be predicted or at least estimated by

geometric considerations?

26

erk Ie

hy ics laboratory, 2

I n m . p r is, un/va slty of californ B. berkeley hugh d . oung, carn eglC m /I unIversity ace lerat,on and deflectio n

f eJe

tro

EF·'

focu ing a d In ten Ity con r J E-2 agnetic deflect· n

f electrons

EF-3

ti n O f electrons

EF-4

and the nlagnetron condition

EF·5

helic al m vacuum diode

me raw-hili n w york

st

I OLIIS

s · n fran is rI(

a y

kc

b rn

0

Usseldorf nr.r.ln.n.

electrons and fields Copyright 1971 by McGraw-HilI, Inc.

All rights reserved. Printed in the United States of America.

No part of this publication may be reproduced, stored in a

retrieval system, or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher.

Library o/Congress Catalog Card Number 79-125108 07-{)50485-7

1234567890 BABA 79876543210

The first edition of the Berkeley Physics Laboratory

copyright 1963, 1964, 1965 by Education Development

Center was supported by a grant from the National Science

Foundation to EDe. This material is available to publishers

and authors on a royalty-free basis by applying to the

Education Development Center.

This book was set in Times New Roman, printed on

permanent paper, and bound by George Banta Com­

pany, Inc. The drawings were done by Felix Cooper;

the designer was Elliot Epstein. The editors were Bradford

Bayne and Joan A. DeMattia. Sally Ellyson

supervised production.

electrons and fields

INTRODUCTION

summary

In this series of experiments we shall study the motion of charged particles (electrons) in electric and magnetic fields. In these experiments the electrons behave as classical particles whose behavior is governed by Newton's laws of motion. That is, the speeds are always small compared to the speed of light (3.00 x 10 8 m/sec) so that no relativistic corrections are needed, and the dimensions ofthe experiment are large compared with atomic dimensions, so quantum effects need not be considered.

The subject areas of the experiments can be summarized as follows:

EF-1

Acceleration of electrons by an electric field, and deflection of a beam of electrons by a uniform transverse electric field.

EF-2

Focusing of a beam of electrons by a nonuniform electric field, and controlling the intensity of the beam.

EF-3

Deflection of a beam of electrons by a transverse magnetic field.

EF-4

Helical motion of electrons under the action of an axial magnetic field.

EF-5

Motion of electrons in a magnetron under axial magnetic and radial electric fields.

The central instrument in these experiments is an electron beam tube whose operation is quite similar in principle to that of a television picture tube. The tube is commonly called a cathode-ray tube (abbreviated CRT). This name originated in the midnineteenth century in the course of investigations of the conduction ofelectricity through gases at low pressures. In such experiments, excitation of atomic energy levels by electron bombardment causes the emission from the region near the cathode of bluish rays which were originally called "cathode rays." In addition to providing a convenient experimental arrangement for the study of electron motion, the cathode-ray tube is also the most important component ofthe cathode-ray oscilloscope, an invaluable instrument in many areas ofexperimental physical and biological science. Thus studying electron motion in the cathode-ray tube contributes to the understanding of the oscilloscope, which will also be used in many later experiments in this and other courses. The motion of electrons in an electric field B closely analogous to the motion of objects in a gravitational field; this analogy is illustrated by Figs. 1 and 2; Fig. 1 shows two examples of motion of a mass under the earth's gravity, and Fig. 2 shows the analogous cases of electron motion. A cathode-ray tube contains (1) an electron gun, which emits electrons, accelerates them to a definite speed, and focuses them into a beam; (2) a deflection system consisting of two pairs of plates; (3) a fluorescent screen to indicate the point of impact of the electron beam on the end of the tube. All

II

1:1

IiI 1',

ii

i 'I:

electrons and fields

FIGURE 1

(8)

g = 9.8 m/sec 2

A mass m near the surface of the earth experiences a gravitational force Fy = mg. (c)

After the mass has fallen for a time t, it has acquired a momentum mvy = Fyt and a kinetic energy !mv/ equal to the work mgh done by the gravitational field.

If the mass m is released from rest, it accelerates downward at a uniform rate. (d)

,

If the mass is given a horizontal initial velocity Vo,

(e)

the gravitational force deflects it downward. After a time t the mass acquires a transverse momentum mvy = Fyt, 2

and the direction of its velocity is changed by an angle ().

introduction to electrons and fields

FIGURE 2

An electron of charge e between a pair of charged grids experiences a force Fy = -eEy = eV/h.

If the bottom grid is at a posItlve potential, the electron is accelerated downward at a uniform rate.

t

J

After the electron has accelerated for a time t, it has acquired a momentum mvy = F,t and a kinetic energy tmv/ equal to the work eEyh = e V done by the field.

If the electron is given a horizontal initial velocity Vo,

the electric-field force deflects it downward. After a time I the electron acquires a transverse momentum mvy = Fyi

and the direction of its velocity is changed by an angle (J.

3

electrons and fields

these are enclosed by a glass envelope from which most ofthe aids evacuated; this is necessary to avoid scattering of the beam by collisions of electrons with air molecules. The arrangement of components is shown in Fig. 3. In Top view

FIGURE 3

Horizontal deflection plates

Cathode

First anode Second anode

\

Fluorescent screen

Electron gun

order for it to be unlikely that an electron collides with even one gas molecule during its travel of the length of the tube, the gas pressure in the tube must be no greater than about 10- 6 atm; ordinary mechanical pumps cannot attain this high a vacuum, and diffusion pumps must be used. The electron gun is shown in more detail in Fig. 4. The source of electrons is the cathode, labeled K in the figure. A thin cylinder is heated to about FIGURE 4

]200 K by passing a current through a twisted heater wire inside the cylinder, electrically insulated from it by a ceramic sleeve. The end of the cylinder forming the cathode itself is coated with barium and strontium oxides; when these materials are heated, some of the electrons in them acquire enough energy to break away from the surface and thus become free to move in the vacuum surrounding the cathode. This process is called thermionic emission. Coaxial with the cathode are four cylindrical electrodes containing baffles with circular apertures, as shown in cross section in Fig. 4. Electrode G1, called the control grid, is operated at a potential of about 5 to 20 V negative with respect to the cathode; the resulting electric field tends to push electrons back toward the cathode. Hence by varying this potential it is possible to 4

introduction to electrons and fields

limit the number ofelectrons emerging from the aperture of G l' thus control­ ling the intensity of the beam. Electrode G2 is connected internally to A 2 , and both are operated at a potential V2 of several hundred or even several thousand volts positive with respect to K. The resulting field accelerates electrons along the axis of the electrodes. Electrode A 1 is held at a potential V l (with respect to K) intermediate between those of K and G 2 • The resulting fields between G2 and A 1 and between A 1 and A 2 serve to focus the beam, so that electrons emerging from G 1 in various directions are brought together in a small parallel beam whose diameter is determined principally by the diameter of the aperture of Gl ' Proper focusing depends chiefly on choosing Vi and V 2 so that the ratio V 1/V2 has a certain critical value. The electron beam passes through two pairs of deflection plates as shown in Fig. 3. Applying a potential difference between the two plates ofeither pair produces a transverse electric field which deflects the beam sideways; this deflecting action is studied in detail in Experiment EF-l. Finally, the electron beam strikes the end of the tube, which is coated with a phosphor which glows visibly when struck by the electron beam. This effect results from collisions of electrons with the atoms of the phosphor, raising some atoms to energy levels higher than their usual states. When the atoms return to their normal energy levels or "ground states," they emit energy in the form of visible light. The inner surface of the glass tube is covered with a conducting coating of graphite (Aquadag), which has several functions. It is connected electrically to the second anode A 2 , and thus serves as an extension of it; it also helps shield the electron beam from stray electric fields which may be present. It collects secondary electrons emitted from the phosphor by electron bombard­ ment, and it also prevents stray light from shining on the inside surface of the phosphor and reducing the contrast of the screen image.

CAUTION

Because of the high vacuum and the large flat surface area of the screen face, the tube is dangerous to handle. Any weakening of the glass envelope which might result from a mechanical shock or a scratch may cause a severe explosion, resulting in widespread scattering of glass and screen material. The tube used in the laboratory has a protective cover, and it should be handled only with this cover in place. Never handle the tube while wearing a diamond ring, which could easily scratch the glass and make it susceptible to cracking.

5

experiment

EF-1

acceleration and deflection of electrons

introduction In this experiment you will observe the acceleration and deflection of e1ectrons by electric fields. To describe this motion we use a rectangular coordinate system in which the z axis is along the tube axis, the x axis is horizontal in the plane of the tube face, and the y axis vertical in the plane of the tube face. An electron emitted from the cathode and passing through the various apertures of the electron gun emerges from anode A z with a velocity Vz in the z direction; the magnitude of Vz is determined by the potential differences V 2 = VB + Vc between K and A z. In traveling from K to A z the electron loses potential energy e V 2; thus if it leaves the cathode with negligible initial kinetic energy, its kinetic energy t mvz z after emerging from A z is determined by the relation (1)

The electron now passes through the spaces between the deflection plates. If there are no potential differences between the plates, it passes straight through and (assuming the electron gun is properly aligned) strikes the center of the screen, making a small bright spot. But now suppose a potential difference Vd is applied between the vertical deflection plates (the pair with planes horizontal). Then there will be a transverse field Ey between the plates; the resulting force on the electron gives it a transverse velocity Vy without changing the axial component V z ' and it emerges from the deflection plates in a direction at an angle () with the axis, determined by the relation tan () =

~

(2)

as shown in Fig. 5. FIGURE 5

+ +++++ ++

---------_L

VY~

--------------------

-----To

~vz~.~t~=f~----~--------L----------------

1 1

---L--- ------­

Screen 7

.

I!

electrons and fields

If the deflecting potential and the plate dimensions are known, all these quantities can be calculated, as we now show. First, a potential difference Va between two plates separated by a distance d, as in Fig. 5, produces a transverse electric field Ey = Vld, and a transverse force whose magnitude is F, = eE, = eVa/d. During the time At the electron takes to pass between the plates, this force gives the electron a transverse momentum mvy equal to the impulse of this force.

=F

mvy

y

8t

8t

= eVa -d

(3)

That is, e Va vy =--8t m d

(4)

But the time interval8t is also the time the electron takes to travel along the z-axis a distance I equal to the length of the plates, at axial velocity Vz• Thus I = Vz 8t. This relation can be solved for At and the result inserted in the impulse-momentum relation, Eq. (4). The result is V=

,

e Va I -­ m d Vz

(5)

Finally, the deflection angle (J is given by tan

(J =

~

v.

=

e Vi

dmv/

(6)

Substituting the energy relation of Eq. (1), we finally obtain tan (J =

V:V 2dI

(7)

This equation shows that the deflection increases with deflecting potential Va' as might be expected, and also increases with the length I of the plates. With longer plates the deflecting field acts for a longer time and causes greater deflection. The deflection is inversely proportional to d: The more closely spaced the plates, the greater the deflecting field for a given total potential difference. Finally, reducing the accelerating potential V 2 = VB + Vc increases the deflection by reducing the axial velocities of the electrons, per­ mitting the deflecting field to act for a longer time. In addition, with smaller axial velocity we obtain a greater angular deflection for the same transverse velocity. After the electron beam leaves the deflecting region, it again travels in a straight line, tangent to the path at the point where it left the deflecting region. Thus, the bright spot on the screen is deflected vertically a distance D determined by the relation D = L tan (J, where L is the distance from the plates to the screen. (We neglect the slight curvature of the screen.) A more detailed analysis of the motion between the plates shows that L should be measured from the center of the plate to the screen. Thus, we have (8) 8

acceleration and deflection of electrons

EF- 1

experiment 1

electrical connections The electrical connections between the power supply and the CRT are shown in Fig. 6. The leads from the tube are color-coded to facilitate proper con-

FIGURE 6

Black

".f

;

;::

c.;l

:;0

~

~

e

:;0

J

14

I

I I

2

• •

3BPl 3

10

+ 11111---- - - I 4~V VI =Vc V2 =VC +VB R=47k!l

nections. The heater is energized by the 6.3-V ac supply at the terminals H-H of the power supply. The cathode should be connected to the negative side (C-) of the C supply. The positive side (C+) of the C supply is connected to the focusing anode, thus providing the potential difference Vi = Vc. The focusing anode is also connected to the negative side (B - ) of the B supply, the other side of which (B + ) is connected to the accelerating grid and last anode. Thus, the total accelerating potential V z between K and A z is V z = Vc + VB' as shown also in Fig. 4. The control-grid (G 1) voltage is supplied by a small biasing battery con­ nected to make the grid negative with respect to the cathode, as shown. For the accelerating potentials used in this experiment, a bias of 4! V should be sufficient, and it may be necessary to reduce the bias at low accelerating potentials, in order to obtain a bright enough spot on the screen. 9

electrons and fields

CAUTION

Never operate G 1 with zero bias voltage; an excessively bright spot causes excessive heating of the screen by electron bombardment, leading to local destruction of the phosphor. As a precaution, a wire should be connected between B + and terminal G on the power supply, which is connected to the metal case of the instrument. In turn, G should be connected to the common building ground on the bench. This procedure ensures that the poten­ tials at the deflection plates are close to ground potential and thus avoids a shock hazard. With this setup a negative potential of several hundred volts, with respect to ground, appears at the biasing battery; contact with any terminal of this battery or the associated leads could cause a severe and possibly lethal shock.

The deflection plate connections are made through polyethylene-insulated twin line (similar to TV antenna lead-in wire). Vertical and horizontal leads may be identified by checking terminals on the tube base. The resistors labeled R in Fig. 6 are mounted permanently on the connectors for the deflection-plate leads. A variable voltage for the deflection plates may be obtained conveniently by means of a center-tapped 45-V battery and a potentiometer, as shown in Fig. 7. Be sure you understand how this circuit

works and the limits between which this voltage can be varied. The voltage is measured with a voltmeter connected as in Fig. 7. After checking all your connections, the power supply may be turned on to the STANDBY position; this energizes the low-voltage heater supply but not the high voltage. Notice the red glow from the heater wires behind the cathode. After a minute you may turn the power supply switch to the ON position. With the B supply control near the top of its range, adjust the C supply for a small, well·focused spot. You are now ready to observe deflection of the beam. 2 measurement of deflection Keeping the accelerating and focusing voltages constant, measure the deflection D as a function of deflecting voltage Vd • Measure or obtain from manufacturer's data the length L from center of deflecting plates to screen. Compute tan f) for each value of Vd , and plot a graph of tan f) as a function of Vd• What shape should the curve have? Why? Be sure to record the values of VB and Vc. 10

acceleration and deflection of electrons

EF- 7

3

alternate beam voltages Change the value of VB' readjust Vc for optimum focus, and again observe tan () as a function of Vd and plot. Can you predict how this graph will differ from the first? Repeat for at least two more values of VB'

4

graphical analysis Multiply each value of tan () in each series of observations by the correspond­ ing value of V 2 Vc + VB and plot the product V 2 tan () as a function of Vd for all the observations on the same sheet. Using Eq. (7), can you predict what the result should be?

5 determination of lId From the result of the above graphs, can you determine the value of the ratio lid? How does the value compare with the value obtained by direct measure­ ment of the plates? Because of the "fringing fields" which extend beyond the geometrical limits of the plates, their effective length is substantially greater than their actual length. Your results may be compared with the results of Experiment F-3 .

d

.s 'S

.e ·e a it

questions 1

What is the voltage sensitivity for vertical deflection; i.e., the deflecting voltage needed for unit deflection distance? How does it vary with accelerating voltage?

2 Would you expect the vertical and horizontal voltage sensitivities to be the

same or different? Why? If you have time, you may want to check this prediction. 3 What is the purpose of the resistors labeled R in Fig. 6? 4 What is the speed of the electrons in the beam, when the accelerating voltage is V 2 = 500 V? How much time is required for an electron to travel from

cathode to screen?

ge 5

)n ot

Ie. ,no ra

he

What would happen if an ac voltage were applied to the deflection plates?

6 Why are the deflection plates flared instead of completely flat? 7

Is the force of gravity on the electrons of any importance '!

B In the discussion of the electron motion we have not considered interactions

between electrons in the beam. How can this omission be justified?

.he )rn ~n.

on

les 11

experiment

EF-2

focusing and intensity control

introduction The first experiment in this series was concerned with deflection of an electron beam by a uniform electric field. In this experiment you will investi­ gate the way in which the electron beam is controlled and focused by the various electrodes which make up the electron gun of the cathode-ray tube. The action ofthe focusing electrodes is quite analogous to that of a converg­ ing lens in focusing a beam of light, and this analogy will be discussed in detail below. In principle, an electron beam can be produced by the system shown in Fig. 8; by making the apertures sufficiently small we can make the beam FIGURES

Beam

Apertures

arbitrarily narrow. The practical difficulty is that electrons leave the heated cathode in various directions; only a very small fraction will emerge in exactly the correct direction to pass through the apertures in the anodes. Most of the electrons will strike the anodes rather than the screen; as a result, the spot on the screen may be too dim to be seen. Fortunately, it is possible, using a properly shaped electric field, to redirect electrons whose initial velocities are not along the axis, thus producing a much more intense beam and a brighter spot. The situation is analogous to the condenser lens and mirror system of a slide projector, shown in Fig. 9. This system "gathers" light emerging from the projection bulb in various direc­ tions and concentrates it to pass through the slide and the projection lens to the screen. If the condenser lenses are removed, the screen image is still seen, but with greatly reduced intensity. Figure 10 shows the various electrodes in cross section. The accelerating and focusing fields exist chiefly in the regions between electrodes; in the interior regions of Al and A z there is practically no field, since these regions are almost completely surrounded by equipotential surfaces. To understand 13

electrons and fields

FIGURE 9

Slide

qualitatively the focusing action of the fields at the two ends of AI' we con· sider the region between A1 and A 2 • This region is shown in enlarged cross section in Fig. 11, which also shows cross sections of several equipotential surfaces and several electric·field lines of force. An electron emerging from A 1 with a component of transverse velocity Vr away from the axis moves into a region where the transverse component of field pushes it toward the axis. At the same time the axial component E z of E accelerates the electron; as it moves toward A2 and into the region where the transverse component Er of E tends to push the electron away from the axis. Because it is moving faster in this region, the outward impulse is smaller than the previous inward impulse, and the electron is given a net deflection toward the axis. It is interesting to note that if the polarity of the potential difference is reversed, the focusing action is still present. In that case, the beam is first driven outward, then decelerated, then driven inward as it approaches A 2• The electrons now spend more time in the second region than the first, and so the net effect is again to drive the electrons inward. The basic geometry ofthe focusing action is shown in Fig. 12, which shows a simplified version of the entire electron trajectory, assuming the path between focusing electrodes is small compared to the total path length. Suppose electrons emerge from the first accelerating electrode G2 with nearly the same axial component of velocity Vz but with different radial components Vr' They arrive at the focusing region with various distances r from the axis, proportional for each case to the respective values of Vr' The function of the focusing region is to change the various vr's just enough so the electrons reconverge on the screen, and so the final values of Vr must also be propor­ tional to r. Thus, the essential condition for focusing is that for each electron the change in radial velocity L\vr must be proportional to r. The situation is analogous to the focusing of diverging rays of light by a converging lens. In Screen

FIGURE 10

Electrostatic lens 14

focusing and intensity control

EF-2

FIGURE 11

- - - Field lines -- -- Equipotential lines

practice, the length of the focusing region is not negligible compared to the total path length, and so the analogous optical situation is that of a thick lens. This analogy is explored in more detail in Figs. 13 and 14. As we shall see, there is a lens equation for electrostatic focusing which closely resembles the corresponding optical relation, and there is even a "refractive index" for the electrostatic lens. To analyze the electron motion in more detail, we represent the electric field E in the region between At and A2 in terms of an axial component E z FIGURE 12

__- - - - - - - -_________~~________~z Focusing electrodes

~~------------------------~---

Cathode

Screen 15

lectrons and fields

6

FIGURE 13

( a)

(II )

s

p

sin 8

1 - .- = n Sin 8 2

A Light rayon entering a more opticaJly dense medium i refracted toward the surface normaJ . (c)

The angles of incid ence and re­ fraction are related by Snell's law.

(d)

. () _

sm

l-

r

Xi

+"R

For paraxiaJ rays we can make the above approximations.

Substitu ting into Snell's law, we obtain the lens formula fo r a single refracti ng surface.

(f) /I

II

s

If" is Ie

than one, or .. ,

If R is negative, the sUlf ace is divergent.

focusing and intensity control

EF-2

FIGURE 14 (a)

(I>

An electron on entering a region of higher electrical potential is re­ fracted toward the normal to the equipoten tial surface. (c)

The angles of incidence and re­ fraction are related by the continuity of the transverse velocity.

(d)

r ]"

,\SSS'

\' I

\

t

]

\ ,

ss

r._ ( I'--J)" ( ,,+I) -XlI + - - -!!taR --Xl

Two region ' of COI/Sfant potential are separated by a complex inter­ mediate region.

The len formula thu take a more complex form.

(f)

;!,

-L+ ~ = (n. -l )(" +l ) . 1'2

If 11 is les than one the is still con vergent.

urface

• nil

Thus an electrostatic lens in charge­ free pace i always convergent. 17

electrons and fields

and a radial component E,. as in Fig. 11. We assume that E z is nearly uniform except in the end regions, and that E, is substantially different from zero only in the end regions. We shall find that these two components are not independ­ ent; if Ez is uniform, E, must be proportional to the distance r from the axis, and this relation is essential for the focusing action. We represent the electron velocity v in terms of its axial and radial components, V z and v" respectively. Let Vi be the potential of Ai and V 2 the potential of A 2 , relative to the cathode. Then, assuming the electrons leave the cathode with no initial velocity, their speeds inside Al and A2 are given by the respective energy relations and

(9)

An electron thus enters the region between A I and A 2 with axial velocity VI'

Suppose it is initially a distance r from the axis. The time I1t taken to traverse

. the end of length I is given by I1t = Ilv1' During this time the radial field

force - eE, gives the electron an impulse -eE,l1t

=

eE,1

(10)

This impulse is equal to the change in transverse momentum m I1v" and thus the change in radial component of velocity of the electron is given by e E,I I1v, = - - ­ m vl

(11)

Now the electron traverses the central region in a uniform axial field E z given by (12)

which accelerates the electron from axial velocity Vl to V2 without changing its radial component of velocity. Finally, in the second end region the radial component of velocity is changed by an amount (13)

obtained from Eq. (11) by substituting V 2 for

Vl'

Thus the net change in v, is (14)

Now, as indicated above, the component E, is related to the axial compo­ nent E z • The simplest way to obtain this relationship is to apply Gauss' law to a cylinder of radius r coaxial with the electrodes. As shown in Fig. 15, most of the flux lines through the center of the cylinder leave through the end regions of length I. The flux through a cross section of radius r in the center of this cylinder is nr2E", and the radial flux through an end region is 2nrlE,. Since these must be equal, we have E _ r E _ r V2 - Vl '-21 z-21 2R 18

(15)

focusing and intensity control

FIGURE 15

EF-2

Field lines

which shows that E, is indeed proportional to r. Using this result in Eq. (14), we find (16)

This shows that Av, is also proportional to r, as required for focusing. The quantity V 2 - VI is related to the speeds VI and V 2 by the energy relations, Eqs. (9), which give (17) in terms of which Eq. (16) becomes

Av, =

r

SR

(V2

2 -

2(1 v1)

VI )

VI

-

(IS)

2

Finally, we relate Av, to the object and image distances Z I and Z 2 shown in Fig. 12. We see that the initial value of V, is rvi/z 1 and the final value - rv 2/Z 2 so the change is (19) Combining this with Eq. (1S) and introducing the abbreviation

n= VI

(~:y/2

(20)

(n - 1)2(n + 1) 8nR

(21)

we finally obtain 1

n

Zt

Z2

-+

We see that the form of this equation is very similar to that of the correspond­ ing equation for formation of an image by a refracting surface, as in Figs. 13 and 14, except that the expression for the "focal length" depends on n in a more complicated way for the electrostatic lens. An exactly similar analysis can be carried out for the focusing region between G 2 and A l' Together these two regions act like the two surfaces of a thick lens, and the composite focal length and the positions of the principal planes can be calculated from the individual focal lengths. In practice, the proportions may be chosen so that the electrons move nearly parallel to the 19

electrons and fields

axis within the focusing anode. Thus, the effect of the A I-A 2 region is to focus a parallel beam on the screen. The electron gun in the 3BPl and similar tubes is designed for a ratio VdV 2 or a refractive index of about n 2. Inserting this value in Eq. (21) together with ZI 00 and Z2 16 cm (the approximate distance from A2 to the screen), we find that the effective value of R is about 1.5 cm. Returning to Eq. (21), we see that the focal length of the A l-A2 region has the desired value of 16 cm whenever n satisfies the equation

±

n

16 cm

(n - 1)2(n

+

1)

8n(1.5 cm)

or (n

(22)

This is a cubic equation, and it therefore has three roots. As just pointed out, one of the roots is n 2. This fact may be used to find the other two; dividing Eq. (22) by the factor n - 2, gives a quadratic which can be solved for the other two roots. One is negative and therefore meaningless. (Why?) The other is approximately n = 0.58, which gives the alternate voltage ratio (23)

for focusing. The intensity of the electron beam is determined by the electron flux emerging from the aperture in electrode G I ' which shields the cathode almost completely from the effect of the potentials on the other electrodes. Ordinarily the potential of GI is negative with respect to K, and the resulting field tends to push electrons back toward the cathode. Some have enough energy from thermal agitation to surmount this barrier, but the more negative GI is, the fewer electrons emerge. GI is usually called the control grid, following the nomenclature for ordinary vacuum tubes. Since the shielding is not complete­ ly effective, the beam intensity and the negative voltage required to cut off the beam depend to some extent on the accelerating voltages.

experiment

20

1

focusing conditions The focusing condition may be checked using the same circuit as used for Experiment EF-1 (see Fig. 6). The alternate focusing condition may be checked experimentally with the circuit shown in Fig. 16. Note that the second anode is now connected to the common C+ -B- terminal, and that a jumper connects this terminal to ground. The focusing anode Al is connected to the B+ terminal through a 180-V battery; the cathode is connected to C­ through a similar battery. Find the second focusing condition, measure the voltages, and compare with the above discussion.

2

control grid To study the effect ofelectrode GI , return to the circuit of Fig. 6. Increase the negative voltage on G1 until the beam is completely cut off, and measure the

focusing and intensity control

EF-2

voltage necessary for cutoff. Now change the accelerating and focusing voltages and repeat. Does the cutoff voltage depend on the other voltages? FIGURE 16

Black

l80V

=

180 V

-=-

3

+

IIII ~Gree---ln 41v 2

VI = VB + Vc + 360 V

V2 = Vc + l80V R =47kD

CAUTION

Do not operate the tube with zero voltage on G1" The electron beam will be so intense that electron bombardment of the screen will heat the impact point enough to change the phosphor chemically, destroying its phosphorescence and leaving a "dead spot" on the screen.

questions 1

The electrostatic lens system of the electron gun forms an image on the screen of the tube. What is the object for this image?

2 Why is the negative root of Eq. (22) not physically meaningful? 3

Why is the focusing condition n

= 2 chosen, rather than the alternate value,

n = 0.58, for practical operation of the tube? 21

electrons and fields

4 Physically, how could you construct a situation in which the potential changes suddenly at a plane surface, as shown in Fig. 14a?

5 How is the focusing condition affected by the fact that not all the electrons

emerging from G1 have exactly the same speed? 6 What would happen if a sinusoidally varying voltage were applied to the vertical deflection plates, and a smaller sinusoidal voltage with the same frequency added to the G1 voltage?

22

experiment

EF-3

tnagnetic deflection of electrons

introduction The first two experiments in this series have been concerned with the motion of electrons in electrostatic fields, produced by electrostatic charges on electrodes. In this experiment we consider the effect of a magnetic field on the motion of electrons. Just as the electric field represents the interaction of two charged particles at rest, the magnetic field represents the interaction of charged particles resulting from relative motion. The electric field can be defined as the force on a unit charge, and the magnetic field in terms of the force on a unit current element; this comparison is illustrated in Fig. 17. A magnetic field can be produced by a current in a conductor; the field in the vicinity of a long straight conductor is represented by field lines en­ circling the conductor, as shown in Fig. 18, which also gives the magnitude of the field at a distance r from the conductor. The field, in turn, exerts a force on a charged particle moving through it, according to the Lorentz law of force. This law states that the magnitude of the force is given by

F = qvB sin

(J

(24)

and the direction of the force is perpendicular to B and v, as shown in Fig. 19. Alternatively, the force may be expressed in terms of a vector product F=qvxB

(25)

The vector product of two vectors A and B, we recall, is defined to be a third vector whose magnitude is equal to IAI . IBI . sin (J and whose direction is perpendicular to the plane of A and B, in the direction a right-hand threaded screw advances if A is turned until it points in the direction ofB. Thus an electron moving through a magnetic field is accelerated by a force with magnitude F proportional to the component of velocity perpendicular to the field, in a direction always perpendicular to both the field B and the instantaneous velocity v. This relation between the directions of F and v has an immediate and important consequence: The magnetic field force never does work on the particle, since the particle always moves in a direction perpendicular to the force acting on it. For this reason a particle moving in a magnetic field must move with constant kinetic energy and thus with constant speed. The direction ofthe velocity may, ofcourse, change; in this experiment you will observe the deflection of an electron beam by a magnetic field oriented perpendicular to the direction of the beam. We consider the situation of Fig. 20. Electrons emerge from the electron gun with a speed v determined by the energy relation (26) 23

is and fields

FIGURE 17 (b)

( a)

n

- " ----J

I

- q" ~- --..!-- -- ---.B--rq - .-= F F ~ 1=-.

---::.

-

F

-JI 11.,­ F

I

~

I'

I

Like charges repel with a force F _ _ 1_ ql q2 -

4 1[6

0

,.2

Like currents attract with a force per unit length F = ~ 21112 47r r (d)

(e)

F

The electric field is the force per unit charge.

If)

(e)

Thus the electric field near ql is

E

The magnetic field is the force per unit current, per unit length.

= _

l _ !!. .

41[6

0

,.2 '

the magnetic field near B = 110 211 471: r

I;

is

EF-3

magnetic deflection of electrons

FIGURE 18

I

FIGURE 19

B

F (out of plane when q>O)

FIGURE 20

I

",'"

~l--~~

/'"

I I

/'"

",/

I 'I

F

~

I

....'

®

I

®

ir*I,

I

I F

",'"

'" /'"

II

",'"

I '"

1~~(j)A" "'" ""'.... I

I®

1--•.-----.V I (j) .... ® ® Electron gun I

L..-_ _ _...J

I

I I

I I

®

I~B (out of plane)

I I I

I

I

, I I I

25

electrons and fields

FIGURE 21

A particle moving in a circle of has an acceleration a toward the radius R with constant speed v center of the circle, always perpen­ dicular to the velocity v, and with magnitude v2 / R.

To produce this motion a force The force on an electron in a uniform is required, with constant magnitude magnetic field has these properties. mv 2,/R and directed always perpen­ Its magnitude is evB. dicular to the velocity v. just as in Experiment EF-1. Now the beam enters a region oflength v in which there is a uniform magnetic field B (the source of which will be discussed later) oriented perpendicular to the plane of the figure, pointing out of the page. The resulting magnetic field force has magnitude F = evB and is always perpendicular to the velocity, as shown. Furthermore, since the acceleration produced by this force is at each instant perpendicular to v, its effect is to change only the direction of v, not its magnitude, as discussed above, and the particle moves with constant speed. Now the conditions just described are exactly those needed for a particle to move in a circle with constant speed. In uniform circular motion the acceleration has constant magnitude and is always directed perpendicular to the velocity, toward the center of the circle, as illustrated in Fig. 21. Thus the electron moves in a circular arc under the influence of the magnetic field force. The radius R of the arc can be obtained easily by recalling that the 26

EF-3

magnetic deflection of electrons

centripetal acceleration is v2 / R, and the force producing the acceleration (sometimes called the centripetal force) must be mv 2 / R. But the force is also equal to evB, as mentioned above; so

evB

R

mv

R

and

(27)

eB

After leaving the region of the magnetic field, the electrons again travel in a straight line, as shown, deflected by an angle 0 from the original axial direction. Reference to Fig. 22 shows that the angle is given simply by sin 0 = I = leB

R

(28)

mv

FIGURE 22 I

I

1

I

I 1

I

I

I" \ \

I

~l---~~l·~-------------L------------~~;

I

~\

1

R

IR

'\

I

v

I

I 8 \ I

___•

Ii

D

I

Screen

I

\

\

1

8

10000 __

I 0 0 0 T~

• 0 ____ I 000 I

1 I

+-_f:__

I

1

and the transverse displacement a at the point of emergence from the field is

a=R

R cos 0

mv

= -- (I eB

- cos e)

(29)

Finally, the beam strikes the screen at a point displaced a distance D from the undeflected beam position. As the figure shows, the total displacement is given by (30) D=LtanO+a When the above expressions for 0 and a are substituted into this equation, the result is somewhat complicated. It can be simplified considerably by using the fact that in the present experiment the angular deflections are small, and we can approximate sin 0 tan 0 = 0 and cos 0 = 1 0 2 /2. We then find the total deflection D:

.)

'1

(31) Using the energy relation, Eq. (26), we obtain

leB D = (2meV )1/2 (L 2

1

+

"2/ )

(32)

As this expression shows, the beam deflection is proportional to the magnetic field B, as might be expected. It depends inversely on the square root of the 27

electrons and fields

accelerating potential; this is in contrast to the electrostatic deflection situation of Experiment EF-I, where the deflection varied inversely as V2 itself, not its square root. The difference is that here we have an additional velocity dependence because of the nature of the magnetic-field force. Now, how can we produce a magnetic field having the characteristics described above, that is, uniform in a certain region and zero outside that region? Clearly, the field of a long, straight wire, discussed earlier, does not have this property, but varies smoothly from point to point. However, if this wire is bent into a circular loop, the flux lines are concentrated inside this FIGURE 23

loop, as shown in Fig. 23, and the field is much stronger in the center of the loop than outside. The field at a point on the axis of the loop is B

Jio/.

=~sm

3 ()

2a

(33)

This effect is enhanced by using several loops distributed along a cylinder, as in Fig. 24. Such an arrangement is called a solenoid. Clearly, the field along FIGURE 24

the axis of a solenoid can be calculated by considering it as a series of loops and adding the contributions of the individual loops. The result of this calculation shows that inside a solenoid which is long compared with its diameter, the field is very nearly uniform along the axis and across a cross section of the cylinder, and is given by B = Jio;1

(34)

where N is the total number of turns and L the total length. It is useful to remember that a long solenoid with I A-turn/em produces a field of approxi· 28

magnetic deflection of electrons

EF-3

mately 10- 4 T (Wb/m2) or about I G (more precisely, 1.26 x 10- 4 T (Wb/m2) or 1.26 G). It would be possible to build a solenoid into the cathode-ray-tube, but it is easier and just as effective to use two solenoids in the arrangement shown in Fig. 25. The field lines spread somewhat, as shown, and so this arrangement FIGURE 25

x

B

Electron beam deflected downward

z

v

(a)

(b) produces a somewhat weaker field than that given by Eq. (34), over a some­ what larger area than that of the individual solenoids. Thus we should not expect too precise quantitative agreement with our theoretical predictions. However, the dependence of the beam deflection on the solenoid current and on the accelerating potential should not be changed by these differences.

experIment 1

magnetic deflection The electron gun electrodes are connected just as in Experiment EF-l (see Fig. 6). The deflection plates are not used in this experiment, but they should be connected to B + as shown. This prevents any accumulation of static charge on the plates, which might cause spurious deflections. The resistance of each solenoid is about 50 to lOOn, so the solenoids may be most conveniently energized by a low-voltage transistor-regulated power supply. They should be connected in series, as shown in Fig. 26, making sure that the fields from the two solenoids add. Rather than measuring the solenoid current, it may be easier to measure the voltage V. as shown. Assuming the coils obey Ohm's law, V. is proportional to the solenoid current and therefore to the magnetic field. Measure and plot the beam deflection as a function of solenoid voltage for several values of accelerating potential V2 • Can you predict in advance what the general appearance of the graphs will be? Next, replot the deflections using as independent variable (horizontal axis) the quantity V.I( V2)1/2. What should the result be, and what does it mean? 29

electrons and fields

FIGURE 26

Solenoid

Solenoid

Low-voltage power supply 0-35 V 200mA

2 earth's magnetic field You may have observed in this experiment and in Experiment EF-I that in the absence of any deflecting fields the position of the spot on the screen changes when the accelerating potential changes. One reason for this effect is the earth's magnetic field. By marking the face of the tube with a pencil, can you find an orientation ofthe tube for which there is no deflection? In this position, what is the relation between the axis direction and the direction of the earth's magnetic field? Now try to find a direction for which the deflection is maximum. U . Eq. (32), compute the value of B. Note that in this case I is the total distincei from A2 to the screen, and L O. The electrodes are made of nickel, is ferromagnetic and acts as a magnetic shield, so there is no apI)re4jatllel deflection of the beam until it leaves A 2 • Compare your determination of the magnitude and direction of the earth's field with handbook values.

1 Suppose the electron charge were positive instead of negative. What would have to be made in the CRT, and how would the deflection of the in a magnetic field change? 2 Why is the particular combination of variables Vs /(V2 )1/2 suggested for graph of the deflection data? 3 Show that the deflection of the beam is proportional to the solenoid even if the magnetic field is not uniform, provided the deflection angle is 4 Do your results for the earth's magnetic field agree with handbook values

If not, what are some possible reasons for the discrepancy? 5

What would happen if the experiment were arranged so the magnetic field parallel to the axis of the CRT? Could this be done with the apparatus in this experiment?

6 If in addition to the magnetic field a voltage is applied to one pair of """"'''''''VII

30

magnetic deflection of electrons

EF-3

plates, the two deflections can cancel each other out. Which pair of plates should be used, and what should the polarity be? If the condition for zero net deflection is satisfied, and then the accelerating voltage is increased, what will happen? 7

Could a magnetic field be used to focus the electron beam? Can you think of a magnetic field arrangement that would function in a way analogous to the electrostatic focusing discussed in Experiment EF-2?

31

experiment

EF-4

helical motion of electrons

introduction In Experiment EF-3 we observed the deflection of an electron beam by a magnetic field perpendicular to the beam direction. We found that in a uniform field the beam is bent into an arc of a circle, with radius R related to the electron charge e and mass m and to the magnitude B of the field intensity by the equation (in MKS units) R

=

mD

(35)

eB

where D is the speed of the electrons, which is constant because the magnetic field exerts no force along the direction of motion. In this experiment we study the motion of an electron beam when the magnetic field is nearly parallel to it. As we shall see, the electrons describe a spiral or helical path around the direction of B; by studying this motion in detail we can duplicate in principle an experiment which has been used for a very precise determination of the charge-to-mass ratio (elm) of the electron. As in Experiment EF-2, it is convenient to represent the velocity of an electron in terms of its axial and radial components, denoted by Dz and Dr. respectively, as in Fig. 27. The z axis coincides with the direction of Band FIGURE 27

z

•laB Ia

Screen

with the axis of the cathode-ray tube used for the experiment. We observe first that if the velocity is parallel to B (Le., if Dr = 0) the magnetic field exerts no force on the electron. Only the component of v perpendicular to B, namely Dr' produces any force; the magnitude of the magnetic field force is equal to eDrB. The direction ofthis force is perpendicular to both the radial component of v and B, and hence never has any component in the direction of B. Thus Dz is constant, if no forces other than the magnetic field force are present. We have already considered in Experiment EF-3 the case in which Dz is zero; in this case the electron moves in a circular path in a plane perpendicular to B, with radius R given by Eq. (35) and Dr substituted for D. The angular 33

electrons and fields

velocity of this uniform circular motion is called the cyclotron frequency because it is also the angular frequency of motion of charged particles in a cyclotron. It is given by Vr

eB

R

m

W= - = -

(36)

Thus the angular velocity is independent of radius of the circle. The period of motion T, the time required for one complete revolution, is given by

2n

2nm eB

T=-= ­ W

(37)

Now if in addition to the initial radial velocity Vr the electron is given an initial axial velocity V z , the result is to add to this circular motion perpen­ dicular to B a uniform motion parallel to B. A little thought shows that in this case the electron moves not in a circle of radius R, but on the surface of a circular cylinder of radius R, and that in fact it describes a helix on this surface. as shown in Fig. 28. In the time T required for one revolution , the electron FIGURE 28

moves axially a distance equal to vzT; this distance, called the pilCh of the helix, is given by

2nmv. eB

p=vT= - -Z

(38)

If the electron travels a total axial distance L after acquiring its radial velocity v" it rotates through a total angle ¢ given by (39)

Comparison of Eqs. (38) and (39) shows that ¢ and p are very simply related:

2nL

¢= ­

p

(40)

This relation can also be derived directly from geometric considerations. In this experiment, ¢ and L can be measured directly, so this equation provides 34

helical motion of electrons

EF-4

a way to determine p, which cannot be observed directly since the electron beam inside the tube cannot be seen. Figure 29, which is a view of the screen looking back toward the electron gun, shows the geometry of the situation. Point Pis the actual position of the FIGURE 29

spot on the screen; the origin 0 of the coordinate system is the position of the spot when Vr = O. The circle of radius R is the end view of the helix. Now suppose we use a fixed value of Vr but change B; how does the spot move? To answer this question it is convenient to introduce the polar coordinates rand as shown in Fig. 29. The unconventional choice of 0 is made so that 0 = 0 corresponds to the starting point on the helix, where
o

r = 2R sin.t

(41)

2

Note that
= 2R sin 0

(42)

We have not yet eliminated B because R depends on B. But by combining Eqs. (36) and (39), we can express R in terms of
R

= ~ = v,L Vz

Rv z

LVr

LVr

-=

vz
(43)

2v z O

Finally, Eq. (42) becomes

r=

Lv, sin 0 -Vz 0

(44) 35

electrons and fields

This is the equation of a spiral curve called a cochleoid; Fig. 30 is a graph of this equation. Values of 4J for several points are shown. We note that each time () becomes an integer multiple of n, corresponding to 4J an integer mUltiple of 2n (Le., an integer number of turns on the helix) r becomes zero, and the beam returns to the undeflected position. As 4J increases, correspond­ ing to increasing B, the beam makes smaller and smaller excursions away from this position, as shown in Fig. 30. y

FIGURE 30

4>=0

r

4> =

= Si~8

211', 411', •••

Now at last we can put the pieces together to obtain an expression for elm in terms of measurable quantities. First we solve Eq. (38) for elm: e

m

= 2nvz

(45)

pB

Next we solve Eq. (40) for p and substitute in Eq. (45): e m

v:z4J

-=­

(46)

BL

Now, the axial velocity v. is determined by the accelerating voltage V2 just as in Experiment EF-I, Eq. (I): (47)

Solving this for finally obtain

v., substituting in Eq. (46), ~ m

=

squaring, and rearranging, we

(!t)2

2 24J V2 = 2V2 L2B2 LB

(48)

Now 4J can be observed directly, at least for certain particular values; Land V2 can also be measured directly, and B can be computed from the dimen­ sions and current of the solenoid. We discuss this calculation next. To obtain B, we use the same considerations as in Experiment EF-3. If B were produced by an infinitely long solenoid, it would be uniform across the cross section and along the length, and would be given (in MKS units) by

B = J1.oN'I 36

(49)

helical motion of electrons

EF-4

where I is the current and Nt the number of turns per unit length. In fact, the solenoid used in this experiment is not infinitely long, and a correction must be made for the variation of the field along the axis of the solenoid. Fig. 31 FIGURE 31

- 1.5

-1.0

-0.5

o

0.5

1.0

1.5

'1./8

shows this variation for the solenoids used in this experiment. In this figure Bo is the field given by Eq. (49), S is the length of each solenoid, and z is the

distance along the axis from the midpoint of the solenoid system. Using this graph, you can obtain an average correction factor for the field produced by the solenoids; this procedure is an approximation, of course, but to try to analyze the electron motion in a nonuniform B field would be extremely complicated. The experimental arrangement described above was first used in 1922 by Professor H. Busch of Jena (Germany) for a precise determination of elm for the electron. Using a specially designed tube, he was able to obtain a pre­ cision of a few tenths of a percent. His account of his experiment was pub­ lished in Physikalische Zeitschrift, volume 23, page 438 (1922). A recent description of a Busch tube is given in an article by Professor H. V. Neher in the American Journal of Physics, volume 29, page 471 (1961).

experiment 1

electric and magnetic fields

To begin the study of helical motion of electrons apply accelerating poten­ tials to the CRT using the same arrangement as in Experiment EF-l, Fig. 6. Establish an axial magnetic field inside the CRT by placing over the neck of the tube a pair of solenoids connected in series to the low-voltage power supply. Make sure the solenoids are connected so that their fields add rather than cancel! Apply a deflecting potential to the vertical plates and gradually increase B. The value of B for any point can be obtained by measuring the potential 37

electrons and fields

across the solenoids, using the results of Experiment EF-3 for the relation of potential to current, and applying Eq. (49), with the correction factor dis­ cussed above. Make a table of values of qy observed, and the corresponding values of potential across the solenoid. Compute a conversion factor to convert from these potentials to the corresponding values of B, compute the values of B, and add these to the table. 2

determination of elm Plot a graph showing qy as a function of B. Draw the best straight line through the points and find the slope ofthis line. From the slope and Eq. (48) compute the value of elm. Compare with the accepted value; what is the percent difference? What do you think were your principal sources of error? What would you do to achieve a more precise measurement?

3

magnetic focusing You may have observed that when qy is an integer multiple of 2n, i.e., when L is an integer multiple of p, the electron spot was especially sharp. This suggests that a uniform magnetic field canfocus an electron beam. Remove the deflecting potential Vd and slightly defocus the electron beam by changing the focusing anode voltage VI' Now increase B and observe the periodic focusing and defocusing of the beam. With reference to Fig. 30, can you explain qualitatively how the magnetic field focuses the electron beam? Use the focusing condition to obtain an improved value of elm. What distance L should be used for this calculation?

questions

38

1

What is the direction of B in Fig. 30?

2

What is the approximate percent variation in B along the axis ofthe solenoids? Approximately what percent error would you expect this to contribute to the final results?

3

How can the direction of B in your experiment be determined by tracing the solenoid wiring? How can it be determined without tracing the wiring, but by observing the spot?

4

Between what two points should the distance L be measured? Explain.

5

If the magnetic field B is held constant but the deflecting potential is changed, how does the helix change? For example, how do the radius and pitch change?

6

The accelerating and focusing electrodes in the CRT are made of nickel, which is a ferromagnetic metal. What effect will this have on the experiment? .

7

In Experiment EF-3 the earth's magnetic field was considered in discussing the motion of the electrons. Is this consideration necessary in this experi­ ment? Explain.

S

Could a short solenoid (i.e., one with a nonuniform field) or a combination of short solenoids be used to form a "lens" analogous to the electrostatic lens of Experiment EF-2? Explain.

EF-S

experiment

vacuum diodes and the magnetron condition

introduction In this experiment we consider the motion of electrons in a vacuum diode. The basic principle of a diode is shown in Fig. 32. Two active electrodes, FIGURE 32 Anode

Heater,

Vacuum

!~

I

(a)

Anode

~/..-vacuum

(h)

usually called the cathode and the anode, are sealed in an evacuated envelope or tube. The cathode is heated to a high temperature, the order of 2500 K, either directly by passing a current through it, or indirectly by means of a separate heating coil, as shown in the figure. The heated cathode emits electrons by a process called thermionic emission. Electrons have to surmount a potential-energy barrier to escape from the surface of the metal, and thermal agitation gives some ofthe electrons the needed energy. The hotter the cathode, the more electrons can escape per unit time. The process is completely analogous to the evaporating of water molecules from a liquid water surface to form water vapor. The higher the temperature, the more rapidly the water evaporates. Once free of the cathode, the electrons are free to move in the vacuum inside the envelope. If a potential difference is imposed as shown in Fig. 32, the resulting electric field between cathode and anode drives the electrons to the anode, from which they travel around the external circuit back to the cathode. The resulting current can be measured, and its dependence on the other variables can be explored. Depending on conditions, the magnitude of the current is governed by one or the other of two different effects. If the potential between cathode and anode is relatively small, emitted electrons tend to accumulate near the 39

electrons and fields

cathode, forming what is called a space charge. This negative charge alters the electric field near the cathode, so that it tends to push emerging electrons back to the cathode. As the potential increases, the electrons on the fringe of the space charge move more rapidly toward the anode; hence the space charge decreases and the current increases with increasing potentiaL In this case we describe the current as being space-charge-limited. For the simple planar geometry shown, and also for a geometry in which the anode and cathode are coaxial cylinders, it is possible to show that the current I is actually proportional to the three-halves power of the potential V. That is, when the current is space-charge-limited, I

(const)V3/2

(50)

This important relation is called the Langmuir-Child law, or the "three­ halves-power law." As V is increased, the space charge decreases until a point is reached where there is practically no space charge. In this case no electrons are driven back to the cathode after emission, and the current is determined entirely by the rate of emission from the cathode. When this point is reached, further in­ crease of V does not increase I. The maximum emission rate increases rapidly with cathode temperature, as might be expected. The operation of a typical diode is shown in Fig. 33, in which current is plotted as a function of voltage for several values of cathode temperature. The figure shows that for each value of T, I is space-charge-limited for sufficiently small V, but that at FIGURE 33

40

T

= 2600

K

30

..::

e

~

~

;j

'-'

~

20

0

T=25OOK

~

10 T= 2400 K T=23OOK

50

100 Anode volts

150

200

larger values of V, I becomes emission-limited. As T increases, the transition to emission limiting occurs at higher and higher values of V. The maximum (emission-limited) current is given by the Richardson-Dushman equation (51) 40

vacuum diodes and the magnetron condition

EF-5

where A is a constant, T is the absolute temperature, is the work function for the material (the energy required for an electron to escape from the surface), and k is Boltzmann's constant. The quantity kT characterizes average energies associated with thermal agitation; thus it is reasonable that the emission should depend on the ratio /kT. One obvious practical application of the vacuum diode is that it acts as a rectifier; electrons can flow from cathode to anode but not in the reverse direction. A more esoteric application is found in the magnetron, a high­ frequency oscillator. Some aspects of magnetron operation will be investiga­ ted here. The diode used in this experiment, the Ferranti GRD7, has the geometry shown in Fig. 34. The cathode is a tungsten wire 0.125 mm in diameter, FIGURE 34

Anode Guard rings

directly heated by a current passing through it, and the anode is a cylinder coaxial with the anode, with inner diameter 0.65 cm. The two cylinders on the ends are called guard rings; they are kept at the same potential as the center section but are not connected to it electrically. Their function is to minimize "fringing" of the field near the ends of the center section, and thus to make the field inside the center section very nearly the same as for an infinitely long cylinder and wire. Now suppose that in addition to the radial electric field produced by im­ posing a potential V between cathode and anode, we impose a uniform magnetic field B along the axis of the cylinders. The electron motion becomes more complicated but also much more interesting. As electrons leave the cathode and begin to accelerate toward the anode, they experience the magnetic field force F = -ev X B. The effects of this magnetic force is shown in Fig. 35; clearly its effect is to make the electron paths curve as shown. If the magnitude ofB is increased, the force increases proportionately, and the electrons curve more sharply. Finally, at some critical value ofB, the electron paths are bent back toward the cathode, without reaching the anode at all. When this condition is reached, a sharp drop in anode current occurs. If the potential is increased, we expect the cutoff value of B to change, but it is not obvious in which direction. Increasing V increases the electrical force 41

pushing the electrons to the anode, but it also increases their speed, which increases the magnetic field force tending to curve them back to the cathode. However, the electric field increases in direct proportion with V, whereas the electron speeds increase only as V 1 / 2 , since the kinetic energy! mv2 is pro­ portional to V. Hence when V is increased, we expect that a larger value of B Anode

FIGURE 35

Small B

Larger B

B out of plane

will be required for cutoff; a more detailed analysis substantiates this expectation. It turns out that the cutoff value of B is related to V by the equation B

(52)

where b is the radius of the anode. Unfortunately, there is no particularly simple way to derive this equation. A derivation is given below, but for readers who are not ready to cope with this derivation, Eq. (52) can be taken on faith and used to interpret the experimental observations. The derivation of Eq. (52) is basically a problem in classical mechanics. We need to find the maximum distance from the cathode (r = a) the electrons reach before they tum back toward it again. At this point r stops increasing, and thus;' is instantaneously zero. In particular, we want to find the critical b) since this is value of B for;- to be zero just at the surface of the anode (r the condition for cutoff. The program will be to use the equation of motion I: F ma to express the angular component of velocity in terms of the instantaneous radius r, and then use the energy relation to find a relation between ;- and r. Finally, we will find the value of B needed for f to become zero at the distance r b. We use cylindrical coordinates r, 9, and Z, as shown in Fig. 36, to describe the position of an electron. We also express the components of velocity, acceleration, and the various fields in terms of their components in the r, lJ,

=

42

vacuum diodes and the magnetron condition

EF-5

y

FIGURE 36

----+-------+-~~--_r-L----_+----x

::

:!! ,i~

{~

:jl

iE

II and z directions. In terms of the cylindrical coordinates, the components of velocity and acceleration are given by f

v,

a,

= ;: -

Ve = r()

ae = ri.J

vz

az = Z

i

r0 2

!I!

il!

Jt

'::

+ 2fO

(53)

m

These relations are derived many elementary mechanics texts. Since there :1 is no initial velocity or force in the z direction, V z and az are always zero. The magnetic field is in the +z direction, as shown, and we can work o u t i l the components of the magnetic field force F

=

-ev X B

We find F,

eveB

Fe

eV,B

Fz

0

- eBr() eBf

(54)

The only force in the () direction is the () component of the magnetic field force. Thus the () equation of motion Fe mae becomes eRr = m(ri.J

+ 2fO)

(55)

We multiply both sides of this equation by r and rearrange terms as follows: m(r 2 i.J

+ 2rfO)

eBrr = 0

(56)

Inspection of this expression shows that the left side is simply the time derivative of the quantity (mr 2() - teBr2). Since the time derivative is zero, the quantity itself must be constant. Denoting this constant value by L, we have (57)

We note in passing that the first term of this expression is the angular momentum of an electron about the axis. Thus, this equation shows that 43

electrons and fields

angular momentum is not conserved in this motion, but that angular momen­ tum plus the term containing B is a quantity which is conserved. To evaluate L, note that the electron leaves the cathode (r = a) with negligible initial velocity, so at r = a we must have = O. Inserting these values in Eq. (57), (58) L = -!eBa 2

e

Combining this result with Eq. (57) and solving for

e=

eB 2m

(1 _

e

2

a ) r2

(59)

We now turn to the energy relation. From the velocity components given in Eqs. (53) we find that the kinetic energy is !m(,2 + r2()2). The potential energy is - e V(r), where V(r) is the electrostatic potential, a function of r. Assuming V is defined to be zero at r = a, the total energy is zero, and we have the energy equation (60)

which, with the help of Eq. (59), becomes m [ ,2 + 2"

2 r2 (eB)2 2m ( 1 - ar2 )2] - e V(r) = 0

(61)

Now, as observed above, the field strength B which just achieves cutoff is determined by the requirement that, just becomes zero at r = b; when this condition is satisfied, the electrons just barely (or do not quite) reach the anode. Inserting these values, together with the fact that at r = b the poten­ tial V(r) is simply the total potential V rise from cathode to anode, we obtain m [0 2"

+

2 (eB)2 ( a b 2m 1 - b 2 )2] 2

-

eV

=0

(62)

Since in the present situation b is much larger than a, we make only a very small error if we drop the term a 2 /b 2 compa,red to unity, obtaining 2 -b m (eB)2 -eV=O 2 2m

(63)

Solving for B, we find that the critical field required to cut off the anode current is given by

B

= (8mV')1/2 eb 2 )

(64)

in agreement with Eq. (52). Electron motion of this type is used in a type of electrical oscillator called a magnetron, used to generate extremely high-frequency (the order of 10 10 Hz) electromagnetic waves. In the magnetron, the electrodes form part of an enclosure called a resonant cavity, and the rotating electrons induce electro­ magnetic oscillations in this cavity. 44

vacuum diodes and the magnetron condition

EF-5

experiment 1 properties of diode

To investigate the properties of the diode, the circuit shown in Fig. 37 is suggested. The filament (cathode) is heated by the 6.3-V heater supply, and FIGURE 37

25W

its temperature can be controlled by the rheostat in series with the supply. The current through the center section of the anode can be determined by measuring the voltage drop across the 100-0 series resistor.

CAUTION

The B+ terminal is grounded so that the case of the voltmeter will be at ground potential. However, this places the filament supply, including the resistors, at a negative potential of up to 300 V, so be careful!

Set the anode potential at about 100 V and increase filament temperature until the central anode current is about I rnA. If an optical pyrometer is available, determine the filament temperature. Measure I as a function of V, and plot. Return the anode potential to about 100 V, increase the filament temperature until the central anode current is about 5 rnA, and repeat the above measurements. Finally, repeat once more at the maximum filament temperature. 45

electrons and fields

2

Langmuir-Child law Examining your data, pick a set of I vs V at a temperature which gives emission limiting at a moderate voltage (say 100 to 150 V) and plot the data on log-log paper. Over what voltage range, if any, is Eq. (50) obeyed?

3

Richardson-Dushman equation For each temperature, determine as precisely as possible the maximum emission-limited current. Plot a graph of ImaJT2 as a function of liT, using semilog paper with ImaJT2 on the logarithmic axis. If an optical pyrometer is not available, approximate filament temperatures can be obtained from the filament voltages, using Fig. 38. If Eq. (51) is satisfied, you should obtain

FIGURE 38

1

2000

2250 Temperature, K

a straight line. (Why?) From the slope and vertical intercept you should be able to obtain the constants A and
magnetron condition To observe the "magnetron" motion and the current cutoff, use the same circuit as used above, but place the diode inside a solenoid powered by the transistorized power supply, as in Experiments EF-3 and EF-4. The magni­ tude B is obtained from the solenoid current as described in these experi. ments. For a solenoid 4 in. long, with an inner diameter of 3t in. and an outer diameter of 5 in., the field at the center is about 0.7 of that expected for an ideal, very long solenoid. Determine the cutoff field for several values of potential across the diode. Plot B2 as a function of V to check Eq. (52). Use this graph to determine the ratio elm.

questions

46

1

Does a diode obey Ohm's law? Explain.

2

Is Eq. (50) obeyed if the current is emission-limited? Explain.

I

vacuum diodes and the magnetron condition

EF-5

3 In the emission-limited situation, the potential between the cylindrical electrodes of the diodes is given by VCr)

=

V In(r/a) In(b/a)

Derive this expression. Is it still valid when the current is space-charge­

limited? Explain.

4 What is the function of the two 100-0 resistors in the filament circuit in

Fig.40? 5

What happens if the direction of the magnetic field in Fig. 35 is reversed?

6 Does the validity of Eq. (64) depend on whether the current is space-charge­

or emission-limited? Explain. 7 How would you expect the cutoff effect to change if the guard rings were

removed from the diode? 8 How would the behavior of the diode change if some air were permitted to leak in? That is, why must it be evacuated? What maximum residual pressure can be tolerated for proper behavior?

47

~

berkeley physics labora t ry, 2d edi iqn

alan m. portis, univerSIty of ca/lfornlo , b ,J.:eley hugh d. y oun carneg ie -m ellO, ur w et ~It t

resistance-capac itance circuits

E -1

resistance-inducta n c e cir c Its

EC-2

LAC circuits and o scillations

EC -3

co upled oscillator periodic structures and transmission line

london

Sf louis

E­

0

an

san francisco

duss 111

mcgraw-hill boo k new york

EC-4

f

matI MS-' MS-2 MS-3 MS-4 MS-5 MS-6

mec. M-1 M-2 M-3 M-4 M-5

I

I

,. electric cIrcuits Copyright © 1971 by McGraw-Hill, Inc. All rights reserved. Printed in the United States of America. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Library o/Congress Catalog Card Number 79-125108

07-050486-5 1234567890 BABA 79876543210

The first edition of the Berkeley Physics Laboratory

copyright © 1963, 1964, 1965 by Education Development

Center was supported by a grant from the National Science

Foundation to EDC. This material is available to publishers

and authors on a royalty-free basis by applying to the

Education Development Center.

This book was set in Times New Roman, printed on

permanent paper, and bound by George Banta Com­

pany, Inc. the drawings were done by Felix Cooper;

the designer was Elliot Epstein. The editors were Bradford

Bayne and Joan A. DeMattia. Sally Ellyson

supervised production.

"

C

t

n Ir

"

tI­ n­

---------

electric Circuits

INTRODUCTION

In this series of experiments we study the behavior of a variety of electric circuits in which the voltages and currents vary with time. In these studies the most important instrument is the cathode-ray oscilloscope, incorporating a cathode-ray tube very similar to the device used to study electron motion in the experiments titled Electrons and Fields. As we shall see, the oscilloscope is an extremely useful instrument for observing and measuring rapidly vary­ ing voltages and currents in electric circuits. The circuits to be considered in these experiments contain various com­ binations of resistors, capacitors, and inductors, batteries, and sinusoidal or square-wave voltage generators. It is well to review briefly the characteristics of these devices. An ideal resistor has the property that when a potential difference Vis applied between its terminals, the resulting current / is directly proportional to V; the proportionality constant, denoted by R, is called the resistance of the device. That is, for an ideal resistor, V= /R

(1)

This relation is called Ohm's law. In the MKS system of units, where V is measured in volts and / in amperes, the unit of resistance is the ohm, abbreviated Q. The units ldl(103 Q) and MQ( I06 Q) are also commonly used. In addition to resistance, every resistor has a power rating, which is the maximum power (/2 R) which may be dissipat­ ed in the resistor. Exceeding this power rating will change the resistance in an unpredictable manner or destroy the device completely. Resistors of the type used in these experiments are common in electronic circuitry and typically have power ratings of! or I W (watt) (the maximum power that can be dissipated without overheating). These resistors are usually made of a carbon-clay mixture which is fired to form a hard ceramic material; the resistance can be controlled by varying the proportions of ingredients. Resistors are labeled according to the color code shown in Fig. I. For example, a resistor whose resistance is 56,000 Q ± 10% would have green, blue, orange, silver bands (reading inward from the end). Capacitor A capacitor maybe thought of as a charge-storing device. When a charge Q

is added to one plate and a charge - Q to the other, the resulting potential difference V between the plates is proportional to Q; this proportionality is expressed by the relation

Q = CV

(2)

where C is a constant characteristic of the device, called its capacitance. In MKS units Q is measured in coulombs, and the corresponding MKS unit of capacitance is the farad (abbreviated F). One farad is an extremely large unit 1

electric circuits

FIGURE 1

CO D E

Color

1st digit

BASIC VALlIES

2£1 digit

Multiplier

VaiuB

lst color

2£1 color

Silver

lO-2

10

Brown

Black

Gold

10- 1

12

llwwn

Red

lOo

15

Brown

Green

1

10 1

18

Brown

Gray

2

2

Black Brown Red

0

2

10

22

Red

Hed

3

27

Red

Violet

Orange

3

3

10

Yellow

4

4

104

3,'3

O range

O ra nge

5

39

Onmge

White

Green

5

,5

10

Blue

6

6

lOB

47

Yellow

Violet

Violet

7

7

10 7

56

Green

Blue

8

lOS

68

Blue

9

9

82

Gray

Gray White

8 9

10

ray Red

~ ~~:~n±5% __

Silver ± 10% No hand ± 20%

of capacitance; the units ,uF( 10 - 0 F) and pF( 10 - 12 F) are commonly used. Capacitors are often made from a I ng narrow sandwich of aluminum foil and Mylar film (an insulating plastic) rolled into a cylinder and encased in plastic, I n addition to capacitance ca pacitors are a lso ra ted according to maximum instantaneous voltage which may be applied. Exceeding this rating will lead to dielectric breakdown and perforation of the insulating material, causing a sho rt circuit which renders the device u eless. The Mylar insulation for a capacitor rated at 100 V is about 1O.u (microns) thick . Induc tor

An inductor is a coil of wire, which mayor may not have a powd red iron or ferrite core. A changing current in the coil produces a changing magnetic flux through it; this in tum induces a voltage V between the terminals which is proportional to the rate of change of current dI/dt. Th is relati nship is expressed by the equation

dJ

V=L -

dt

(3)

where L is a constant characteristic of th e device called the inductance. The M KS unit of inductance is the henry, abbreviated H. The units mH(l 0 -3 H) and .uH(10 - 6H) are also commonly used. B attery 2

An idealized battery is a devic which produces a potential difference between its terminals which is constant and independent of the current through the

introduction to electric circuits

device. The most familiar batteries are made from the same kind of carbon­ zinc dry cells as are used in flashlights. Each such cell has a potential of about 1.5 V. Thus a 45-V battery has 30 cells in series. In real batteries, the potential is not completely independent ofcurrent, but the behavior can be represented as an idealized battery with constant potential, in series with a fixed resistance, called the internal resistance of the battery. The internal resistance of a dry cell is the order 0.1 n when new, but increases with age and use. The basic physical principles used in the analysis of circuits are Kirchhoff's circuit laws. The voltage law, or loop law, states that the algebraic sum of the potential differences around any closed loop must be zero. The current law, or point law, states that the 'algebraic sum of the currents into any junction must be zero. These two laws, together with the characteristics of the circuit devices described above, provide a complete theoretical basis for the analysis of circuit behavior. In addition to the circuit devices discussed above, two other instruments will be used, a voltmeter and a signal generator. The voltmeter, as its name implies, measures the potential difference between two points in a circuit. It is important to understand that when a voltmeter is connected in a circuit, it becomes part of the circuit, and the current flowing through the voltmeter must often be taken into consideration. The amount of current flowing through the voltmeter is determined by its internal resistance. Ordinary volt­ meters typically have an internal resistance of the order of 10 3 to 105 n, depending on full-scale reading and details of construction. Vacuum-tube voltmeters achieve much higher internal resistance by using electronic amplification. The vacuum-tube voltmeter used in these experiments has an internal resistance of 11 Mn (megohm).

3

experiment

EC -1

resistance-capacitance circuits

Introduction In this experiment we investigate the behavior of circuits containing a resistor and a capacitor in series. First, we study the behavior of the circuit with a constant applied voltage, and then its response to a sinusoidally varying applied voltage. We consider first the circuit of Fig. 2 containing a battery, a resistor, a capacitor, a voltmeter (labeled V) and a switch. When the switch is closed, FIGURE 2

+

the capacitor quickly charges to the potential Vo ofthe battery; the magnitude of charge Q 0 on either plate is, according to Eq. (2), Qo

C~

~

When the switch is opened, the initial situation is that shown in Fig. 3. The potential difference across the capacitor also appears across the voltmeterFIGURE 3

I

+Q -Q

•

1

v I

resistor branch, causing a current to flow in this branch. This current acts to decrease the charge on the capacitor, which decreases its potential and thus also decreases the current. Thus the charge Q decreases rapidly at first, and then more slowly; correspondingly, the current has a relatively large initial value immediately after the switch is opened, but then tapers off and approaches zero after the capacitor becomes nearly completely discharged. The charge on the capacitor thus varies with time in the manner shown in Fig. 4. 5

electric circuits

FIGURE 4

o

2

1

3

tIRe This circuit may easily be analyzed more quantitatively. We let the instantaneous charge, c}lrrent, and potential be Q, I, and V, respectively, noting that all three of these quantities are variables, that is, functions oftime. First, since the current I is due entirely to the discharging of the capacitor, and since current is just the rate of transfer of charge, we have dQ 1= - ­ dt

(5)

The current, in tum, is related to the instantaneous potential V and the circuit resistance. Denoting the total resistance of resistor and voltmeter in series as R, we have 1= V

R

(6)

Finally, the potential V is related to the charge Q on the capacitor at any instant by V= Q C

(7)

Equating the right-hand sides of Eqs. (5) and (6) and substituting the expression for V of Eq. (7), we find dQ dt

Q RC

(8)

showing that the rate of decrease of charge at any instant is proportional to the charge remaining at that time, as previously stated. The only function having the property that its rate of change (Le., its derivative) is proportional to the function itself is the exponential function. In particular, the function which satisfies Eq. (8), along with the condition that at time t = 0 the charge has the initial value Q 0' is Q = Qoe- t /RC

(9)

The reader is urged to verify by differentiating Eq. (9) and substituting into Eq. (8) that it does indeed satisfy the equation. Figure 4 is a graph of Eq. (9), 6

resistance -capacitance circuits

EC - 1

in which the scales show values of the ratios QIQ 0 and tlRC rather than Q and t themselves. The advantage of this procedure is that these ratios are dimensionless ; that is, they are pure number s, without units. The product RC is c lled the time constant, or the relax ation time of the circuit. As Eq. (9) shows, after a time t equal to RC the charge has dropped to a fraction QIQo = e - 1 = 0.368 or 36.8% of its original value. A related quantity, usually ea sier to measure experimentally, is the time required for Q to drop to one-half its original value. D enoting this time by T1 /2' we have (10) T aking natural logarithms o f both sides and rearrangin g, we find T i ll =

RC In 2 = O. 693RC

(11)

T his time may be called the half-life, a term which is also used in the descrip­ tion of radioactive decay processes.

ELE CTR OM ECHA NIC AL ANALO GS

FIGURE 5

There are many interesting and useful analogies between electric circuits and mechan ical systems. One of the simplest is the relaq on of the RC circuit to the mechanical system shown in F ig. 5, which is a simplifi ed version of a

Fl li\! I

o

'x

El~ll\ ()~ !~ Oil V

I

I

o

Xu

- -

x

door-closer mechanism. The piston is perforated, and the oil must flow through the holes when the piston moves. As a result, there is a resisting force which depends on velocity, because of the viscosity of the oil. F or moderately small velocities, this force is proportional to velocity a nd may be expressed as F = - bv, where b is a proportionality constant a nd the negative sign indicates that the force always opposes the motion. The spring also exerts a force on the m oving piston. When displaced a distance x fro m its equilibrium position, the spring exerts a force F = - kx, where k is the fo rce constant of the spring. The sum of these two forces on the piston must be equal to the mass ofthe piston times its acceleration, acco rding to N ewton's second law. If the mass is negligibly small, the sum of the two force s is zero, a nd we have

- kx - bv = 0

or

dx dt

k

- -x b

(12)

T his differential equation has exactly the same/arm as the equation for the charge on the capacitor, Eq. (8). Disp lacement x corresponds to charge Q, velocity v to current l. There are similar relationships between the parameters of the corresponding system components ; the damping constant b corre­ sponds to the resistance R, and the force constant k of the spring to the reciprocal of the capacitance C. 7

electric circuits

Thus this analysis shows immediately that when the door closer is given an initial displacement Xo from its equilibrium, it approaches equilibrium exponentially, according to the equation (13)

with a time constant equal to b/k. If an additional time-dependent driving force F(t) is added to the door-closer mechanism, the situation is analogous to the RC circuit with an external time-dependent voltage V(t), as shown in Fig. 6. Ifthe mass of the piston is not negligible, it must be included in the analysis, and the presence of mass introduces the possibility that the piston may "overshoot" its equilibrium position and undergo a damped oscillation. In fact, as we shall see in Experiment EC-3, the damped harmonic oscillator is completely analogous to an electric circuit containing a resistor, a capacitor, and an inductor in series. The behavior of the RC circuit described above, sometimes called charge relaxation, can be observed directly with a voltmeter, provided the time constant RC is sufficiently long, say, the order of a few seconds or more. Often, however, it is useful to use values for which RC is very much shorter than a second, and then this method breaks down. Because of the mechanical properties of the meter movement (inertia and1damping), it cannot respond to extremely rapid fluctuations in voltage or current; even if it could, the human eye could not follow this motion.

THE OSCILLOSCOPE

8

Thus a better instrument for observing and measuring rapidly varying voltages is needed. The oscilloscope is just such an instrument, and we now discuss the principles of its operation. The basic idea is to use the deflection of an electron beam in a cathode-ray tube as a voltage indicator. We recall from Experiment EF-I that the deflection of the spot on the screen is pro­ portional to the deflecting potential on the deflection plates. Furthermore, the time offlight of an electron through the tube is the order of 10- 8 sec so the response of the electron beam to a change of deflection potential is corre­ spondingly rapid. Even so, though, the human eye cannot follow extremely rapid motion of the spot on the screen. This difficulty is overcome in an ingenious manner by using both pairs of deflection plates. The voltage V to be observed is applied to the vertical deflection plates, either directly, or through electronic amplification, and a voltage is applied to the horizontal plates which increases uniformly with time. Thus, the vertical deflection of the beam is proportional to the applied voltage, the horizontal deflection to time, and the spot traces out a graph of Vas a function of t! Even if this trace occurs in a very short time interval, the image persists on a screen for a time, just as a fluorescent lamp continues to glow for a fraction ofa second after the power is turned off. The trace on the screen can be viewed with the eye or photographed for more detailed study. A further important refinement in this technique is possible. Instead of observing a single charging and discharging of the capacitor, we can turn the • battery on and off in a cycle, with a definite frequency; the capacitor voltage will then vary as in Fig. 7, in which T represents the time for one cycle. Similarly, the voltage V. applied to the horizontal plates can be varied in a cycle, so it increases uniformly during the time interval T, then quickly returns to its initial value and repeats the cycle. This voltage is shown as a

resistance -capacitance circuits

EC-1

FIG URE 6

(b)

(a)

3)

~l~ } . )

ng us in

Oil

x

is, ay In . is

or, A dashpot is li ke .. .

an RC circuit.

ge me

reo

(c)

ter cal nd he

(d)

F.1V~~ } ~v

ing

dQ-I

ow

dt ­

dx - v dt ­

ion ~all 0­

re, he

to charge and current.

Di splacement and veloci ty con'e­ spo nd . . .

-re­

ely (e )

(f)

' of .cal (la

ith ied lof the Ii to the .dy. . of the ilge ~le.

na ly .s a I

~_F ~ F

= bv + kx

Force correspond s ...

o---i~ v V= IR

+ £c

to voltage.

electric circuits

FIGURE 7

function of time in Fig. 8. Assuming these two variations occur with exactly the same frequency, the effect will be to trace out the graph of one cycle, over and over again on the screen. FIGURE 8

The voltage applied to the horizontal deflection plates is called, for obvious reasons, a sawtooth voltage. Applied to the cathode-ray tube, it is also called a sweep voltage or a linear time base, since its function is to sweep the beam horizontally at a constant rate. This sawtooth voltage is generated by elec­ tronic circuitry built into the oscilloscope; the circuit is always designed so as to permit synchronizing the "sweep frequency" exactly with the frequency of the vertical deflection voltage.

summary

To summarize, the essential functional units of a cathode-ray oscilloscope are as follows:

Cathode-ray tube

This is the indicating device. As discussed in the EF experiments, it consists of an electron gun, a deflection system, and a screen for visible display of the electron beam.

Power supply

The power supply must provide suitable potentials for the grids and anodes of the electron gun as well as current to heat the cathode. Typical second-anode accelerating voltage is 2000 V, although 10,000 V is not uncommon. (Television sets frequently use accelerating voltages of 15,000 to 20,000 V.)

Sawtooth generator

The sawtooth generator must provide a voltage which varies with time as in Fig. 8, with a variable frequency, and it must be able to synchronize this frequency with a repetitive input voltage.

Signal amplifier

The voltage required to deflect the electron vertically the radius of the screen is about 200 V. In order to display signals as small as 0.1 V, it is necessary to provide additional amplification, by a factor of several thousand.

A functional block diagram of the oscilloscope is shown in Fig. 9 and a typical oscilloscope control panel in Fig. 10. 10

resistance -capacitance circuits

EC-1

FIGURE 9

P ower supply

Amplifier

Signal input Synchronizing

vol tag

Sawto th generator

Cathode-ray tube

:;tly 'Ver

FIGURE 10 1 M'"E," ((I' ,)))

ACOFF

}O-OCtS

OUS

Bed !am :lec­ oas yof

P OS

aU" ')1I

HOR

@

@

pos

~", ..)~

ope

.f an :am. ~the

lting

mtly

g.8, ith a bout onal

Id a

VER"f 1.·P T

PH SE

FREQ

0

@

VER "B'R

..

+

HOR INP iT

0 GND

1v

p-p

•

EXT SYNC

®.-+ ® G D

11

electric circuits

SINUSOIDAL VOLTAGE

FIGURE 11

The oscilloscope can be used to study another important aspect of the behavior of RC circuits, namely their response to a sinusoidal input voltage. We consider the circuit shown in Fig. II, in which the applied voltage is a R

sinusoidal function of time with maximum value (amplitude) Vo and angular frequency ro. We expect that the current in the circuit will also be sinusoidal, but its amplitude and phase will vary with frequency in an interest­ ing manner. We might expect that if the voltage variation is very slow, so the period of oscillation is much longer than the RC time constant, the charge on the capacitor at each instant should be given simply by Q = C V, much as in the situation where V was constant. At higher frequencies, however, the capacitor may not be able to charge and discharge through the resistor R rapidly enough to "keep up" with the variations in voltage. In this case we expect a phase difference between Q and V; that is, they are "out of step" by . some fraction ofa cycle depending on the frequency. In addition, the maximum . charge Qo may well be less than CVo. These expectations are borne out by more detailed analysis, and we now proceed with that analysis. Applying Kirchhoff's voltage law to the circuit, we find Vo cos rot = IR

Q

+-

C

dQ R dt

= -

Q

+­

C

where we have used the relation 1= dQldt. Following the above discussion, we assume that Q varies sinusoidally with the same frequency as the voltage but with a phase difference. That is, we assume that Q is given by Q = Qo cos (rot

+


where Qo and are as yet unknown constants. Qo is, ofcourse, the maximum value Q attains during a cycle, and is called the phase angle. A complete cycle corresponds to an increase in rot of 2n; if it should tum out that the time . variation of Q is ahead of that of V by a quarter-cycle, we would have tP == n/2, and so on. Now we have to determine the values which Qo and must have in order for Eq. (15) to satisfy Eq. (14), as required by Kirchhoff's loop law. Calculat­ ing dQldt from Eq. (15) and inserting Q and dQldt in the appropriate places . in Eq. (14), we find Vo cos rot = -roRQo sin (rot

+


+

(~) cos (rot +
The next step is to expand sin (rot +
+ B) + B)

sin A cos B

+ CDS A sin B

= cos A cos B - sin A sin B

+
. (16)

using the

resistance-capacitance circuits

E C- 1

Using these and grouping the terms in sin mt and cos mt, we obtain cos mt ( -mQoR sin cP + ':; cos cP - V o)

+ sin mt ( -mQoR cos cP - ':; sin cP)

= 0

(17)

If Eq. (15) is to be a correct description of the time variation of charge on the capacitor, Eq. (17) must be satisfied at every instant. In particular, it is useful to consider the times when mt = 0 and mt = n/2. In the first case, the second term vanishes, and we see that the equation is satisfied only if the first bracket vanishes. Similarly, when mt = n/2 the second bracket must vanish. Setting this second bracket equal to zero, we find immediately tan cP = -mRC

(18)

Similarly, setting the first bracket equal to zero and rearranging,

Q = o

_ mR

Vo sin cP + (l/C) cos cP

Multiplying numerator and denominator by C cos cP and using Eq. (18), we find CVo CVo Qo = CVo cos cP = (tan2cP + 1)1/2 = [(mRC)2 + 1] 112 (19) since cos cP = llsec cP = 1/(tan 2 cP + 1)112. Thus our qualitative predictions are borne out. The charge lags the applied voltage (because cP is always negative) by a phase angle which approaches zero when m is small, but approaches - n/2 when m is very large. Furthermore, for small m and correspondingly small cP, Qo has nearly the value CVo it would have for a constant voltage; as cP increases Qo becomes less than CVo , and it decreases with increasing m. At all frequencies, VR and Vc are 90° out of phase. It is also interesting to observe how the current 1 varies with frequency. Taking the time derivative of Eq. (15) and using the identity cos (A + n/2) = - sin A, we find 1=

~~ =

-mQo sin (mt

+ cP)

= mQo cos ( mt

+ cP + ~)

The maximum value of I, denoted by 10 , is given by mQo. Using Eqs. (18) and (19), we may represent this in a variety of ways, as follows: 10

mQo = mCVo cos cP = - -; sin cP _ _ mCVo - [(mRC)2 + 1]112 - [R 2

+

Vo (llmC)2]112

(20)

We see that in the low-frequency limit 10 approaches 0, and the phase of 1 approaches n/2. In the high-frequency limit, when cP = -n/2, the current is in phase with the voltage, and its amplitude becomes VoiR. That is, at very high frequencies (Le., m » IIRC) the circuit acts just as though the capacitor were not present at all. Conversely, in the low-frequency limit the circuit behavior is the same as though R were not present. At high frequencies a capacitor acts as a short circuit; at low frequencies, as an open circuit. 13

,I

I

.!

l~

II'I.

ctric circuits

experiment 1

charge relaxation In order to observe charge relaxation in its simplest form, assemble the circuit shown in Fig. 12, which repeats Fig. I with suggested component values. The

FIGURE 12

22 Mfl

+ 4.5 V

-=­

O.5~F

v

switch is not necessary; momentary contact with the battery is sufficient to charge the capacitor fully, because of the very small internal resistance of the battery.

CAUTION

A potential of 45 V can be sufficient to cause an uncomfortable shock. Be careful!

Set the voltmeter to the 15-V scale and complete the circuit. Note that the voltmeter reads only 15 V, or one-third of the battery voltage. Why? (Remember that the voltmeter reads only the voltage across its terminals.) Now break the battery connection and observe the discharge of the capacitor. Measure the time required for the meter reading to drop to half of its initial reading. Compute RC from Eq. (11) and compare with the expected value. Be sure to include the meter resistance in R. 2 exponential decay The decay can be observed for a longer time by switching the meter to suc­ cessively lower ranges. The zero adjustment of the meter may shift slightly when this is done. To verify that the decay is exponential, measure the voltage at a succession of times, spaced 2 or 3 sec apart, using a stopwatch and switch­ ing to successively lower meter scales until the lowest range is reached. The errors resulting from shift of the zero point can be corrected approximately by observing the magnitude of the shift on each scale, and making appropriate corrections. Plot Vas a function of t, using semilog paper with Von the log scale. Can you predict the shape of this graph? Why is semilog paper useful? From the graph, determine RC and compare with the value obtained from the circuit components. 3 rapid relaxation To measure more rapid relaxation the oscilloscope may be used, as discussed in the introduction. To obtain a repetitive relaxation, we use a voltage source whose output is represented by Fig. 13. This voltage, called a square wave, is

resistance-capacitance circuits

EC-1

FIGURE 13

-

0

T

2T

t

available from your sine-square-wave generator. Controls on the generator permit variation of the amplitude and the frequency f = liT of the square­ wave output. To gain familiarity with the operation of the oscilloscope and the square­ wave generator, connect the square-wave output to the vertical input of the oscilloscope. Adjust the oscilloscope to obtain each of the patterns shown in Fig. 14. Try changing the amplitude and frequency of the generator, and observe what corresponding changes are needed in the oscilloscope controls. FIGURE 14

(a)

(b)

(c)

(d)

(e)

(f)

After familiarizing yourself with the operation of the oscilloscope and the sine-square-wave generator, assemble the circuit shown in Fig. 15. Note that FIGURE 15

R

c

15

ircuits

this circuit differs from that of Fig. 3 in that the capacitor must both charge and discharge through R. What kind of oscilloscope pattern do you expect? With initial values R = 10 kQ, C = 0.1 jlF, observe one period of the charge and discharge of the,capacitor. Using the known frequency of the square wave, the x axis of the oscilloscope can be calibrated directly in time units. Measure the half-life, and from this compute the time constant RC, again using Eq. (II). Note that the square-wave generator has an internal resistance; typical values are:

Range, V

Internal Resistance, Q

0-0.1

52 52 0-220, depending on control setting

0-1.0 0-10

If R is not much larger than this internal resistance, it is important to add the two to obtain the total circuit resistance. You may also wish to try different values of R and C. In particular, what happens when RC is much larger than the period T of the square wave? When it is much smaller?

4

sinusoidal voltage The same experimental arrangement can be used to observe the response of the RC circuit to a sinusoidal input voltage, by simply using the sine-wave output ofthe sine-square-wave generator. The predictions ofEqs. (18) and (19) can be checked. Note that the oscilloscope measures not the charge Q on the capacitor, but the potential across it, which is equal to Q/C. According to Eq. (19), the peak voltage Vc across the capacitor is simply Vc = Vo cos I/J. The ratio Vc/Vo can be measured easily by connecting the vertical input of the oscilloscope alternately to the capacitor and to the sine-wave generator. Choose a frequency for which this ratio is roughly one-half, measure the ratio as accurately as you can on the oscilloscope, and compute the phase­ shift angle. From this and the known frequency of the sine wave, compute the value of RC. Compare with the value obtained from the circuit components. Remember that the wave-generator scale gives the ordinary frequency f, and w = 2nf

5

phase shift The phase shift between generator voltage and capacitor voltage can be measured by several methods. We discuss three possibilities below.

Sweep Synchronization If the sawtooth voltage which provides the horizontal "sweep" is synchro­ nized with the voltage across the capacitor, then the sweep will start at the same point of the sine wave for all frequencies, and no phase shift will be observed. If, however, the sweep can be synchronized with the generator voltage V, then at different frequencies the sweep will start at different points on the capacitor voltage cycle, because of the phase difference between the two. The appearance of the oscilloscope trace displaying the voltage across C will then be as shown in Fig. 16. By measuring the displacement of the sine wave along the horizontal axis, the phase difference can be measured directly. As the driving frequency is increased, the amplitude of the capacitor voltage

resistance-capacitance circuits

[Je

EC-1

FIGURE 16

l? V

ile tle lle C, lal

Q

Lw frequen

4>.

t of

:or. the lse­ the nts. :lnd

l

be

[1ro­ the be 'alor ints the

oss sine ctly. tage

High fr '10 'II\'.

decreases, and the pattern shifts to the right, indicating that the maxima in charge occur later than the maxima in driving voltage, corresponding to a negative phase angle, commonJy referred to as a phase lag. To synchronize the sweep with the driving voltage, we make use of the fact that the sine- and square-wave outputs from the generator are in phase at all frequencies. Connect the square-wave signal to the external synchronization jack, tum the SYNC SELECTOR switch to external, tum the SYNC AMPUTUDE control fully clockwise and reduce the output of the square-wave generator to the lowest level at which positive, consistent synchronization is obtained. Measure the phase angle at several frequencies and compare with the predic­ tions of Eq. (I 8).

i of !lve 19) the : to

i

Inlt'ml ,(\lah fr 'quenc~

Intensity M odulation

As discussed in Experiment EF-l, the electron beam intensity is controlled by the bias voltage on the control grid G 1 of the CRT. It is also possible to use an external signal to vary or modulate the beam intensity by varying this grid voltage. In the jargon of electronics the beam intensity is called the ' z axis," since in effect it provides a third coordinate in the CRT display. If the beam is modulated in phase with the driving voltage, the phase shift between driving voltage and capacitor voltage can again be measured directly on the screen. As before, the phase angle can be compared with the prediction of Eq. (18). To modulate the beam synchronously, connect the square-wave signal into the z-axis jack and adjust the beam intensity and square-wave output for good contrast. With the SYNC SELECTOR switch on internal, the traces should be similar to those shown in Fig. 17. Note that as the frequency is increased the brightened region appears to move to the left, indicating that the driving voltage is ahead of the capacitor voltage in phase.

The advantage of intensity moduJation over sweep synchronization is that the synchronization characteristics of the sawtooth generator are somewhat frequency-dependent and may introduce spurious phase shifts; this problem is avoided by intensity modulation, which produces results independent of the means of triggering the sweep. 17

electric circuits

FIGURE 17

Q

Low

Lissajous Figures

Iligh

lUh:nn die te frcqueuc,

frequc\1ey

I"rCljlll·)"tC' ••

A final way of determining the phase angle 4> between driving and capacitor voltages is to compare these voltages directly on the scope face by using one for horizontal deflection, the other for vertical deflection, as shown in Fig. 18, and not using the sawtooth sweep generator at all. For this setup, the horizontal frequency selector control is turned to external input; this con­ nects the horizontal deflection plates to the horizontal input terminals instead of the sweep generator.

FIGURE 18

R

cJ.

Under these conditions, the motion of the spot on the screen consists of a superposition of two simple harmonic motions in perpendicular directions, with a phase difference 4>. Any such pattern is called a Lissajous figure. Suppose the deflection amplitudes are adjusted so the maximum vertical and horizontal amplitudes are equal. Then when the two voltages are in phase as is the case at low frequencies, the trace should be a straight line inclined at 45°, as in Fig. 19a. At high frequencies, where the phase angle is 90°, the trace should be a circle, as in Fig. 19c. At the intennediate frequencies it is an ellipse. FIGURE 79

18

(a)

(b)

(c)

Low frequency

Intermedia te frequency

High frequency

resistance-capacitance circuits

E C-1

The simplest way to determine the phase shift from the elliptical trace is to set the amplitudes so the maximum vertical and horizontal deflections are equal, and then make the measurements shown in Fig. 20. To show that this FIGURE 20

1

,_____8_,]", ,in"

method gives the correct phase shifts, we note that the x and Y coordinates are given by X

Xl

cos wt

Y = Yl cos (wt

+ 4»

The distance B is just twice the Y displacement at a time when x wt = ± n/2. At this time, Y

Yl cos (

=

0, such as

; + 4» = ±Yl sin 4>

so B 2Yl sin 4>, as shown in the figure. We also have A = 2Yl' so sin 4> can be obtained. Using this method, measure the phase angle as a function of frequency. Plot a graph showing tan 4> as a function of w. From the slope of this graph detennine the RC, and compare with the value obtained directly from the circuit components.

questions 1

U sing the facts that R has units of potential difference per unit current and C has units of charge per unit potential difference, show that RC has units of time.

2 How could a resistance-capacitance combination be connected to produce an output voltage which is the time integral of the input voltage? 3 How might the internal resistance ofthe square-wave generator be measured?

4 When the RC circuit is driven by a sinusoidal voltage whose frequency w is equal to 1/RC, what is the phase shift? The ratio of capacitor voltage to driving voltage? 5 Starting with Eqs. (18) and (19), derive an expression for the maximum

charge Qo which does not contain 4> but expresses Qo as a function of w. 19

:il

electric circuits

6 In Fig. 19a, why is the line inclined to the left, not to the right? 7 In Fig. 20, does the spot trace out the ellipse in a clockwise or counterclock­ wise direction? 8 What would happen if in the Lissajous figure setup the beam intensity was also modulated with the square-wave output of the generator? 9 Suppose the vertical and horizontal inputs of the scope are both sinusoidal with the same amplitude, but the vertical input has a frequency exactly double that of the horizontal input. Sketch the Lissajous figure which results.

20

experiment

E C -2

resistance-inductance circuits

introduction In Experiment EC-I we studied the behavior of circuits containing a resistor and a capacitor in series. We observed the exponential decay of charge on a capacitor discharging through a resistor, and we studied the response of this circuit to a sinusoidal applied voltage. In this experiment we conduct a similar study of a circuit containing a resistor and an inductor. We shall find many similarities, as well as some important differences, with the RC circuit. FIGURE 21

R

L

We consider first the circuit of Fig. 21. The battery potential is Vo, and if the inductor has negligible resistance, a steady current 10 flows in the circuit" given simply by 10

= Vo R

(21)

At a certain time, say t = 0, we flip the switch, removing the battery from the loop. What happens? First, the current cannot drop instantaneously to zero; the voltage across L is proportional to dl/dt, and this would be infinitely great if the current were to change discontinuously. Thus the current must die out gradually, and we represent its time variation with the notation I(t); i.e., I is a function of t. To find what this function is, we proceed just as with the RC circuit, applying Kirchhoff's voltage law to the RL loop. The voltage drop across R is just IR, and that across L is L dl/dt, so the loop equation is (22) The solution of this equation must be a function whose time derivative is equal to - R/L times the function itself, and it must equal 10 at time t = 0. We see that the function

let) =

10e-(RIL)t

(23) 21

electric circuits

satisfies these requirements; in fact, we can show that this is the only function which satisfies them. Following the outline of Experiment EC-I, we note that in this case the characteristic time, or time constant, is given by L/R; after a time equal to L/R the current has dropped to lie of its initial value. Similarly, the half-life TI12 defined in Experiment EC-I is given by (In 2)

L

L

R = 0.693 R

(24)

As with the RC circuit, we may explore electromechanical analogs. Con­ sider again the dashpot situation, but now suppose the spring is removed and that the piston has a non-negligible mass m. In this case the only force on the piston is the viscous force bv, which is to be equated to the mass times acceleration, m dv/dt. Thus the equation of motion (Newton's second law) is

bv+m

dv dt

0

(25)

Comparison of this equation with Eq. (22) shows that they have exactly the same form; v plays the same role as I, b as R, and L as m. The v-I and b-R analogies are the same as for the RC circuit, and we now find that mass m corresponds to inductance L. Pursuing this analogy, we see that if the piston is given an initial velocity Vo and released, the velocity varies with time according to the equation vet)

=

voe-(b/m)1

(26)

with a characteristic decay time equal to m/b and a half-life of TI12 = (In 2) m/b. Returning to the RL circuit, we continue the outline used for the RC circuit by considering the response of the circuit to a sinusoidal applied voltage, as in Fig. 22. Before analyzing this circuit in detail, it is instructive FIGURE 22

L

)

R

to look at it qualitatively. When the frequency Q) is very small, the current changes very slowly; since the potential drop across L is given by L dl/dt, this is very small; hence the circuit should behave almost as though the inductor were a short circuit, in which case the current should be in phase with the voltage, with amplitude 10 = VoiR. Conversely, at very high frequencies the voltage across L may be much larger than that across R, and R may there­ fore be neglected. In this case the maximum current is much less than VoiR, and there is a phase difference between voltage and current. 22

resistance-inductance circuits

EC - 2

The circuit equation for Fig. 22 is derived from Eq. (22) by adding a term for the driving voltage. If this is given by Vet) = Vo cos wt, the circuit equation is RI

dl

+ L dt

= Vo

cos wt

(27)

We look for a solution having the same frequency w as the driving voltage but possibly a phase difference. Thus we try a solution in the form I(t) = 10 cos (wt

+ tP)

(28)

The procedure for evaluating 10 and tP is completely analogous to the cor­ responding calculation in Experiment EC-I, and the details are left for the student. The results, which may be verified by substituting in Eqs. (28) and (27), are tan tP = 10 =

Vo cos tP R

wL R [R2

+

Vo (WL)2]1/2

(29)

Thus our qualitative predictions are borne out. At very low frequencies (wherewL « R), tP is nearly zero and 10 is nearly equal to VolR,justas though the inductor were short-circuited. At very high frequencies (wL» R) tP approaches -n12 and 10 approaches Vo/wL, just as though the resistor were short-circuited. At intermediate frequencies the current always lags the voltage in phase by an angle between zero and -nI2. The quantity [R2 + (wL)2] 1/2 is usually called the impedance of the circuit, and is denoted by Z. Thus at any frequency we have 10 = Vo/Z. Another useful conclusion to be drawn is that at any frequency, if the same current flows through Rand L, as in the present situation, the voltage across L always leads that across R by a quarter-cycle (nI2). At very low frequencies, L becomes a short circuit; at very high frequencies, an open circuit.

experiment ,

exponential growth It is impractical to observe the exponential decay of current in an RL circuit with an ordinary voltmeter as we observed charge relaxation for the RC circuit because an excessively large L would be needed to obtain a sufficiently

long time constant. We may however use a setup analogous to that shown in Fig. 15. Such an arrangement is shown in Fig. 23. Note that this situation corresponds to exponential decay of current in the RL circuit, or to ex­ ponential growth, in which the current starts from zero and grows to a final value determined by the square-wave amplitude Vo. The time constant should be the same in both cases. (Why?) With initial values R 1 kn and 25 mH, observe one period of ex­ ponential growth of current in each direction. Using the known frequency of the square wave, the x axis of the scope can be calibrated directly in time units. Measure the half-life; from this compute the time constant LIR. To 23

electric circuits

compare this value with the value computed directly from the values of L and R, it is important to note that R includes not only the circuit resistor but also the internal resistance of the square-wave generator, as discussed in Experiment EC-1.

FIGURE 23

L

R

You may wish to try different values ofRand L. In particular, what happens when L/R is much larger than the period T of the square wave? When it is much smaller? 2 sinusoidal response The same experimental arrangement can be used to observe the response of the RL circuit to a sinusoidal input voltage, by using the sine-wave output of the generator. The predictions of Eqs. (29) can be checked. The oscillo­ scope measures the voltage across R. From this the current is easily obtained by using Ohm's law: 10 = VR / R. Alternatively, this relation can be combined with Eqs. (29) to eliminate 10 and obtain

VR Vo

R [R2

+ (WL)2]1/2

The voltage ratio VR/Vo can be measured easily by connecting the vertical scope input alternately to R and to the sine-wave generator. Choose a frequency for which this ratio is roughly one-half, measure the ratio as accurately as you can, and compute the phase-shift angle 4>. From this and the known frequency of the sine wave, compute the value of L/R. Remember that the generator frequency scale gives the ordinary frequency f, and that w 2nf 3 phase shift The phase shift between generator voltage and circuit current (or resistor voltage) can be measured by any of the methods discussed in Experiments EI-4 and EC-I. Measure phase angle at several frequencies above and below the frequency at which VR/Vo = t. Plot a graph showing tan 4> as a function of w. From the slope of this graph determine L/R and compare with the value computed from the circuit component values. 24

resistance-inductance circuits

EC-2

questions 1 Show that the quantity LIR has units of time. 2 How could an RL combination be connected to produce an output voltage which is the time derivative of the input voltage? 3 When the RLcircuit is driven by a sinusoidal voltage with frequencyw = RIL, what is the phase shift? The ratio of inductor voltage to driving voltage? 4 Show that taking the time derivative of a sinusoidal function [such as cos (wt + 4>)] always has the effect of increasing its phase by n12. 5 If the internal series resistance of an inductor is not negligible, how will this change the relative phase of voltage and current for the inductor? 6 If the driving voltage in Eq. (27) were given by Vo sin wt instead of Vo cos wt, how would the subsequent analysis be changed? Would the results given in Eqs. (29) be the same or different? Explain. 7

In the circuit of Fig. 21, suppose the battery is completely disconnected, and then at time t = 0 it is connected; derive an expression for the current as a function of time. Hint: The difference between the final current 10 and the instantaneous value l(t) decays exponentially with time constant LIR.

experiment

E C-3

LRC circuits and oscillations

introduction In Experiment EC-I we studied the exponential discharge of a capacitor through a resistor and the response of the resistor-capacitor system to a sinusoidal driving voltage. We observed that the behavior of this system is analogous to that ofa mechanical system consisting ofa spring and a dashpot, such as a door-closer mechanism. In this experiment we study an electric circuit which is the electrical analog of a harmonic oscillator. To introduce basic ideas, we consider first the circuit shown in Fig. 24, which is very similar to the circuit of Fig. 2 in Experiment FIGURE 24

1 -=-vo

2

c

)

+ Q

L

EC-I except that the resistor has been replaced by an inductor and the battery is connected differently. Suppose the capacitor is given an initial charge Q 0 by closing switch I momentarily, and then at time t = 0 switch 2 is closed. The capacitor begins to discharge through the inductor. However, unlike the case of the RC circuit, the current cannot change instantaneously, since the voltage across the inductor is given by L dI/dt. Instead, the rate of change of current is determined by the condition that the instantaneous potential across the capacitor must be the same as that across the inductor. Defining the direction of current flow as in Fig. 24, we have the relations dQ dt

1= - -

and

Q

C

L dI dt

(30)

Combining these, we find d 2Q L dt 2

Q

C

(31)

This equation has exactly the same form as the equation of motion of a harmonic oscillator with a mass m and a force constant k, for which Newton's second law gives us the equation

-kx

(32) 27

electric circuits

We see that the reciprocal of the capacitance I/C plays the role of the force constant k of the spring, just as in Experiment Ee-l, and the inductance L plays the role of the mass m in the mechanical system. We know that the motion of the harmonic oscillator when given an initial displacement Xo is described by the equation x =

Xo

cos OJot

where the angular frequency OJo (equal to 21if, wherefis the number ofhertz or cycles per second) is given by'

_ (k)1/2

OJo -

(33)

­

m

Pursuing the analogy, we see that the charge on the capacitor also oscillates with time according to the equation Qo cos OJot

Q

with angular frequency

OJo

given by

(34)

(LC)1/2

In the harmonic oscillator, the energy is transformed from potential to kinetic and back again during the motion. At the points of maximum dis­ placement and zero velocity, the energy is entirely potential; at the points of zero displacement and maximum velocity, it is entirely kinetic. Similarly, in the LC circuit at times of maximum capacitor charge and zero current the energy is entirely stored in the capacitor; at times ofzero charge and maximum current it is stored in the magnetic field of the inductor. Thus the electric field energy of the capacitor is analogous to potential energy, the magnetic field energy of the inductor to kinetic energy. These analogies are shown in more detail in Figs. 25 and 26. At this point it does not take very much imagination to see that the electrical analog of a damped harmonic oscillator is an electrical circuit con­ taining a resistance, an inductance, and a capacitance. For the harmonic oscillator Eq. (32) has to be modified by the addition of a term - b dxldt corresponding to the damping force, assumed to be proportional to velocity and in the opposite direction. Thus, the differential equation for the motion of the damped harmonic oscillator is

d 2x m dt 2

dx

+ b dt + kx

0

(35)

Similarly, for the circuit shown in Fig. 27, Kirchhoff's loop law gives the equation

Q

IR

C

=

0

which may be rewritten, in terms of Q (using 1= -dQldt) as d 2Q L dt 2

+

dQ

R

dt +

Q

C = 0

(36)

which has exactly the same form as Eq. (35), as predicted. As before, L is analogous to m, I/C to k, and the resistance R to the damping constant b. Another aspect of the analogy between the damped harmonic oscillator 28

EC-3

LRe circuits and oscillations

FIGURE 25

(b)

(a)

L

x=o

I

m~WdG

s

c :s

Consider a mas pring ...

)

on the end of a

(c)

and consider, also, an LC circuit.

( d)

L

f

1 +QO ~

~-

~f

l~

[ll ~c

~ Potential en rgy

e ic

h =T

Potential energy =

2

r

G::~. , fOOOOoolO . ,

1e

"l

6)

(f 1

Xj o 1./ --~

~.

t

;;0

charging the capacitor.

Stretching the spring is like ...

(e )



=0

~ '----..

~

Releasing the spring is like ...

\ S \

~

t

=0 c

closing the switch.

is b.

pr 29

electric circuits

and the LRC circuit is seen by considering energy relations in the two systems. As already observed, the total mechanical energy of an undamped harmonic oscillator is constant; the effect of the damping force is to continuously FIGURE 26

t=O

KE

LC circuit

Mass- spring system

Simple pendulum

PE

~

IttO\M v=O

8=0

t

=:w

v

' l

tooooooooooooag

M

,

,

\

~

,

tooomooOOOO1g=­

..

\

,'-\

::.:::.--",.­

f)=0 _~~~._~"

M

0=-0

x

=0

v ==

v""",

Q == 0

C

1= - Imax

____ .~~. __~ ___ ~"~~m.a~x+ __ "_...~____.....~.__ ~_~_~~_~_._~~ __~. ___ ._~ _____~ __~ ___•__ +_..__

.____.__

~_

~

t=

---'

~--I=III-'-: C

t

= l; v=O

8= ­

cD _I -Qo

-

C

+

1=0

I

t

=

57r 4w

[

'.~'

i

C

-1­

t --

37r 2w

f)=0

t=

x= 0

v

=

ti max

I'TI ----' ~--IIl--l

Q= 0

C

1= lmax

77r 4w

I :' C

30

•

I

•

LRe circuits and oscillations

EC - 3

L

FIGURE 27

c

R

+Q

c

-Q

decrease the energy, since at each instant its direction is opposite to that ofthe velocity and it therefore always does negative work on the system. Similarly, the total energy of an LC circuit without resistance is constant; the inductor and the capacitor store energy but do not remove electrical energy from the circuit. The addition of the resistance provides a means for the system to lose energy through /2 R power loss, which continuously decreases the electrical energy in the circuit, converting it into heat in the resistor. Of course, a harmonic oscillator which is completely undamped is an idealization which cannot be realized. For example, in experiments with a linear air track one finds that the viscosity of the air layer which supports the glider provides a small but not completely negligible damping force which is approximately proportional to velocity. In the same way, an LC circuit with no resistance is an idealization. Even if no resistor is included in the circuit, the resistance of the inductor coil wire and the connecting wires is never completely negligible. Experience with harmonic oscillators (either on the air track or with a simple pendulum) also shows that the loss of energy due to the damping force is accompanied by a steady decrease in the amplitude of the oscillations, so that successive displacements from equilibrium become smaller and smaller, as shown in Fig. 28. By analogy, we expect the electrical oscillations in the FIGURE 28

'Q

• Or--r----~----~---.~--~~--~----~----~----

/'

./

./

LRC circuit to decrease, the peak charge on the capacitor for any cycle being somewhat smaller than that of the previous cycle. This is just what is meant by the term "damped oscillation." 31

electric circuits

How rapidly the oscillations are damped depends, of course, on the magni­ tude of the damping constant b or the resistor R; a larger value of b or R causes a more rapid decay of oscillations. This relationship can be explored in detail; we discuss here two different approaches to the problem, one an approximate analysis using energy considerations, the other making use of the general solutions of Eqs. (35) and (36). Considering the energy approach first, we ask the following question: If the maximum displacement (amplitude) for a given cycle is xo, how much

. energy does the system lose during that cycle? The instantaneous rate of loss

of energy is the rate of doing work against the damping force, which is simply

the magnitude of the force (bv) times the velocity v, or bv 2. This quantity

varies during the cycle, but the total energy lost is still given approximately

by the average rate of energy loss during the cycle (the average value of bv 2)

multiplied by the time required for a cycle, which is 1

T=f

21t OJ

m)112 21t ( k

(37)

To find the average value of v2 , we note that the average kinetic energy 2 )av for a harmonic oscillator is equal to its average potential energy 2 )av. so each of these quantities must equal haifthe total energy E. Thus we have tm(v 2 )av = tEo The average rate of loss of energy is then


bE

m

(38)

and the energy loss during one cycle is

AE

=

-(!E)(~) =

b -21t (km)1/2 E

(39)

Now dE/dt is not constant over a cycle, but is greatest when v is greatest, and zero when v is zero. If we ignore this varhition and consider how the energy decreases on the average, we see that Eq. (38) is a differential equation for E and that its solution gives the energy as a function of time. The form of this equation and its solution are familiar from the study of the RC circuit, and we see that the solution is (40) That is, the energy of the oscillator decreases exponentially, with a character­ istic relaxation time given by (m/b). Just as in the harmonic-oscillator experiment (Experiment M-5), it is useful to introduce a constant called the quality factor, defined as 21t times the ratio of maximum energy stored in the system to the energy dissipated in one cycle. This constant was called Q in Experiment M-5; here, to avoid confusion with the electric charge Q we denote it as QF. An expression for QF is easily obtained from Eq. (39): 21tE mOJ QF = AE = b

=

(mk)1/2 b

(41)

Having found how the energy of the system decreases with time, we may now ask how the amplitude, which is often more directly observable, decreases with time. Since E at any time is proportional to the square of the amplitude, the variation of Xo with time must be given by a function which is the square 32

LRe circuits and oscillations

E C-3

root of the function describing the time variation of E, that is, by a function of the form

Specifically, the amplitude must be given by (42)

where Xo is the initial amplitude at time t = 0. Thus we see that the relaxation time for the amplitude of the oscillations (the time required for the amplitude to drop to l/e of its original value) is 1"=

)

y 'f s

2m b

(43)

Just as in Experiment EC-I, we can define a half-life Tl!2' during which the amplitude drops to half its original value, and this is given by Tl/2 =

1"

,

In 2

=

(In 2) 2m

b

=

1.386m

(44)

b

In view of the analogies between the damped harmonic oscillator and the LRC circuit, and especially in view of the identical form of the governing differential equations, Eqs. (35) and (36), respectively, it should be clear that this entire discussion can be applied to the LRC circuit by simply making the

appropriate symbol substitutions. Replacing m by L, b by R, and k by I/C, we obtain the following results: ~)

t, le

m

t, 0)

Wo =

(LC)1/2

1"=

2L R

QF= -I (L)1/2 -

R C

(45)

We also note that QF = wo1"/2. The results obtained above can be obtained without resorting to approxi­ mations by making use of the general solutions of Eq. (35) or (36); these solutions also exhibit other interesting characteristics of the motion. Without entering into the details of how to solve such differential equations, we simply state that the most general solution of Eq. (35) is Q

Qoe- t / t cos [(w o2 ­

1"\y/2 t + 4>

J

(46)

~r-

'ul jo

Ie. th ily

~l ) )w

ses :ie, 'Jre

with Wo and 1" given by Eqs. (34) and (44), respectively. It can be verified that Eq. (46) is indeed a solution of Eq. (36), by calculating the appropriate derivatives and substituting, a straightforward, but rather laborious, pro­ cedure. Equation (46) is similar to the prediction of our approximate analysis, a sinusoidal oscillation with an exponentially decreasing amplitude with relaxation time 1", but there is an important difference. The angular frequency is given not by Wo but by (wo 2 - 1/1"2)1/2, a quantity always smaller than the undamped frequency Woo In the limit when b or R -+ 0, this becomes equal to wo, but when damping is significant, the frequency is always smaller, and the oscillations slower, than without damping. Furthermore, Eq. (46) predicts that when the damping becomes great enough so that W1" o = t, the frequency becomes zero; then oscillatory motion no longer occurs, and the decay becomes purely exponential. The 33

electric circuits

condition Wot = 1 is known as critical damping; in tenus of the system parameters, the condition for critical damping is

1/2

2(mk) -'----'-- = I

b

(47)

for the damped harmonic oscillator, or

~(~)1/2

(48)

for the LRC circuit. Stated in other terms, critical damping occurs when the time constant t of the expo.nential decay factor becomes smaller than To/2n, where To is the period the system would have in the absence of damping. With less damping than this, oscillatory motion occurs and the system is said to be underdamped; with more, the motion is exponential and is said to be overdamped. For critical damping, QF t.

RESPONSE TO SINUSOIDAL DRIVING FORCE

Just as with the RC circuit studied in Experiment Ee-I, we can extend our study ofthe LRCcircuit to its response to a sinusoidal driving voltage. In fact, many of the most important practical applications of the LRC circuit make use of its frequency response characteristics. We consider the circuit of Fig. 29, noting that this circuit is identical to that of Fig. 27 except for the addition of a sinusoidal voltage source whose potential is given by (49) v = Vo cos wt where the angular frequency w is in general not equal to the characteristic frequency Wo = 1/(LC)1/2 of the circuit, but is determined by the character­ istics of the generator producing this voltage.

FIGURE 29

In applying Kirchhoff's loop law to the circuit of Fig. 29, the only change from Eq. (36) is the addition of a term Vo cos wt, with proper regard for the signs of the various potential differences. The governing differential equation is now d 2Q

L dt 2

dQ

+ Rdi +

Q C = Vocoswt

(50)

The variation with time of the capacitor charge Q is now described by the function which is the solution ofEq. (50). In finding the solution, we proceed just as with the RC circuit of Experiment EG-l ; we assume that the solution is sinusoidal with the same frequency as that of the driving voltage, but with a phase difference. That is,

Q = Qo cos (wt 34

+ 4»

(51)

EC-3

LRe circuits and oscillations

We then substitute this expression and the appropriate derivatives into Eq. (50) to find the values Qo and 4> must have for it to be a solution. As before, we expand the functions sin (rot + 4» and cos (rot + 4» using the sum formulas, and then collect terms in sin rot and cos rot, insisting that the coefficients must separately vanish for the same reasons as in Experiment EC-l. Carrying out this program, we have

Qoro 2 L cos (rot

+

QoroR sin (rot

4»

+

4»

+ ~ cos (rot + 4» = Vo cos rot

Expanding and factoring,

Qo [(

~-

Lro 2 ) (cos rot cos 4> - sin rot sin 4» - roR (sin rot cos 4>

+ cos rot sin 4»J =

Vo cos rot

Equating the coefficients of cos rot and sin rot, respectively, we find Qo [

(~

Lro

2

)

= Vo

(52a)

RrocOS4>J=o

(52b)

cos 4> - Rro sin 4>J

Qo[-(~-Lro2)sin4>

r

Rearranging Eq. (52b), tan 4>

=

R roL _ l/roC

(53)

Dividing Eq. (52a) by sin 4>, substituting Eq. (53), and solving for Qo, We obtain

Qo

Vo . ~ = --sm 'I'

(54)

roR

and Qo may also be expressed in a form which does not contain 4> explicitly: Qo

=

[R2

Vo/ro - 1/roC)2]1/2

(55)

+ (roL

The significance of Eqs. (53) and (55) can be seen from Fig. 30, which shows graphs of Qo and 4> as functions of the frequency ro of the driving FIGURE 30

l!?.. Wo

¢

W Wo

-1f'

35

electric circuits

voltage. We see that at very low frequencies the phase angle is zero, and the charge Q is in phase with the driving voltage, Just as with the RC circuit. At higher frequencies
when the factor wL - l/wC becomes zero, at which point
+

Aw)L

+

(wo

Aw)C

±R

(57)

This equation can be solved exactly for Aw, but it is easier and more in- l formative to make an approximation based on the assumption that the damping is small enough so the curve is fairly sharply peaked and Aw « woo In this case, Aw ----~--

Wo

+ Aw

Wo

as may be verified by long division. Using this result, together with (I/LC)1/2 in Eq. (57) and simplifying, we find simply Aw

=

R

+= -2L

Wo =

(58) 1:

Equation (58) shows that when R is small, Aw is also, and the response curve drops off sharplY on both sides of the peak. Larger values of R give a flatter, broader peak. We also note that the "width" of the response curve is directly related to the quality factor~discussed earlier. Combining Eqs. (45) and (58), we find QF

=

Wo

2Aw

(59)

Thus, the damping characteristics of the circuit are related very directly to its frequency response; high QFmeans small damping, a long relaxation time, and a sharply peaked response cur ve, and conversely. The current lin the circuit is simply the time derivative of Eq. (51). Its phase always leads that of Q by n/2. Hence, at resonance I is in phase with V, and the current through R is the same as though Land C were short-circuited! Hence, Wo is-also the frequency at which maximum power dissipation in R occurs. It is easy to show that at the frequencies w = Wo ± Aw the power dissipation 36

LRe circuits and oscillations

EC-3

in R is halfits maximum value, and these frequencies are sometimes called the half-power points.

experiment 1 oscillations The decaying oscillations in the LRC circuit can be observed using the same technique as used in Experiment EC-1 to observe exponential decay, in which a square-wave generator produces the same effect as a battery switched on and off periodically. The circuit of Fig. 31 is suggested. The scope measures FIGURE 31

.001 p,F

the voltage across C; the sweep is synchronized to the square wave, and so the horizontal (time) axis can be calibrated, using the known frequency of the square wave. Measure the frequency and half-life of the decaying oscillations; compute (00 and r and compare with the predictions of Eqs. (44) and (45). In this cir­ cuit R represents the resistance of the inductor, which can be measured with the vacuum-tube volt-ohmmeter, and the internal resistance of the square­ wave generator, discussed in Experiment EC-I. 2 critical damping To study critical damping and overdamping, add to the fixed resistance R a 25-kQ variable resistor. Starting with a small value of R, increase R until critical damping is reached. Measure R with the ohmmeter, and compare the critical-damping resistance with the prediction of Eq. (48). What happens when the resistance is larger than the critical-damping value? 3 frequency response To study the frequency response to a sinusoidal driving voltage, the same circuit may be used with the sine-wave generator (Fig. 32). Using the intensity­ modulation technique discussed in Experiment EC-1, measure the amplitude and phase of Q as functions of frequency, and graph the data. Compare the observed resonance frequency with the predicted value. 37

electric circuits

FIGURE 32

.OOlJJ,F

4 quality factor

Find the "half-power" frequencies, determine OJ for the half-power points, and compute the QF of the circuit; compare with the value of QF determined from decaying oscillation. r

questions 1 For the LC circuit with no resistance, show that when the capacitor charge is

maximum the current is zero, and conversely. 2 What is the quality factor of your LRC circuit when the external resistance is zero and R includes only the resistance of the inductor windings? 3 By taking the derivative of Eq. (55) with respect to

OJ and setting the result equal to zero, show that the maximum value of Q 0 occurs exactly at OJ o only in the limit when R is small, but that for larger R the resonance peak occurs at a somewhat smaller frequency. Also show that if R is sufficiently large, the curve has no peak at all, but is a continuously decreasing function of OJ. Find the critical value of R and the corresponding QF. Hint: The calculation is simplified by arranging Eq. (55) so that only OJ2 appears, using a single symbol (say y) for OJ2, and differentiating the quantity 1/Q0 2 with respect to y.

4 With a sinusoidal driving voltage, what is the phase relationship between the

voltage across C and that across L? Between the voltage across C and that across R? 5 Show that at the resonance frequency, the voltages across Land C are equal in magnitude but a half-cycle out of phase, so the total potential difference

across Land C is zero. 6 What problems would be encountered in designing an LC circuit whose resonance frequency is (a) 10- 2 Hz; (b) 10 10 Hz? 7 Show that the resonance amplitude Vc of the voltage across the capacitor may be much larger than the amplitude Vo of the driving voltage, and that it

is given by Vc = (QF)Vo' 38

experiment

EC-4

coupled oscillators

introduction In this experiment we study the behavior of two oscillating systems such as mechanical harmonic oscillators or LC circuits when there is an interaction between the two systems. Although the experimental study will deal with electric circuits, we introduce the basic ideas in the more familiar context of mechanical systems. We consider the system shown in Fig. 33. The masses move on a horizontal FIGURE 33

straight line, and without friction, as on a linear air track. This system was studied qualitatively in Experiments M-4 and M-S. If the spring k' were not present, we would have two identical harmonic oscillators, each of which could vibrate with any amplitude and with angular frequency (60)

The spring k' permits the two oscillators to interact, and changes the behavior of the composite system in an interesting way. To analyze this system in more detail, we write an equation of motion F = ma) for each mass and then try to find solutions of the resulting set of equations. The coordinates Xl and X 2 describe the displacements of the masses from equilibrium. The forces on the first mass are kXI due to the left spring and k'(X2 Xl) from the center spring. Note that when Xl = X2, the center spring is neither stretched nor compressed, so its force is zero. Thus for the first mass, we find

a:::

m

d 2x I

-kXI

+ k' (X 2 -

Xl)

(61a)

Similarly, the equation of motion for the second mass is found to be (61b)

m

Rearranging these two equations and using the abbreviation Xl = d2XI/dt2, mX 1

(k

t

k')x I

+ k'X2 (62) 39

electric circuits

We notice that the equations for the two masses have exactly the same/orm, with Xl and X2 interchanged. This symmetry, of course, results from the symmetry of the physical situation. In finding solutions for these equations we are guided by physical intuition. One obvious possible motion occurs when the two masses oscillate in phase and with the same amplitude; in this case the spring k' never exerts any force, and both masses move with simple harmonic motion with frequency (0 = (k/m) 1/2. Are there any other possible motions in which both masses move sinusoidally with the same frequency [not necessarily equal to (k/m )1/2] 1 To answer this question we try a pair of solutions in the form Xl

Al cos lOt

X2

A2 cos (Ot

(63)

We substitute these trial solutions, with appropriate derivatives, into Eqs. (62) and find whether or not they are solutions, Le., whether or Qot they satisfy the differential equations. Substituting them and dividing out the common factor cos lOt, we find -mw2AI -m(0 2A 2

=

-kAl

+ k'(A 2 -

Ai)

-kA2

k'tA2

A 1)

Grouping the terms in A I and A2 in each equation,

(64)

Thus Eqs. (63) are solutions of Eqs. (62), provided the amplitudes A 1 and A2 satisfy Eqs. (64). Equations (64) form a set of simultaneous homogeneous linear equations. Now homogeneous linear equations always have the property that there are either no solutions at all or else infinitely many. For solutions to exist, the equations must be consistent. One way to test for consistency is to solve each equation for Al and compare the resulting expressions. We find k'/m

-wI _(k .+ k')/m A2 w

2

(k

+ k')/m

A2

(65)

The equations are consistent only if the coefficients of A2 in Eqs. (65) are equal. Clearly, whether or not this condition is satisfied depends on the value of (0. For it to be satisfied, we must have k'/m 2 w - (k + k')/m

which may be rewritten

40

(02 -

(k

+ k')/m

k'/m

coupled oscillators

E C -4

Taking the square root, w2

_

k

+ k'

k' +­ -m

m k

k

m

+ 2k' m

and finally (66)

In this last step we have discarded the negative roots; since cos ( wt) = cos wt, the negative roots give no additional solutions. Introducing the notation w + and w _ for the larger and smaller of the roots, we write them as m

Thus the two masses can move sinusoidally with the same frequency, but only if w is one of the values given by Eqs. (66). There must also be a definite amplitude relation, as shown by substituting the expressions for w+ and w_ in tum into either of Eqs. (65). Choosing the first equation, and substituting w+, we find (k

+

k'/m 2k')/m - (k

+ k')/m A2

A similar calculation for w_ shows that for that root Al

=

A2

We are now ready to interpret these results. The root w_ corresponds to the case discussed previously, in which the two masses vibrate' with the same amplitude, and with the same frequency as that of the uncoupled oscillators. In this case the coupling spring k' has no effect. In the other case, since the amplitudes have opposite signs, the motions are equal in amplitude but a half-cycle out ofphase, and the frequency w + is greater than for the uncoupled systems, since in this case the coupling spring acts as an additional restoring force. Since the equations of motion are linear differential equations, any sum of solutions is also a solution. Incorporating the relations between ampli­ tudes just obtained, we write the most general solution, which includes all physically possible motions of the system, as

+ Bcos w+t

Xl

=

X2

= A cosw_t - Bcosw+t

A cos w_t

(67)

where A and B are arbitrary constants depending on the initial conditions. Each individual motion with a single frequency is called a normal mode; in general, the motion is a combination of normal-mode motions, but if one of the amplitudes A or B happens to be zero as a result of particular initial conditions, then the resulting single-frequency motion of the two masses is called a normal-mode motion. The two normal modes are illustrated in Fig. 34. One case of particular interest, especially when the coupling spring k' is much weaker than the other two (i.e., when k' « k) occurs when the ampli­ tudes A and B are equal. In that case Eqs. (67) can be rewritten in an interest­ 41

electric circuits

FIGURE 34

ing and instructive form. The normal-mode frequencies w+ and w_ are very nearly equal, and it is useful to use the notation w+ Wo

=

+

w

2

w+ - W ~w = - - - -

2

or

(68)

so that Wo is the average of the normal-mode frequencies, and ~w is the amount by which each one dU!ers from the average. Clearly, if k' « k, then ~w « woo We now introduce this notation, together with the assumption A = B, into Eqs. (67) and expand each cosine using the formula cos (a

±

b) = cos a cos b

±

sin a sin b

to obtain Xl =

A cos (wo - ~w)t + A cos (wo

+ !lw)t

= A (cos wot cos ~w t + sin wot sin ~w t + cos wot cos !lw t - sin w ot sin !lw t) The expression for X2

X2

= A cos (wo - !lm)t - A cos (w o + !lw)t

is expanded the same way. Rewriting the two results for Xl

= [2A cos (!lw t)] cos wot

x2

= [2A sin (!lw t)] sin w ot

XI

and

X2,

(69)

The motion is not a simple sinusoidal one, since each coordinate varies with time according to the product of two sinusoidal functions. However, one of the functions varies slowly with time, with frequency ~w, whereas the other varies more rapidly with a frequency Wo midway between the two normal-mode frequencies. Thus we can think of these motions as having a frequency 0)0 , but a variable amplitude which goes from zero to 2A and back. This interpretation is suggested by the brackets in Eqs. (69). Furthermore, we see that when Xl has maximum amplitude (i.e., when cos !lw t = ± I) X2 has zero amplitude, and conversely. Initially mass 2 is stationary and mass I is vibrating with amplitude 2A . Th is amplitude decreases and eventually becomes zero after a time such that !lm t = n/2, when mass 2 is vibrating with amplitude 2A. The motion is shown graphically in Fig. 35, which gives graphs of x 1 and x 2 as functions of time. 42

E C -4

coupled oscillators FIGURE 35

\

,f(

,.:

j'\

~

-;:

I'

If.i

.~'

m.~'n

1\\ m ~~M

f,

~ 'II} \f~

~

~j

!J

~

1M

~

~ ~,

ry

m

~ 'nI I~

II

'"\~

18)

;'1

,~

ro, ~

IW

~

~ ~

1M

~

he en

~~

t

~

~

loI, ~

to

ENERGY RELATIONS

We may also look at this situation in terms of the energy relations. At time t = 0 all the energy is in oscillator 1. Because of the coupling provided by spring k', energy is gradually transferred to oscillator 2 until all the energy i stored in oscillator 2, when it begins to be transferred back to oscillator 1 again. The amount of time required for the energy to go from 1 to 2 and back, which we may call the exchange time lex' is given by the relation 11m lex = n. The angular frequency of the cyclic exchange of energy is given by

2n m = - = 211m ex

~9)

ELECTRICAL ANALOG

ies er,

~~

~

(70)

lex

Using the electromechanical analogs already discussed in Experiments EC-l through EC-4, we may derive an electrical analog for two coupled harmonic oscillators. Each mass-spring combination becomes an inductance-capaci­ tance combination, and the coupling spring becomes a coupling capacitor. Specifically, the electrical analog is the circuit shown in Fig. 36. To verify in detail that it is, indeed, the analog of the mechanical system just discussed,

a

"k. e,

FIGURE 36

X2

L

1

By

.ng S

ves

QJ

T'----~c

L

+

Q I- Q2

+

l~--d

Cf

Q2

--(C,

43

electric circuits

we write out the circuit equations, using Kirchhoff's voltage (loop) law twice, once for each loop. The various charges and currents are labeled as shown in the figure. The charge on the coupling capacitor C' is ± (Q 1 - Q 2) and the corresponding potential is ± (Q 1 - Q 2)/C'. With the definitions of the charges and currents indicated in Fig. 36, the currents are given by II = dQtfdt and 12 = dQ2ldt. The potentials across the inductors are given by L dltfdt = - L d 2Q tfdt 2, and similarly for 12, The circuit equations are _ Q1 C

or

_

2 L d Q1 dt 2

+

Q2

-

Q1 = 0

C

2 -L d Q2 _ Q2 _ Q2 - Ql = 0 dt2 C C 2 L d Ql = _Ql + Q2 - Ql dt 2 C C'

(71)

Comparing these last two equations with Eqs. (61) for the coupled har­ monic oscillators, we see that they are identical in form, with the same electromechanical analogs discovered in the earlier experiments: L +-+ m, l/C +-+ k"Q +-+ x, and 1+-+ v. Thus, everything that has been said above for the coupled harmonic oscillators can be said over again for the coupled LC resonant circuits, including the description of the normal modes, the energy transfer when both modes are present, and the behavior when the two parts of the systems are uncoupled. The case k' = 0 corresponds here to an in­ finitely great coupling capacitance c. Such a capacitor has the property that the potential across it is zero no matter how much charge is added to it. In general for a capacitor V = QIC, so here V is zero for any finite value of Q. Hence C acts as a short circuit, and the two LC circuits become uncoupled. The condition for weak coupling, I1w « Wo, is that C' be much larger than C, which may seem paradoxical until we reflect on the above remarks. The normal-mode frequencies are

w_

W+

=

(_1 )1/2 LC

(72a)

= [~(~ ~)JI/2 L C + C

(72b)

As shown in Fig. 37, w_ corresponds to the case in which the two currents are in the same direction, so there is no current in C'; and w + corresponds to FIGURE 37

(a) Even mode

44

(b) Odd mode

coupled oscillators

EC-4

currents in opposite directions. If C' » C, it is useful to expand Eq. (72) in a Taylor series:

_(1)112( 1 + 2 -C)1/2_(1)1/2( C

- 1+

W+ -

-

C'

LC

The average frequency

Wo

LC

is then given approximately by

Wo

and the exchange frequency

=

LC (1 + 2C'C)(1)li2

Welt

(73)

= w+ - w_ is approximately

(74) In addition to studying the normal modes of the system, we can also study its response when a sinusoidal voltage is added to one of the loops. The analysis for this situation is straightforward but will not be given here in detail. The procedure is to add a term Vet) = Vo cos wIt to one or the other of Eqs. (71), dep~nding on where the voltage source is connected. We then look for solutions which have the form X2

A z cos w't

(75)

with the same frequency w' as that of the applied voltage. In this case the resulting equations for the A's are no longer homogeneous, and they can always be solved explicitly for the A's. As might be expected, interesting things happen when the driving-voltage frequency Wi is close to one of the normal­ mode frequencies w+ and w_. At these frequencies the generator can feed energy into the system with maximum effectiveness, just as a boy pushing another boy on a swing transfers energy most efficiently when he pushes with the same frequency as the natural frequency of the swing motion. This syn­ chronizing of the driving frequency with a natural frequency of the system is called resonance; it was studied in a simpler context in Experiment EC-3.

experiment 1

free oscillations To study the behavior of the coupled LC circuits, the circuit shown in Fig. 38 is suggested. The oscillations are initiated by the square-wave generator,just as in Experiment EC-3. Because of the resistance present, the oscillations do not persist indefinitely but are damped out. The 270-Q resistor provides a small voltage drop which may be observed on the scope. The 50-pF variable capacitor is included, rather than a 25-pF capacitor, to "tune" the system so that the uncoupled oscillators have exactly the same frequency, despite the fact that manufacturing tolerances permit considerable variation in two components which have the same nominal value. For C, an initial value of 500 pF is suggested. To balance the two circuits, adjust the variable capacitor until the difference frequency Aw is a minimum. Measure the exchange frequency on the scope and compare with the value computed from Eq. (74). Also measure Wo by 45

electric circuits

25mH

FIGURE 38

25mH

50pF

C'

270g

counting the number of cycies of oscillation corresponding to each cycle of the square wave. Compare with the value computed from Eq. (74). You may wish to try other values of the coupling capacitor C. Suggested alternate values are 200 pF and 0.001 jlF. What happens if C is short-circuited? Ifit is removed completely? 2 forced oscillations To study the response to a sinusoidal driving voltage, the circuit of Fig. 39 is suggested. Holding the generator voltage constant, measure the output as 25mH

FIGURE 39

25mH

50pF

C'

-a function of frequency. You should find two maxima, one at w _ and one at w+. Measure these frequencies for several values of C, and compare with the predicted values computed from Eqs. (72). Compare the observed frequency difference with the exchange frequency determined from the transient behavior. For the w _ mode the phase relation of the currents is as shown in Fig. 37a; hence the net current in C is always zero. For the w + mode the phase is as in Fig. 37b and the current through C' is double the value for either loop. Hence, a resistor placed in series with C' should damp the w+ mode but 46

coupled oscillators

EC-4

not the OL mode. It is of interest to study the effect on the curve of response versus frequency. Which resonance peak is changed? What effect does this resistor have on the transient response? How might you damp the (J) _ mode? Try it!

questions 1 For the coupled harmonic oscillators of Fig. 33, what set of initial conditions could be imposed to obtain the (J)+ mode? Only the (J)_ mode? Are these sets of initial conditions unique, or are there many possibilities for each mode? 2 Why is the exchange frequency

(J).x

equal to 2 !J.(J) and not just !J.(J)?

3 Why should the condition that !J.(J) be a minimum give the best matching of the two LC circuits? 4

In studying the sinusoidal response of the coupled oscillators, how might one damp the (J) _ mode without damping the (J) + mode?

5 Find a mechanical analog for the circuit shown in Fig. 40. This system has L' FIGURE 40

L

c

L

c

three normal-mode frequencies, one of which is zero. To what physical condition does the zero-frequency mode correspond? 6 Find an electrical analog for the mechanical system shown in Fig. 41. How FIGURE 41

many normal modes does it have? Can you predict the nature or frequency of any of them without doing detailed calculations? 7 The inductors used in this experiment are not ideal, but have some resistance in addition to their inductance. How does this change the transient behavior compared to the idealized behavior? The sinusoidal response? 8 For the circuit of Fig. 39, derive an expression for the current in the right-hand capacitor, as a function of driving voltage amplitude and frequency. Use values L, C, and C, and neglect all resistances. 47

experiment

E C-5

periodic structures and transmission lines

introduction In this experiment we extend the study of coupled oscillators undertaken in Experiment EC-4. Instead of having two coupled mass-spring or inductor­ capacitor systems, we now consider an arrangement in which many of these systems are coupled together to form a repetitive or periodic structure. As we shall see, such structures can be used to provide a time delay in pulses trans­ mitted through them, can serve as filters which pass some frequencies and block others, and have a number of other interesting properties. These systems are often called filters, lumped-parameter filters, delay lines, or simply lines. In this discussion the term line will be used often. It is easier to understand the behavior of systems such as this when we can visualize what is happening; hence we begin by discussing a simple mechanical example of a periodic structure, shown in Fig. 42. A row of identical masses FIGURE 42

m 2

m, constrained to move along a straight line without friction, is connected by a series of identical springs k. The two end masses are m12. These are

included so we may consider the system as a series of sections, each consisting of a spring and two half-masses. In addition, the two ends of the system could be connected together to form a cyclic structure. If a mass at one end of the line is given a displacement and then returned to its equilibrium position, a displacement pulse is propagated down the length of the line. When it reaches the far end it does not disappear, but is reflected, initiating a second pulse traveling back along the line. By connecting the end springs to dashpots rather than rigid walls, we can partially or com­ pletely absorb pulses, eliminating the reflections. In the electrical analog of this system, reflections or the prevention of reflections are often of primary importance. To analyze the motion of the system in detail, we proceed just as in Experi­ ment EC-4. We use 1: F ma to write a differential equation of each mass in terms of its displacement Xn from its equilibrium position. If there are N masses in all, then the index n ranges from I to N. The differential equation for the nth mass will involve not only Xn but also X n - 1 and x n+1 because of the effects of the springs. For a typical mass not at the end of the line, we have

=

m xn = -k(xn - x n -

1)

+ k(Xn+l

- xn) = k(x n-

1 -

2xn + xn + 1 ) (76a) 49

electric circuits

The end masses must be treated separately, and we find

tm Xl =

k(x z - Xl)

(76b) (76c)

To find solutions of these equations, we assume that a pulse initiated at one end of the line propagates with constant speed and without change of shape. This assumption is not universally valid, but conditions for its validity will be discussed later. Stating this assumption symbolically, we assume that for a pulse traveling to the right, if a certain mass n undergoes a displacement which varies with time according to the function Xn(I), then the next mass (n + I) undergoes the very same displacement, but at a later time I + T, where T is the time required for the pulse to propagate a "distance" of one section, as shown in Fig. 43. Thus we assume that Xn-l(t) = x.(1

and

X n+

l(t)

xnCl

+

n

-

T)

(77)

FIGURE 43

To simplify these expressions we express the functions xn(t ± T) in terms of Xn and its derivatives at time t, using a Taylor series expansion, as follows: Xn(t

+

x.(t -

n X.(I) + xn(t)T + t 5i. n(t)T2 + n = x.(t) x.(t)T + 1- X.(t)T2 +

(78)

Similar expressions can be written for X n - 1(t ± T) and x n + 1 (t ± T) by simply changing n to (n 1) or (n + 1), respectively, in the above expres­ sions. In the following analysis we shall assume that all except the first three terms in these infinite series are negligibly small. The validity of this approxi­ mation hinges on the assumption that the time taken for the entire pulse to pass a given point is very long compared to T. If this is true, then each x changes relatively very little in a time interval the difference between x.(t) and x.(t + is then small, and the series converges rapidly. As we shall see, this assumption is closely related to the previous assumption that pulses are propagated with a definite speed and without change in shape. Pursuing our study of solutions of Eqs. (76), we now insert Eqs. (77) into Eq. (76a) and then use the series expansions given by Eqs. (78), as follows:

n

-k[xn(t) + .xn(t)T + 1x.(t)T 2 -k[xn(t)T2] 50

2x.(t)

+ xit) -

x.(t)T + 1xn(t)T2]

(79)

periodic structures and transmission lines

EC - 5

On the right side we have dropped all terms containing T3 and higher powers of T; these are presumably much smaller than the terms retained. This equation does not tell us in detail how the masses move; to find that we must know the shape of the initial pulse. But it does tell us that our initial assumption, Eqs. (77), about the nature of the solutions, is consistent with the laws of mechanics. In addition, since Eq. (79) must be an identity in order to conform to these laws, we find that the time T, the "delay per section," is given by

_(m)112 -

T-

k

(80)

We also note that everything that has been said so far is also valid for a pulse propagating to the left rather than the right. In this case the signs in Eq. (77) are reversed and there are corresponding changes in the first forms ofEq. (79), but the final result is the same. Furthermore, since the equations of motion are linear, any sum of solutions is itself a solution. Hence, the motion may be a superposition of pulses traveling in opposite directions. Indeed, when a pulse is reflected from an end, this is just the situation, and we discuss such reflections next.

REFLECTIONS In the absence of damping forces, the equation of motion for the mass m12, at the right end with coordinate X N , is Eq. (76c). However, as mentioned

previously, we may want to provide a means for energy absorption to try to prevent or change the reflected pulse. Hence, we now add a variable damping force -bXN(t) to this mass, leading to the equation of motion (81)

This equation can be used to analyze the reflection ofa pulse at the end of the line. We consider the situation shown in Fig. 44. The peak of a pulse with FIGURE 44

•

unit amplitude passes point x N _ 1 at time t - T and returns after a partial reflection with amplitude B at time t + T. Meanwhile the displacement of X N is a superposition of the incident and reflected pulses, with amplitude A. Because of this superposition, XN-l at time t is related to the values of x N 51

electric circuits

both at time t + T and at time t - T. If there were no reflected pulse, we would have simply

(82) in analogy with Eq. (77). If we considered only the reflected pulse, we would have XN-1(t) =

(~) xN(t -

(83)

T)

The factor l/A in this expression appears for the same reason as the factor A in the previous equation. Finally, recognizing that X N - 1 must actually be the superposition of these two contributions, we solve each of the two above equations for x N -1 (I) and add, obtaining finally the relation

(84)

Thus the relation between XN and x N-1, analogous to Eqs. (77), is AXN- 1(t) = xN(t

+

T)

+ BxN(t

(85)

- T)

To determine the constants A and B, we insist that this geometric relation be consistent with the dynamic equation for the end mass, Eq. (81). Proceed­ ing as before, we use the Taylor series expansion for the terms on the right side of Eq. (85), obtaining AXN_ 1 (t) = (1 + B)xN(t) + (l - B)TXN(t) + t (l + B)T 2 N(t) +

x

(86) Substituting this result in Eq. (81),

1"

2 mX N

=

-kXN

+k

I+B 1 B. A xN + k A TXN

+

kl+BT2" bO x N - XN

2A

(87) If our assumption about the relation between initial and reflected pulses is to be consistent with the equation derived from F = ma, Eq. (87) must be satisfied no matter what the shape of the pulse is. This can be true only ifthe coefficients of each derivative on the two sides are equal; that is, Eq. (87) must be an identity. Equating coefficients, we obtain the relations

k 1

+ B T2 2A

O=k

I-B T-b A

(88)

1+ B 0= - k + k - ­ A

The third equation tells us that A = B + 1, as we might have guessed from the geometry of the situation. This result, combined with the first equation, gives T2 = m/k, which we already knew. Eliminating A from the second equation gives

k~T 1

52

+B

b

or

B = 1 - b/kT 1+

(89)

periodic structures and transmission lines

EC - 5

Finally, eliminating T, we obtain B =

I

bl(mk) 1/2

-----:-=

I

+ bl(mk)li2

A

=

B

+

1

(90)

Thus, the reflection coefficient B is given in terms of the basic circuit param­ eters. Several particular cases are of interest. If the damping constant b is zero, we find B = I, and the pulse is reflected completely. In this case A = 2, and the displacement of the end mass is twice as great as the incoming pulse amplitude. Second, we note that when b = kT = (mk)l/2, B is zero. In this case the pulse is completely absorbed by the damping, there is no reflected pulse, and the peak displacement of the end mass is the same as the others. Finally, when b is very large, the reflection coefficient B approaches the value I, and A becomes zero. That is, the reflected pulse is inverted, and the end mass does not move at all. Hence we see that the reflection depends critically on the magnitude of b.

ELECTRICAL ANALOG We are now ready to construct the electrical analog of this system, guided

by the analogies between m and L, k and I/C, band R, already explored. From these it is clear that a reasonable guess for an analog is the circuit of Fig. 45. We now establish that this is indeed a valid analog. FIGURE 45

Using the notation indicated in the figure, we apply Kirchhoff's loop law to the loop containing capacitors Qn-l and Qn. We find Qn-l _ Qn =

C

C

Li n

(91)

Similarly, for the loop containing Qn and Q,,+I' Qn _ Q,,+l _ C C -

Li

n+l

(92)

Kirchhoff's current law applied to the junction above Q" gives or

(93)

Solving the loop equations for the current derivatives and substituting in Eq. (93), we obtain I C(Q"-l

(94) 53

electric circuits

Comparison of this equation with Eq. (76a) shows that they have exactly the same form, showing the validity of the analog. To complete the investigation of the electromechanical analog we would need to obtain the equations for the ends and compare them with Eqs. (76c) and (81). These calculations are not needed for our later work, and they will not be given here. The student is invited to work them out for himself. The most significant results of the mechanical analysis above are Eqs. (80) and (90). We can now translate these into the electrical analog language, obtaining T B

=

I I

(LC)1/2 R/(L/C) 1/2

+ R/(L/C)1/2

(95) (96)

In particular, the condition for complete absorption of the pulse at the end terminated by the resistor R is R

(~y/2

(97)

The quantity (L/C) 1/2, which obviously must have units of impedance, is called the characteristic impedance of the system; thus, we have the important result that there is no reflection from the end of the line when, and only when, the terminating resistance is equal to the characteristic impedance. I rather than n = N) have Reflections from the other end of the line (n not been discussed, but a little thought shows that there is nothing inherently directional about this system. A reflected pulse traveling from right to left in Fig. 45 can undergo a second reflection at the left end of the line, with the reflected amplitude determined by the effective impedance of the circuit to which the line is connected. This may be the internal impedance of the source feeding pulses into the line, or that in combination with whatever additional resistors are connected. Thus, in general, if the ends of the line are not terminated in a resistance equal to the characteristic impedance of the line, multiple reflections occur, with an attenuation at each reflection determined by Eq. (96).

DISPERSION

Now we return to the question of whether a pulse always travels through the line with constant speed and without change of shape, independent of what that shape is. This question is most easily discussed in terms of the propagation of sinusoidal waves. A pulse can always be represented as a sum of sinusoidal components, using the language of Fourier analysis. If all sinusoidal components travel with the same speed, then we expect all parts of the pulse to travel at this same speed, which is just what we have assumed. But if it turns out that the speed of the sinusoidal waves depends on their frequency, then it will not be true, in general, that a pulse propagates with a definite speed. Thus the program is to assume a sinusoidal solution of Eq. (94), with constant phase differences between sections, to test whether or not it is a solution, and to evaluate the time delay T per section. The trial solution is (98)

The nth charge is assumed to be delayed by a time nT with respect to the 54

periodic structures and transmission lines

EC-5

first one, and Q 0 is an amplitude constant, the same for all charges. Inserting this in Eq. (94), -LCw 2Qo cos w(t - nT) = Qo cos wet - (n + I)T] - 2Qo cos w(t - nT)

+

Q 0 cos wet - (n - I)T]

(99)

We expand the cosine functions using the sum formulas to obtain factors which are cosines of the separate quantities wT and w(t - nT), and divide out the common factor Qo cos w(t - nT), as follows: - LCw 2Qo cos w(t - nT) = Qo cos w(t - nT) cos wT

+

Qo sin w(t - nT) sin wT

- 2Qo cos w(t - nT)

+

Qo cos w(t - nT) cos wT

- Qo sin w(t - nT) sin wT LC w 2 = 2(1 - cos wT)

.

2

(100)

wT

= 4 SIn­

2

The last form is obtained using a half-angle formula. Thus there are solutions in the form of Eq. (98) in which each charge oscillates sinusoidally with the same frequency, but with a constant phase difference between sections, given by ¢ = wT. However, the time delay T per section is not given by Eq. (95) except as an approximation, and in general it is not independent of frequency. If the period corresponding to w is long compared with the delay T per section, then the phase difference between adjacent sections is very small, since wT is very small. In this case the sine function in Eq. (100) may be expanded in a power series. If only the first term of the series is kept, we obtain or

T = (LC)1/2

Thus in the low-frequency limit, and only then, is T given by Eq. (95) independent of frequency. Conversely, since the sine function can never exceed unity, there is an upper limit on the frequency, given by or

2

w = (LC)1/2

(101)

If the frequency is higher than this critical value, there are no sinusoidal solutions of Eq. (94) in the form of Eq. (98). Fig. 46 shows a graph of T as a function of w, obtained from Eq. (100). As the figure shows, T is equal to (LC)1/2, independent of w at small fre­ quencies; but as w increases, T also increases, reaching the value n(LC)1/2/2 at the cutoff frequency. The period corresponding to this frequency is 2n/w or n(LC)1/2, which is equal to twice the delay time per section at the cutoff frequency. Thus, a sinusoidal wave cannot propagate through the structure when its frequency is greater than that corresponding to a period equal to twice the delay per section. It is also amusing to note that the cutoff frequency is equal 55

•

electric circuits

FIGURE 46

T 1r (LC)II2/2

I I

I I ----I-­ I I I

I I I L----------'---:-:-c--- W 2/(LC)1!2

o

to twice the resonant frequency of a simple LC circuit with the same values of Land C as the line. The foregoing discussion shows that a pulse is propagated on the line without distortion only when its various frequency components are small compared to the cutoff frequency. Otherwise some distortion occurs, and if the principal components are above cutoff; the pulse will be greatly distorted and attenuated. The whole phenomenon of distortion and attenuation of the pulse resulting from the dependence of propagation speed on frequency is usually called dispersion. There are many examples of analogous phenomena in other branches of physics involving wave phenomena, including acoustic waves, optics, and quantum mechanics.

TRANSMISSION LINES

FIGURE 47

From Eq. (101) it is clear that the cutoff frequency can be increased by decreasing the Land C of the individual sections of the line. For example, if both Land C are reduced to one-half their former values, the cutoff frequency increases by a factor of 2 while the characteristic impedance is unchanged. This observation suggests the possibility of a line whose inductance and capacitance are distributed continuously along its length; such a line would presumably have no high-frequency cutoff and would be completely without dispersion. Such a system is indeed possible and practical, within certain limits. In principle, any pair of conductors forms such a system, which is called a distributed-parameter line or a transmission line. For example, for a pair of straight parallel conductors, there is a certain capacitance per unit length between the conductors, and also a certain inductance per unit length associat­ ed with the flux change between two elements of the two conductors. These ideas are illustrated in Fig. 47. In practice, lines are often made in the form of

I.....-... L\.x---j

I

I

II L i

56

-T- I I J

coaxial cylinders; the outer conductor then forms an electrostatic shield for the space between conductors, so that the line is unaffected by stray fields from other conductors in the vicinity. Such a line is called a coaxial line, often shortened to coax. Our analysis of the lumped-parameter line can be taken over directly to

periodic structures and transmission lines

EC-5

the distributed-parameter line by simply reinterpreting Land C as the inductance and capacitance per unit length. The characteristic impedance is still given by Eq. (97) and the delay time Tbecomes the delay per unit length. The reciprocal of this quantity is then the actual propagation speed in length per unit time. In the ideal case the line is completely free of dispersion and the propagation speed completely independent of frequency. Real lines never achieve this ideal behavior for two reasons. First, the conductors always have some resistance, which leads to dissipation of energy. Second, for mechanical support part of the space between conductors must be occupied by a dielectric materiaL Dielectric properties are always frequency-dependent, and at high frequencies a dielectric dissipates energy, becoming equivalent in a sense to a shunting resistance. Thus even coaxial lines always have high-frequency cutoffs, but by careful choice of materials this frequency can be raised to 10 10 Hz and higher. It can be shown, using elementary electromagnetic theory, that for a line in the form of coaxial cylinders of inner and outer radii a and b, respectively, the inductance and capacitance per unit length are given by L =

~ln b 2n

a

C

2m; In (bja)

(102)

Thus the propagation speed is U

=

---:-=

(103)

This is independent of the dimensions of the conductors and depends only on the electric and magnetic properties (8 and 11) of the material between the conductors. In particular, if 8 and 11 are close to the values for vacuum (80 and 110) then the propagation speed is close to the speed of light in vacuum, c = (11080) - 1/2. But if the dielectric constant is substantially greater than unity, that is, if 8 is considerably larger than 8 0 , then u is less than c. The characteristic impedance R (LjC)1/2 is given by b R = 1 (11)1/2 In- 2n 8 a

(104)

Thus by varying the ratio bja of radii of the conductors, we can easily change the characteristic impedance of the line. The entire analysis of the distributed-parameter line can also be carried out from the standpoint of the fields between the conductors, and propaga­ tion of waves in this region. The analysis is then quite different in detail, but the final conclusions concerning the propagation speed and characteristic impedance are the same.

experiment 1 pulse propagation

To experiment with pulse propagation on a line, we need a source of pulses. A simple way to produce pulses is to use a square-wave generator and an RC circuit, as shown in Fig. 48. The behavior of this setup can be analyzed in detail as in Experiment EC- L If the RC time constant is very short com­ 57

1

i

•

!

periodic structures and transmission lines

EC - 5

the distributed-parameter line by simply reinterpreting Land C as the inductance and capacitance per unit length. The characteristic impedance is still given by Eq. (97) and the delay time T becomes the delay per unit length. The reciprocal of this quantity is then the actual propagation speed in length per unit time. In the ideal case the line is completely free of dispersion and the propagation speed completely independent of frequency. Real lines never achieve this ideal behavior for two reasons. First, the conductors always have some resistance, which leads to dissipation ofenergy. Second, for mechanical support part of the space between conductors must be occupied by a dielectric material. Dielectric properties are always frequency-dependent, and at high frequencies a dielectric dissipates energy, becoming equivalent in a sense to a shunting resistance. Thus even coaxial lines always have high-frequency cutoffs, but by careful choice of materials this frequency can be raised to 10 10 Hz and higher. It can be shown, using elementary electromagnetic theory, that for a line in the form of coaxial cylinders of inner and outer radii a and b, respectively, the inductance and capacitance per unit length are given by

L=l1ln~ 2:n:

a

C =

2m; In (bja)

(102)

Thus the propagation speed is u

=

I (LC)I/2

I

=
(103)

This is independent of the dimensions of the conductors and depends only on the electric and magnetic properties (e and 11) of the material between the conductors. In particular, if e and 11 are close to the values for vacuum (eo and 110) then the propagation speed is close to the speed oflight in vacuum, c = (l1oeo) -1/2. But if the dielectric constant is substantially greater than unity, that is, if e is considerably larger than eo, then u is less than c. The characteristic impedance R = (LjC)I/2 is given by 1 (11)112 b In 2:n: e a

R = -

(104)

Thus by varying the ratio bja of radii of the conductors, we can easily change the characteristic impedance of the line. The entire analysis of the distributed-parameter line can also be carried out from the standpoint of the fields between the conductors, and propaga­ tion of waves in this region. The analysis is then quite different in detail, but the final conclusions concerning the propagation speed and characteristic impedance are the same.

•.

"il experiment 1 pulse propagation To experiment with pulse propagation on a line, we need a source of pulses. A simple way to produce pulses is to use a square-wave generator and an RC circuit, as shown in Fig. 48. The behavior of this setup can be analyzed in detail as in Experiment EC-l. If the RC time constant is very short com­ 57

~

electric circuits

c

FIGURE 48

R

pared to the period of the square wave, current flows in the resistor only for a very short time at the beginning of each half-cycle, and the result is a series of pulses of alternating polarity. Clearly, the pulse duration is proportional to RC. If it is essential to have pulses of only one polarity, a diode can be inserted in series with the resistor. To study transmission and reflection of pulses, the circuit shown in Fig. 49 is suggested. This network feeds the input pulse, the output pulse, and the FIGURE 49

5.BkQ lOOpF

Output

Input

Delay

line

G

G

reflected pulse to the vertical input of the scope, whose sweep is synchronized to the square-wave generator. The resistor at the line input is chosen approxi. mately equal to the characteristic impedance, to eliminate reflections from this end of the line. Measure the total delay time (NT) for the line. From this and the number of sections, determine the delay time T per section. Adjust the terminating resistor RL until it is matched to the line, that is, until reflections are eliminat­ ed. Remove RL and measure its resistance with an ohmmeter to obtain the characteristic impedance. From this measurement and the delay per section, calculate the values of Land C. 2 variation with RL Vary the terminating resistance RL and measure output amplitude as a function of R L • Plot a graph of output amplitude as a function of R L • Repeat for the reflected pulse. Compare with the results predicted by Eq. (96). 3 multiple reflection Change the 560-0 input resistor to a considerably smaller or larger value to observe reflections from the input side of the line. Can you observe multiple 58

periodic structures and transmission lines

E C-5

reflections? You may want to disconnect the S.6-kQ resistor feeding the input signal to the scope, for this observation. How do the phase relations for the reflections depend on the terminating resistance? You may find that the transmitted and reflected amplitudes are somewhat smaller than expected. Why should this happen? 4 voltage attenuation

We can show that the voltage attenuation per section is proportional to the ratio of the series resistance per section to the characteristic impedance; this resistance is associated principally with resistance of the inductor windings. Measure the series resistance of the line; from this value and the number of sections, find the resistance per section and compute expected attenuation to within a constant factor. By comparing with the observed attenuation when the termination is matched to the line, estimate the value ofthe proportionality constant. 5 cutoff

Compute the high-frequency cutoff of the line. Can you observe cutoff experimentally? 6 coaxial line

All the above experiments can be repeated with the coaxial line. Observe transmission and reflection with several values of terminating resistance, both less than and greater than the characteristic impedance. What similarities with the lumped-parameter line do you find? What differences?

questions 1 Suppose both ends of a line are terminated in resistances much larger than the characteristic impedance Z. Discuss the polarities of successive pulses arriving at the output end after 0, 2, 4. 6, ... , reflections. What if both terminating resistances are very small compared to Z? If one is large, the other small ? 2 Show that the quantity b(mk)-1/2 in Eq. (90) is dimensionless.

•

3 Show that the quantity (LjC)1/2 has units of resistance. 4 In the-iumped-parameter line used in this experiment, the inductors are often all wound on a common base. Thus, there is some inductive coupling be­ tween sections. What effect might this have on pulse propagation?

5 Derive the expressions for the inductance and capacitance per unit length, Eqs. (102), for a coaxial line. 6 Show that the delay T per section has the value n(LC)1/2j2 at the cutoff frequency. 7 Show that in MKS units the characteristic impedance of a coaxial line is given by Z = 60 In (bja) n. Is the coefficient 60 an exact number or an approximation? If the latter, by what percent does it differ from the exact value? 59

electric circuits

8 Derive an analytical expression for the shape of one pulse formed by using the RC circuit with the square-wave generator. Does the actual pulse have precisely this shape? What happens if the "corners" of the square wave are not perfectly sharp? 9 For a lumped-parameter line, are the phase and group velocities for pulses

the same or different? Which is greater? Why?

60

a

• l I

am

I

EIH

4

t

f:m

,

1

• I

7

-

6

:=­ e

!

,

lEE

1 9

b

8 I 4

5

4

-

,

,

I

I

OOFR ffR I I I I

berkeley physics laborator ,2d edition

alan m. portis, university of california, erkol y hugh d. young, carnegie-mellon universlIY

acoustic waves

AF·1

acoustic diffraction and Interference

AF·2

a coustic Interferometry

AF-3

fluid f low

AF-4

viscous flo w

AF-5

turbulent fl w

AF-6

mcgravv-hill book company new york

sf louis

sen francisco

dUsseldorf

acoustics and fluidS Copyright 1971 by McGraw- Hill, Inc.

All rights reserved. Printed in the United States of America.

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Library of Congress Catalog Card Number 79-125108

07-050487-3

1234567890 BABA 79876543210

The first edition of the Berkeley Physics Laboratory

copyright © 1963, 1964, 1965 by Education Development

Center was supported by a grant from the National Science

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Education Development Center.

This book was set in Times New Roman, printed on

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pany, Inc. The drawings were done by Felix Cooper;

the designer was Elliot Epstein. The editors were Bradford

Bayne and Joan A. DeMattia. Sally Ellyson

supervised production.

lts

rwork nental . units enient ations : table rs are

c c

m

/mole

'eber

acoustics and fluids

INTRODUCTION

The first three experiments in this series are concerned with wave propaga­ tion in fluids, chiefly in air. Several wave phenomena which can be demon­ strated with visible light or microwaves can also be observed with acoustic waves, which of course are not electromagnetic in nature. We shall note several similarities, as well as some important differences, between acoustic waves and electromagnetic waves. To obtain waves with wavelength of a convenient magnitude, it is desirable to work at frequencies of about 40 kHz (40,000 cycles/sec). The upper limit of audibility to the human ear is about 20 kHz, so these waves are classified as ultrasonic. Nevertheless, they are also referred to loosely as sound despite the fact that they are not heard by humans. Some animals have a hearing frequency range which extends far higher than that of man. Several interference and diffraction effects with sound are completely analogous to the same effects with electromagnetic waves. Sound waves in gases, being longitudinal rather than transverse, do not exhibit the phenom­ enon ofpolarization, and this is an important difference from electromagnetic­ wave phenomena. Acoustic waves in solids can be either longitudinal or transverse or a combination, and in this case polarization effects can be present. . The last three experiments in this series involve the flow offluids, principal­ ly air and water. We begin by considering an idealized fluid, in which all effects associated with viscosity are completely absent. From here we proceed to viscous flow and finally to turbulent flow. At each stage we analyze the behavior in terms of the basic physical principles involved.

experiment

A F-1

acoustic \Naves

introduction It is well known that a gas can serve as a medium for propagation of a

mechanical wave, in which the disturbance from equilibrium consists in temporary pressure fluctuations away from the equilibrium value, with cor­ responding regions of compression and rarefaction. Wave propagation in gases is discussed in many introductory texts and will not be treated in detail here. For an ideal gas in which the compressions and rarefactions occur adiabatically, the propagation speed u is given by u=

(r:~1/2

(1)

where r is the ratio of specific heats (y = Cp/Cv ), R the gas constant, T the absolute temperature, and M the molecular mass of the gas. For these experiments we use a wave having a frequency of about 40 kHz and a wavelength of the order of 1 em. Such waves are easily produced using a sine-wave generator and an electroacoustic transducer. An ordinary loud­ speaker is such a transducer, but for reasonable efficiency in this frequency range we use a specially designed transducer in the form of a small disk about 2 em in diameter. The characteristics of this transducer are investigated in detail in Experiment EI-S. As a detector of ultrasonic waves we simply use another transducer. At relatively short range (Le., less than 1 m) the wave amplitude is large enough to produce a signal in the detector which can be fed directly to an oscilloscope input. If the source and detector transducers are placed "face-to-face," separated by a certain distance L, each acts as a partial reflector, and a standing wave can be established in the space between them. The standing-wave amplitude is maximum when each transducer is near a node, which will be the case when the distance is an integer number of half-wavelengths. That is, (2)

where n is an integer. Thus by changing the distance and observing the peri­ odic change of amplitude, one can measure the wavelength directly. An alternate procedure for measuring the wavelength is to connect the source transducer to the vertical oscilloscope input and the detector to the horizontal input. The two signals differ in phase by an amount directly proportional to the distance between the transducers. If we locate a detector position for which the two signals are in phase and then move the detector away until the signals are again in phase, we have moved it a distance of exactly one wavelength. 3

acoustics and fluids

experiment 1 amplitude variation Connect the transducers as shown in Fig. 1. The 10-kQ resistor serves to short out any stray 60-Hz hum pickup which might otherwise find its way to the FIGURE 1

If-- I L

.~

lOkQ

scope input. It will be necessary to use shielded leads for the transducer con­ nections in order to avoid electromagnetic coupling between the two circuits. The transducers have fairly sharp resonance peaks in their response, in the vicinity of 40 kHz. With the two transducers face to face, vary the frequency in the vicinity of 40 kHz until the resonance frequency is found, and leave the generator set at this frequency. It may be desirable to check periodically to make sure the setup has not drifted off resonance. Now move the detector transducer along the axis, observing the periodic changes in response. Measure the wavelength as accurately as possible, and from it and the frequency compute the speed of the wave. Compare your result with the prediction of Eq. (I). 2 phase change Change the oscilloscope connections to the arrangement shown in Fig. 2. Tilt the detector transducer slightly so the reflected wave does not return to the source. Moving the detector along the axis, observe the changing relative phase of the two signals. Again measure the wavelength, and compare your result with the previous on.e. You may wish to move the detector a distance of several wavelengths for better accuracy.

questIons 1 Is electromagnetic coupling between circuits a more serious problem at

40 kHz or at 60 Hz? Why? At which frequency is shielding of leads more important? 4

acoustic waves

FIGURE 2

I I ~L~

AF-1

lOH2

2 Explain how this setup could be used as a thermometer. If you could measure 10 wavelengths with a precision of 0.2 mm (i.e., 0.02 mm per wavelength), what is the smallest temperature change you could detect, assuming a constant frequency? 3 Some organ pipes are designed to produce a longitudinal standing wave with an antinode at each end of the pipe, so that the length of the pipe is a half­ wavelength. How does the pitch (frequency) of such a pipe change with temperature? At ordinary temperatures what temperature rise would be needed to change the pitch a semitone (about 5% change in frequency)? 4 With reference to Question 3, the metal of which the organ pipe is made also

expands with increasing temperature, thus changing the pitch. Is this effect more or less important than the change in sound velocity? Explain. 5 Under what conditions would you expect the compressions and rarefactions in sound-wave propagation not to be adiabatic? In this case, how will the sound speed differ from the prediction of Eq. (1)? 6 What gas has the greatest sound speed? By what factor is it greater than that of air? 7 Air is principally a mixture of two diatomic gases. What values of')' and M

should be used in pq. (1) for air?

5

experiment

A F -2

acoustic diffraction and interference

introduction Many classical diffraction and interference experiments which were originally performed with light can be done very simply with ultrasonic waves. One of the simplest is the two-slit experiment with which Thomas Young in 1802 convincingly established the wave nature of light. The basic setup is shown in Fig. 3. For values of () for which the difference in path lengths from the slits FIGURE 3

~I

d

a

to the detector is zero or an integer number of wavelengths, constructive interference occurs and an intensity maximum is observed; when it is a half­ integer number of wavelengths, a minimum occurs. Thus the respective conditions are: Maximum

d sin () = nA.

Minimum:

d sin ()

=

(n

(3a)

+ tM

(3b)

where n is zero or an integer. An alternative arrangement is the "Lloyd's mirror" setup shown in Fig. 4. The reflecting surface forms a virtual image of the source, as shown. The detecting transducer may again be used to explore the node pattern in the interference pattern formed by the initial and reflected waves. Diffraction effects can also be observed. The simplest example is the radiation pattern of the transducer itself, as shown in Fig. 5. This pattern results from the fact that even if all points in the transducer surface moved in phase (which is actually not the case), the radiation at points off the axis is a superposition of waves that have traveled different distances from the source 7

•

acoustics and fluids

FIGURE 4

Mirror

FIGURE 5

FIGURE 6

~I

and hence have phase differences. Actually calculating the interference pattern would be rather complex, but it can be investigated experimentally. A more straightforward diffraction experiment consists of using a single narrow slit for an aperture, as in Fig. 6. For a slit of width a, destructive interference occurs when the radiation from one-half of the slit interferes 8

acoustic diffraction and interference

AF-2

destructively with the radiation from the other half. Thus, the condition for destructive interference is

~ sin e =

(n

+ t)A

n = 0, ± 1, ± 2, ...

(4)

experiment Connect the source transducer to the sine-wave generator and the detector to the vertical oscilloscope input, just as in Experiment AF-1. Shielded leads should be used for both transducers. Make sure the generator frequency is adjusted to the resonance of the transducers. 1 radiation pattern

The setup shown in Fig. 5 may be used to measure the radiation pattern from the disk transducer. To obtain reliable measurements you will need to keep the detector transducer always oriented toward the source and at a constant distance. The experiment should be performed as far as possible from reflecting surfaces, in order to minimize spurious effects from reflections. A convenient arrangement can be made by taping the receiver to a ruler, which may be rotated about a post. A protractor can be laid under the ruler to measure angles. Is the pattern symmetric about the axis? Alternatively, the source transducer may be rotated. 2 single-slit diffraction

Use the same arrangement as above, but add a slit aperture, as shown in Fig. 6. You may wish to start with a fairly wide slit, say a = 3A, and then observe how the diffraction pattern changes as the slit is made narrower. Locate the minima in the pattern and compare their positions with the pre­ dictions of Eq. (4). 3 two-slit interference

For interference, use the two-slit arrangement of Fig. 3. In order to reduce complications resulting from diffraction at the slits, it is simplest to have the individual slit width the order of a wavelength or somewhat less, with the slits separated by a few wavelengths. A strong interference pattern should be readily observable. Measure the positions of as many maxima and minima as possible, and compare with the predictions of Eqs. (3). Alterna­ tively, one may use these measurements to compute the wavelength of the r~diation, as in the original Young two-slit optiCal experiment. 4

Lloyd's mirror

For the Lloyd's mirror experiment, position the source a few wavelengths away from a vertical reflecting plate and measure the nodes of the interference pattern, as in Fig. 4. Alternatively, the table top may be used as a mirror. Judgingfrom the positions of the nodal lines, is the virtual source produced by the mirror in phase or out of phase with the real source? Displacements in the individual waves are purely longitudinal; from the observed node pattern, what is the direction of the displacement at the reflecting surface? 9

acoustic diffraction and interference

A F-2

destructively with the radiation from the other half. Thus, the condition for destructive interference is (n

+ t)A

n

0,

± 1, ±2, ...

(4)

experiment Connect the source transducer to the sine-wave generator and the detector to the vertical oscilloscope input, just as in Experiment AF-1. Shielded leads should be used for both transducers. Make sure the generator frequency is adjusted to the resonance of the transducers. 1 radiation pattern

The setup shown in Fig. 5 may be used to measure the radiation pattern from the disk transducer. To obtain reliable measurements you will need to keep the detector transducer always oriented toward the source and at a constant distance. The experiment should be performed as far as possible from reflecting surfaces, in order to minimize spurious effects from reflections. A convenient arrangement can be made by taping the receiver to a ruler, which may be rotated about a post. A protractor can be laid under the ruler to measure angles. Is the pattern symmetric about the axis? Alternatively, the source transducer may be rotated. 2 single-slit diffraction

Use the same arrangement as above, but add a slit aperture, as shown in Fig. 6. You may wish to start with a fairly wide slit, say a = 3;', and then observe how the diffraction pattern changes as the slit is made narrower. Locate the minima in the pattern and compare their positions with the pre­ dictions ofEq. (4). 3 two-slit interference

For interference, use the two-slit arrangement of Fig. 3. In order to reduce complications resulting from diffraction at the slits, it is simplest to have the individual slit width the order of a wavelength or somewhat less, with the slits separated by a few wavelengths. A strong interference pattern should be readily observable. Measure the positions of as many maxima and minima as possible, and compare with the predictions of Eqs. (3). Alterna­ tively, one may use these measurements to compute the wavelength of the radiation, as in the original Young two-slit optiCal experiment. 4

lloyd's mirror

For the Lloyd's mirror experiment, position the source a few wavelengths away from a vertical reflecting plate and measure the nodes ofthe interference pattern, as in Fig. 4. Alternatively, the table top may be used as a mirror. Judging from the positions of the nodal lines, is the virtual source produced by the mirror in phase or out of phase with the real source? Displacements in the individual waves are purely longitudinal; from the observed node pattern, what is the direction of the displacement at the reflecting surface? 9

acoustics and fluids

[s this a general boundary condition for a reflecting surface? What experiment could you perform to demonstrate directly that sound in air is longitudinal and not transverse?

questions 1 Do all points on the source transducer surface move in phase? Do they move

with the same amplitude? Compare to the behavior discussed in Experiment E[-S.

2 In the single-slit experiment, if the slit is too narrow, no minimum in intensity is observed. What is the smallest value of a for which a minimum occurs between 0° and 90° ? 3 When a sound wave is reflected from a rigid surface, is the reflected wave in phase or opposite in phase to the incident wave, at points very near the surface? Explain. 4 Ifthe detector transducer were rotated 90° about an axis perpendicular to the

disk surfaces, would its behavior change? Compare this situation with the comparable one for an electromagnetic wave detector. 5 How might the acoustic wave setup be used to make a Michelson inter­

ferometer?

10

experiment

A F -3

acoustic interferometry

introduction By using more than one source transducer, or more than one detector, or both, we obtain new and interesting possibilities for acoustic phenomena. The simplest example is shown in Fig. 7, which is, in principle,just a variation FIGURE 7

of the two-slit setup used in Experiment AF-2. We can explore the radiation pattern in the plane of the figure and can also look at points above and below this plane. An interesting variation is to excite the two source transducers with signals of slightly different frequency, say W t and {J)2' At a point equidistant from the two sources, the resulting signal will show the familiar beat phenomenon as shown in Fig. 8, which is a plot of the resultant signal as a function of time. FIGURE 8

-~

~·------------------~w

I

,,....-~

I

I

/

/

.I.

/

.....:.

11

acoustics and fluids

The signal varies rapidly with a frequency equal to the average of W 1 and W 2 , that is, t(Wl + W2), while the frequency corresponding to the envelope curve (shown as a broken line) is the difference frequency ~w = t(Wl - W2)' Now suppose we move the detector so that it is a half-wavelengtli closer to one source than to the other. Again we observe the characteristic beat pattern, but the phase of the envelope curve is shifted by one-quarter cycle, since the times at which the two signals were instantaneously in phase at the equi­ distant point coincide with the times at which they are a half-cycle out of phase at the new point, leading to cancellation. Similarly, a point can be found at which the two envelope curves are a half-cycle out of phase.

TWO DETECTORS

FIGURE 9

Now we take an additional step and add the outputs of two detectors, with their individual path differences for the two sources. The interesting feature added in this situation is that the maxima and minima in the resultant signal from the two detectors are determined by the relative phase of the envelope curves, not the individual signals themselves. Even if the phases of the oscillators which produce the signals W 1 and W2 were to change randomly, the relative phase of the two envelope curves would not change. The resultant envelope curve can be observed directly by means of a standard detector circuit, as shown in Fig. 9. The Rand C values are chosen so that the output contains the low-frequency envelope voltage but not the high-fre­ quency variation corresponding to the original frequencies. 10 kQ

IJ ~F 11------6---------6-------<>----------' 27kO.

01

Still a further variation is to obtain the envelopes of the two signals separately, and then examine the relative phase of these signals. An arrange­ ment for making this observation is shown in Fig. 10. A variation of this scheme forms the basis of the Brown-Twiss interferometer, with which Brown and Twiss in 1954 were able to measure the angular diameter of stars which are strong radio-frequency sources by comparing the relative phases of low-frequency intensity variations as measured at separate receivers.

experiment 1

beating

With two transducers driven by separate sine-wave generators with slightly different frequencies, examine the resultant radiation pattern with a single detector, and observe the beating of the two signals, as shown in Fig. 8. How does this pattern vary with the path difference for the two sources? 12

acoustic interferometry

A F-3

lOkQ

FIGURE 10

~I

lOkQ

27kQ .01 jJ,F

-2

superposition of beats Using two receivers in the scheme shown in Fig. 11, observe the superposition of the two beating signals. In particular, observe the variation in amplitude of the envelope curve when the distance between the two detectors is changed. When they are close together, is there constructive or destructive interference? Does this depend on the polarity with which the transducers are connected?

FIGURE 11

3

envelope detection Observe the shape of the resultant envelope curve by detecting the sum signal from the two receivers, using the circuit of Fig. 9. To observe the intensity variations, bring the two receivers Rl and R2 close together and sweep the two sources S 1 and S2 in front of the receivers. By increasing the separation 13

acoustics and fluids

between R1 and R2 you should find a separation for which the intensity variations from the two sources just cancel. The envelope may be seen on an oscilloscope, or it may be measured with a voltmeter. Calculate the separa­ tion between Sl and S2' 4 Brown-Twiss interferometer

The Brown-Twiss interferometer is shown in Fig. 10. The two receiver signals are detected separately, and their relative phase is obtained by displaying them with the vertical and horizontal deflections of the oscilloscope. Note that when the receivers are close together, the two envelope variations are in phase, as indicated by a diagonal line on the scope. As the separation of R1 and R2 is increased, the line opens into an ellipse. When they are just 90° out of phase, the ellipse axes are vertical and horizontal. Use your results to calculate the separation between Sl and S2'

questions 1 When two signals with frequencies W 1 and W 2 are added, the envelope frequency is t(w 1 - w 2 ), but the intensity varies with a frequency twice as

great,

(W1 -

w 2 ). How are these two observations reconciled?

2 In these experiments, is it necessary for the signals emitted by the two sources

to have the same amplitude? What happens if they do not? 3 If the polarity of one of the transducers in Fig. 9 is reversed, how do the experimental results change?

14

experiment

A F -4

fluid flow-

Introduction In the next three experiments we shall investigate the flow offluids, principally air and water, and their interactions. We begin with the simplest case, the flow of water under conditions where we may neglect the viscosity of the water and also assume that it is incompressible. In Experiment AF-5 we shall take up viscous effects, and in AF-6 airflow, where compressibility is important. We begin by proving Bernoulli's theorem, which establishes the relation between velocity of flow and pressure in a fluid. We consider the situation of Fig. 12a, where water flows out of an orifice in the side ofa tank. Figure 12b FIGURE 12

(a)

(b)

shows the flow in more detail. The lines are flow lines, and we consider a tube bounded by flow lines as shown. At the first section the tube has cross-sectional area AI' and at the second (within the orifice) A 2 • While the fluid at point 1 advances a small distance ds I , that at 2 advances a distance ds 2 , and if the fluid is incompressible, the two corresponding volumes must be equal. Thus, we have (5) During this displacement, the net work done on the fluid within the tube between these sections, given by P dV in each case, is (6)

where PI and P2 are the pressures at the two ends of the tube. This work is to be equated to the net change in energy (kinetic p[us potential) of the fluid in the tube. Denoting the two flow velocities by VI and V2, we find that the kinetic energy of the fluid entering the tube at section 1 is tm 1v 12 !pA l ds i V1Z. The kinetic energy of the fluid leaving at section 2 is given by 15

acoustics and fluids

a corresponding expression, so the net change of kinetic energy during this displacement is given by

dE = tpA2 dS 2 v/

tpAI dS l v/

(7)

Similarly, the change in potential energy, determined by the change in the vertical coordinate y, is given by

dV

= m19Y2

migYI

= pgA2Y2 dS 2 -

pgAIYI dS I

(8)

The total energy change is the sum of Eqs. (7) and (8). Equating this sum to Eq. (6) and using Eq. (5) to divide out the common factor A ds, we obtain PI + tp v 1 2 + pgYI = P + tpv/ + pgY2 (9) 2

A more general way of stating this result is that the quantity P + tpv 2 + pgy

(10)

is constant along a stream line. If we can assume that all stream lines intersect the upper surface of the water normally, it is a simple matter to show that the quantity shown in Eq. (10) is, in fact, constant through the entire fluid. How might you check this assumption? To prove this result we simply note that anywhere on the upper surface the quantity given in Eq. (10) has the value

PI

+ tp (

dy d/

)2 + pgYl

(II)

where dytfdt is the rate at which the surface level is dropping and PI is atmospheric pressure. For any point in the fluid Eq. (9) gives

(P

PI)

+ tpv 2 +

pg(y - YI) = tp

(~Ir

(12)

TRAJECTORY Finally, we apply Eq. (12) to find the trajectory of the fluid after it passes

through the orifice. Differentiating with respect to time, we obtain

dv dt

. = - (g) ;; dy dt = g sm ()

(13)

where () is the angle between the direction of flow and the horizontal, as shown in Fig. 13. In addition, we shall need an equation for the rate at which the velocity changes direction. With reference to Fig. 13, the transverse force FIGURE 13

16

fluid flow

AF-4

on a unit length of the tube is pg cos e. The transverse acceleration may thus be written as (14)

pg cos 6

PV (d6/dt)

where we have assumed that no transverse forces are applied across the boundaries of the tube segment shown in Fig. 13. Now, the combination of Eqs. (13) and (14) is exactly the same as for a free particle. That is, the trajectory of a stream and of water droplets will be identical, provided they have the same initial velocity. Thus, if Yo and xo are the coordinates of the orifice and the stream has an initial velocity Vo

= [ 2g(YI

- Yo)

dYl)2JI/2 + ( dt·

(15)

in the horizontal direction, the subsequent motion will be given by x = Xo

+ vot

(16)

Eliminating the time, we obtain for the equation of the trajectory (x - x o)

2

2Vo2

+-

g

(y

Yo)

= 0

(17)

Equation (17) represents a parabola. Our first measurement will be of the trajectory of the fluid leaving the orifice. From this we can determine the speed of efflux Vo for various surface heights Yl' A second way of testing Eq. (15), which is known as Torricelli's law, is by studying the relaxation of the level of the fluid in the storage tank after it has been raised to some initial level and the syphon turned off. We may solve for the expected behavior by eliminating Vo between Eqs. (I5) and (28) (in the Experiment section) to obtain :

=

~: [2g(y

- Yo)] 1/2

(18)

where we have assumed Ao to be small compared with A l' We may integrate Eq. (18) to obtain for the heighty as a function of time: (19)

where Y 1 is the height of the surface at t = O. Note in particular that the time for (y - Yo) to drop to half its initial value will be given by t

=

(J2

1) Al

Ao

(Yl ­

YO)1/2 g

(20)

SURFACE TENSION You will find in the experiment on relaxation that the surface level does not

relax all the way down to Yo, but instead levels off at a height about a centi­ meter above Yo. What do you think causes this? This variation arises from surface tension and is the same phenomenon which produces a rise in fluid level in a capillary tube. With reference to Fig. 14, let us assume that the 17

acoustics and fluids

FIGURE 14

p

water surface at the orifice has a radius of curvature R, where r is the radius of the orifice. Then there will be an excess pressure in the fluid given by

p

2y R

=

(21)

where y is the surface tension or, what is equivalent, the energy per unit area of the water-air interface. Now, the excess pressure must be balanced by a rise in fluid in the storage tank: (22)

Equating Eqs. (21) and (22), we find that the radius of curvature is given by R

=

2y pg(Yl - Yo)

(23)

Note that as Yl is raised the radius R gets smaller. Finally, when R is suffi­ ciently small a stream fonos and the situation shown in Fig. 14 is no longer stable. We may also use the following argument to find the threshold for flow. Let us assume that at the threshold vo is sufficiently small that the kinetic energy of the fluid can be neglected. Then the only additional energy is the surface energy of the fluid. If the fluid has a radius r, then the surface energy per unit length is U

=

2nry

(24)

The work done on the fluid per unit length is

W = nr 2 p

(25)

Equating Eqs. (24) and (25), we obtain P

=

pg(Yl - Yo)

2y = -

r

(26)

e xperiment 1 trajectory Using the fluid system provided, insert an aperture of 2 mm diameter at the lowest position provided. Using the air-assisted syphon, raise the level of water in the storage tank to some convenient height Yl as shown in Fig. 15. Let Y2 designate the surface of the water in the receiving tank and X 2 the position at which the stream strikes the surface. 18

fluid flow

FIGURE 15

AF-4

Y

YI

Yo

-----~--------~------_+----~----- x

From Eq. (17) one may solve for Vo to obtain

Vo

9

=

(x z - x o) [ 2(yo - Y2)

J1 /2

(27)

Obtain Vo for several values of YI' N ote that if Al is the area of the storage tank and Ao is the area of the orifice, then

dY l

Ao

----;[( = Al Vo

(28)

For the experimental arrangement provided Ao/ Al is about 10- 3 (verify this) so that dYt /dt may be neglected . Repeat your measurements with the orifice at other available positions and determine v. Finally, plot your data for Vo as a function of (Yt - Yo) using log-log paper. D o your data lie on a straight line? What is the slope of the line ?

2 relaxat ion U sing the system shown in Fig. 15, transfer water to the storage tank up to some initial level Yt. Using a stopwatch, determine the time for YI - Yo to drop to half its initial value. Using the expression given in Eq. (20), compute A o. How does your value compare with a visu al estimate? Why might the computed Ao be somewhat smaller than the geometrical area of the orifice ? You may wish to use another starting value of Yl and repeat your measure­ ment. Again determine Ao. Do you find any systematic variation in Ao ? 3 surface tension Find the height (Yl - Yo) at which water stops flowing. Estimate the rad ius of curvature R and use Eq. (23) to compute the surface tension y. Alterna­ tively, raise the height Yl until water just begins to flow. Use this conditio n to determine y. The accepted value of the surface tension of the air-water interface at 18°C is 73.05 dyn/ cm. How do your resu lts compare with this value? W hich of your determinations is more reliable ? Why? Once water has started to fl o w it will continue even though the pressure drops below the value given in Eq. (26). Why? You may wish to repeat your measurement of surface tension with water to which a wetting agent has been added . (About 0 .6 cc in a liter of water will produce an easily detectable change in y.) 4 droplet formation You will note that after the water stream has fallen for some distance it appears to brea up into a fine spray. Actually, the stream is forming large 19

acoustics and fluids

droplets, but the drops are moving too fast to be seen clearly. By using a stroboscopic light one can "stop" the motion of the water and easily observe the formation of droplets. You may wish to observe the droplet formation for various heights of the aperture and flow rates. Why does the water stream break up into droplets? 5 w h irlpool Note that when there is sufficient airflow to nearly drain all the water from the receiving tank, a whirlpool forms at the exit tube; and as the water moves toward the hole, its velocity increases. Neglecting viscous forces, we may expect that the angular-momentum density L = pvr

(29)

will remain constant so that the azimuthal velocity will increase as 1/,.. In addition, there will be a radial velocity, which must also increase as 1/1' as the FIG URE 16

stream lines converge (see Fig. 16). We may write, then, that the kinetic energy of the water is proportional to l /r2: E =

1

2PV 2

C = - r2

(30)

From the Bernoulli equation, we expect that the sum of kinetic and potential energy will be a constant:

(31) where Y2 is the elevation of the free water surface. Eliminating v between Eqs. (30) and (31), we obtain for the equation of the water surface

r

=

k ------:-=

(Y2 - y)1 /2



(32)

Looking through the side of the receiving tank, measure the contour of the surface and compare with Eq. (32). How do the radial and azimuthal 20

fluid flow

AF-4

energies compare? Do you find that the fluid consistently rotates with one sense? Why?

questions 1

Is water truly incompressible? Look up a value for the bulk modulus of water and estimate the maximum percent volume change in this experiment.

2 Would you expect the surface tension to vary with temperature? Explain. 3 Very hot water flowing out of a faucet and splashing in a sink has a different sound from cold water. Why? 4 It has been claimed that the swirling of water into a whirlpool has something

to do with the rotation of the earth. Could the Coriolis effect have anything to do with this phenomenon? Make a rough estimate of the magnitude of the Coriolis force on an element of water (say 0.01 cru l ) moving toward the drain. Can you think of alternative explanations for the origin of the angular momentum involved? 5 Does the surface tension of water vary appreciably with temperature? Is this

effect significant in your measurement of surface tension?

21

experiment

A F-5

viscous floW'

Introduction We know from our study of the damping of a glider on an air track that Newtonian fluids resist shearing motion with a force proportional to the velocity gradient. Thus, the glider moving along an air track experiences a force given by dv

(33)

F= -'I1Ady

where A is the contact area, dv/dy the velocity gradient, and '11 the viscosity of air which has the value at 18°C of 187.2 x 10- 6 P (poise). Water at 20°C has a viscosity 1'/ = 0.01002

P

about 50 times that of air. In this experiment we shall begin by studying the flow of water through a tube and in this way determine its viscosity. We show in Fig. 17 a section of a FIGURE 17

(a)

(b)

tube through which water is flowing. If P is the pressure at the input and L the length of the tube, we have the following expression, which corresponds to Eq. (33) for the case of cylindrical symmetry: rP = -I'/L ~ (r dV) dr dr

We may solve Eq. (34) to obtain for the velocity profile of the fluid (a 2 r2) v= 4'11L

(34)

(35)

The total flow through the tube may be obtained by integrating over the cross section of the tube: a. nPa4 (36) Q 0 v2nr dr

f

81'/L

Equation (36) is called the Poiseuille formula. 23

acoustics and fluids

We shall use the same experimental arrangement as in Experiment AF-4. The difference is that we shall replace the orifice by a tube. Since the flow is proportional to the pressure) we may expect that the height of the fluid in the storage tank will decrease exponentially with the time. We write for the pres­ sure P = pg(y - Yo)

(37)

and for the flow Q

= -A dy dt

(38)

Eliminating P and Q in Eq. (36), we obtain dy dt

y - Yo

(39)

T

FIGURE 18

where T = 8f1LA/(na4 pg) is the relaxation time. It should be apparent from the observation of flow through a tube that the Bernoulli equation cannot be applied to this situation. Since the cross section of the fluid is constant, the velocity must be uniform in the tube. In contra­ diction to the Bernoulli equation, we may have a pressure drop with no velocity increase. It is clear that this discrepancy arises from the fact that we neglected the force of vis ous damping in deriving Bernoulli's theorem. It is a simple matter to add the pressure drop due to viscous forces to the Bernoulli equation. Let us consider the situation shown in Fig. 18. The pressure drop in the tube is obtained by solving Eq. (36) for P: 8f1LQ na

P = -4­

Adding this term to the Bernoulli relation, we obtain for the velocity of flow from .an orifice at Yo (40)

The flow is given by (41)

and we may rewrite Eq. (40) a, (42) 24

viscous flow

AF-5

Now if we direct the stream upward, it will rise to a height Y2 given by

!V o2

g(Y2 ­ Yo ) =

(43)

Eliminating Vo between Eqs. (42) and (43), we find for the viscosity

pa 2 g(Yl - Y2) " = 8L [2g(Y2 - Yo)] 1/2

OSCILLAT IONS

(44)

With a tube of the form shown in Fig. 19, one may observe oscillatory behavior of the fluid. If there were no viscous damping the equation of motion of the fluid surface would be

d 2y M dt 2

=

- 2pAgy

(45)

FIGURE 19

where M = pAL. This equation is just the same as for a mass on a spring (see Experiments M-4 and M-5) and has the solution y

= Yo cos 2nft

_ 1 (29)1/ 2 21t L

f -

(46)

lfwe now consider the effect of viscous damping, there will be an additional fo rce resisting the flow of the liquid given by

F

=

dy dt

- 81t1'/L ­

(47)

In the absence of gravitational restoring force, the liquid motion would relax with a characteristic time M

pA 8m1

T=--= -

81t1'/ L

(48)

If we start the fluid into oscillation, the amplitude of oscillation should decay exponentially. T he oscillator may be characterized by a quality factor (see Experiments M-4 and M -5)

Q = 21tf r

(49)

One may determine the Q by counting the number of cycles n for the ampli­ tude of oscillation to drop to 1/.)2 of its initial value u sing the relation

Q = 8.86n

(50) 25

acoustics and fluids

THEORY OF AIR-SUPPORTED We are now ready to analyze the operation of the air-supported glider used in GLIDER Experiments M-l through M-5. We may use this theory to detennine how the

separation between the glider and the air track depends on the mass of the glider and the inlet pressure to the track. First, we must make the point that if the Bernoulli equation described the flow of air between the glider and the track, then the glider could not float. Since the air under the glider flows to atmospheric pressure, the pressure would have to be everywhere less than atmospheric. And if this were the case, the glider could not float. It is clear that the situation is more like the flow of viscous fluid down a tube, in which the inlet pressure is in excess of the outlet pressure. In order to discuss the glider problem, we shall simplify the geometry to the situation shown in Fig. 20. We imagine that the airflow is from a pair of FIGURE 20

(b)

(a)

lots midway along the track rather than from small holes. In this way we can treat the flow as purely one-dimensional. We let the total area of the slots covered by the glider be A o and the area of the glider be A. The distance between the glider and the track is denoted by d. The condition for equilibrium of the glider is given by Mg

=

1 J2 [i(P 3

1 -

P2)A ]

(51)

The pressure PI above the slot may be related to the flow by noting that the velocity at the slot satisfies the relation ~pV2

+ PI = Po

(52)

so that the total flow may be written as

Q = Aov

= Ao [

2(P O - Pl )]1/2

p

(53)

Finally, we may write an expression for the flow between the glider and the track in terms of the pressure difference (PI - P 2) : (54) 26

viscous flow

AF-5

We may set Eqs. (53) and (54) equal to eliminate Q and then eliminate the intermediate pressure Pl between this equation and Eq. (51). Our final result for the distance of separation dis (55)

We take the representative values Po

P 2 = 1.5 psi

A

=

14 in. 2

L

I in.

=

M = 120 g

Ao = 0.0035 in. 2 (10 holes of diameter 21 mils) and substitute them into Eq. (55) (taking proper account of the units). We obtain for the separation d

13

mils

which is of the order of the observed value. Note that Eq. (55) indicates that there is a threshold for the glider to float:

Po - Pz (4t) Mg:::::: 0.035

psi

For higher pressures the separation d increases rather slowly, as the sixth root ofthe pressure above threshold. Note that if we assume the glider is accurately parallel to the track, the threshold pressure is independent of the area of the slots. Actually, the glider is not perfectly plane, nor is it perfectly balanced; as a result, the threshold pressure is considerably greater than the value obtained above and is, in fact, close to I psi. This imbalance also has the effect of increasing the damping, since most of the damping will come from those portions of the glider for which d is smalL

experiment 1 relaxation

Insert a tube of length 15 cm into the lowest position on the plate separating the storage and receiving tanks. Fill the storage tank to some initial level Y l' Determine the time t 1/2 for y Yo to drop to half its initial value. The relaxa­ tion time will be given by t l/Z

.. =

till = 0.69315

=

1.4427tl/2

Use your computed relaxation time to determine the viscosity 1'/. By working with a partner you may wish to take data every second and plot the distance y - Yo as a function of time on semilog paper. Are you satisfied that the drop is exponential? 27

acoustics and fluids

2

trajectory With the connector at some convenient height, insert a 15-cm length of flexible tUbing. With the stream directed upward determine Yo, Y l ' and Y2 as shown in Fig. 18 and compute the viscosity.

3

oscillations Excite oscillations in your siphon tube, either by pumping water or by dis­ placing the tube. Determine the frequency of oscillation and compare with Eq. (46). Determine the Q and compute the relaxation time 1:. Using Eq. (48) determine the viscosity 11.

4

viscosity modifiers The viscosity of water may be increased substantially by the addition of microscopic particles. Two materials generally used are:

Methyl cellulose

This is a high-polymer coiled-chain molecule consisting of cellulose that has been treated to replace hydroxyl surface groups with methyl groups.

Silica

These are flame-formed colloidal silicon dioxide particles. Prepare a suspension of methyl cellulose in water. Twenty cubic centimeters per liter of water will produce a substantial increase in viscosity. Repeat the measurement described in the first section above, plotting y - Yo as a function of time on semilog paper. Do you obtain a straight line? How is the behavior of the fluid modified? What modification might you make in the derivation ofEq. (39) to yield nonexponential behavior? Fluids for which the shear stress is not strictly proportional to the rate of shear strain are called non-Newtonian. You may wish also to determine the viscosity of a suspension of silica particles. About 50 g of colloidal silica in 1 liter of water will produce a substantial increase in viscosity. Is silica in water a Newtonian fluid?

5

rise of bubbles A spherical object moving slowly through a viscous Newtonian fluid experiences a drag given by the Stokes' law: F

=

61C'f/aV

(56)

where a is the radius of the sphere and v is the velocity. A bubble will experi­ ence a buoyant force F = pgv =

4na 3 pg 3

(57)

Equating Eqs. (56) and (57), the velocity with which a bubble rises should be V

2 pg 2 a 9 11

=-

(58)

That is, the velocity should increase as the square of the radius. Determine the velocity of rise of bubbles introduced by the air stone into your methyl cellulose suspension. Can you verify Eq. (58) for the small bubbles? From your observation of flow through a tube, what deviation might you expect from Eq. (58)? 28

viscous flow

6

AF-5

turbulent flow

For large velocities and large spheres, Eq. (58) breaks down as a result of turbulent flow behind the sphere. A measure of the ratio of kiRetic energy to shear energy is the Reynolds number pv 2 '1v/a

R

pv '1a

(59)

Only for Reynolds numbers smaller than unity can we expect Eq. (58) to hold. Eliminating '1/p between Eqs. (58) and (59), we have for the velocity

C~gay!2

v

(60)

Requiring R to be less than one, we have the relation for nonturbulent motion

ga)112

2

v< ( -

(61)

9

In your experiments with bubbles, at what radius does turbulent flow develop? In pure water, how small would a bubble have to be for its rise to be non­ turbulent? Eliminating the velocity between Eqs. (58) and (59), we have for the Reynolds number of a rising bubble R

=

2: (~y

a

3

(62)

Substitute the value of 1/ for pure water and compute the upper limit for the radius such that R is less than one. You may be able to make some measure­ ments on small air bubbles introduced into pure water by your air stone. You may easily verify that the drag is considerably greater than predicted by Eq. (56). In Experiment AF-6 we shall study air drag, where the Reynolds number is far in excess of one.

questions 1

For maximum fluid flow through a given cross-sectional area, should one large tube or many small tubes be used? Why? What are the relative merits of round and square tubes?

2

Suppose in a certain non-Newtonian fluid the viscous drag force is not given by Eq. (56), but instead is proportional to v2 . In that case is the bubble velocity still proportional to a 2 , as in Eq. (58), or to some other power of a?

3

For a glider on an air track, is the airflow velocity with the glider stationary large or small compared with typical glider velocities?

4

How does the viscosity of water vary with temperature? What common phenomena can you cite to illustrate this variation?

29

experiment

A F -6

turbulent flO\N

introduction In this experiment we shall investigate the rate of fall of a light object with a large surface area in order to determine under what conditions the viscous retarding force is given by Stokes' law F

=

(63)

61tl1aV

and to determine the nature of the deviation from this law. By referring to the damping of a glider as discussed in Experiment M-3, we may get some idea of the conditions which limit the validity of Eq. (63). We noted in Experiment M-3 that the effect of viscous damping was to produce a force v

(64) F= ma = -l1A­ d

where A is the area of contact, m the mass of the glider, and d the separation between glider and track, that is, the distance over which the velocity changes. We may solve Eq. (64) and obtain the result that a glider with initial velocity v will travel a distance x given by x =

md l1A

v

(65)

before stopping.

SMALL REYNOLDS NUMBERS In this experiment we study a situation similar to that shown in Fig. 21

where a sphere is moving through a gas. Portions of the gas must be FIGURE 21

v

31

acoustics and fluids

accelerated to a velocity v of the order of the velocity of the sphere in order to go around the sphere. The momentum of the gas must be dissipated by viscous forces. We may make an analogy between the glider and the gas as follows:

Then the distance x in which the gas stops will be given by

x

(66)

A measure of the extent of turbulence of the gas is the ratio of x to the radius of the sphere; this is the Reynolds number R

x a

pav

(67)

'1

We see that what is actually important in establishing the Reynolds number is not the viscosity but rather the ratio of viscosity to density. This quantity is called the kinematic viscosity and has the units square centimeters per second (cm2/sec) in cgs units. The cgs unit of kinematic viscosity is called the stoke. In Table I we list a number of common materials, their viscosity, density, and kinematic viscosity at room temperature.

Viscosity, P

P Density, g/cm 3

14.9 0.0001827 0.01002

1.260 0.0012047 0.9982

'1 TABLE 1

Glycerin (20°C) Air (18°C) Water (20°C)

R

Kinematic viscosity, stokes 1.183 0.1516 0.01004

Note that in terms of kinematic viscosity, air is more viscous than water. Thus we should expect that for an object of a given size, the threshold for turbulent flow will be higher for motion through air than for motion through water.

LARGE REYNOLDS NUMBERS Now, the expression given by Eq. (63) describes the damping force only so

long as the kinetic energy imparted to the gas is negligible, which is to say that the Reynolds number must be small compared with unity. In the limit that the Reynolds number is large compared with unity, we may neglect the vis­ cosity of the gas in computing the damping and simply consider the momen­ tum transferred to the gas. Let us imagine that an object of area A moves through a gas and imparts its own velocity to the gas in its way. The retarding force is the rate of transfer of momentum

F

dp dt

-=

-v

dm dt

(68)

Actually the force will be somewhat smaller than this because some air will slip around the edges of the object. In this experiment we shall regard A as 32

turbulent flow

A F-6

an effective cross-sectional area, which may be less than the geometrical area. Before discussing the experimental measurement of the effective area A, we shall solve the equation of motion of a mass falling with a damping force of the form given in Eq. (68):

dv dt

2

M- = Mg - pAv

(69)

This equation may be solved exactly by integration to obtain for a mass starting from rest: gt (70) v = Vo tanh­ Vo

where Vo = (Mg/pA)1 12. For short times the hyperbolic tangent is of the order of its argument and we obtain (71) v = gt which is the expression for free acceleration. In the limit of long times the hyperbolic tangent approaches unity and the velocity approaches a limiting value:

=

v

(72)

Vo

For intermediate values of the time we may expand the hyperbolic tangent. To second order tanh z '" z

(1 - ~ + ... )

(73)

Expanding Eq. (70), we obtain v - gt

,

[1

Integrating 1 t2 x"" zg

[1

- -(gt/VO)2] _____-

6

(74)

/ V-: , -O)_2 ] - -=-(g-,-t 12

(75)

The time to fall a distance x from rest is then

2x)1/2 [ 1+ (gt/vo) 2] =~ (2X)112 [ 1+ (g 24 g

t"'~

(76)

Substituting the value of Vo from Eq. (70), we obtain

(~y/2 (1

t =

+

ft;)

(77)

Note that the correction term is the ratio of the mass of air swept out by the area A to the mass M of the falling object.

experiment 1 parachute fall

In order that the effects of air drag be as large as possible we shall want an object of large cross-sectional area and small mass. In addition, the distance 33

acoustics and fluids

available for fall should be large. An ideal object for these studies is a toy para­ chute. To get an idea of the distance through which the parachute must fall before approaching terminal velocity, determine the mass of the parachute, estimate its cross-sectional area when filled with air, and compute the distance 12M/pA. The parachute filled with air should be permitted to fall through this distance before a timing measurement is made. Drop a para­ chute from as high as possible above the floor and determine the time for the parachute to fall from a point I m above the floor to the floor. Using the expression for the terminal velocity

, _(lVlg)li2

to ­

pA

compute the cross-sectional area A and compare with the geometrical area. You may wish to add additional mass; split lead shot provides convenient additional mass. Add a known additional mass and determine Vo. Can you verify that Vo is proportional to the square root of the mass? (Note that for Stokes' -law damping, the terminal velocity increases linearly with the mass.) 2

Reynolds number Finally, we compute the Reynolds number for the case that we have been discussing. Substituting the expression for the limiting velocity into Eq. (67), we obtain R

=

~(p~gyJ2

(78)

Taking M = 5 g, we obtain R = 7600. We may also notice that the ratio of the force due to drag to the Stokes' law force is given by pAv 2

pav

61tlJav

6'1

(79)

which is just one-sixth of the Reynolds number.

questions

34

1

Discuss the extent of turbulence in the motion of air under a glider on an air track. To what extent is Stokes' law valid?

2

Qualitatively, is the shape of an object moving through a fluid of greater importance in determining the drag force when the flow is highly turbulent, or when it is not turbulent? Explain.

3

Discuss the degree of turbulence in the fall of raindrops through the air.

4

Does the term kinematic viscosity imply that the ordinary viscosity is not kinematic? Explain.

5

To what extent do you think the flow of water around a moving submarine is turbulent?

~i,:,

"'~"-~C::'

,;,;--,,"-==_7#7

57

77' --P"'"

ttin :rr S"' ziT "7WtmrN rrilldiw rW't t! NT I
i_a.~

··"i,?",';;;;-·,m......·i·.--iOriIiiI_•• ,,~ _____'"

ber

Y h y ics I

ratory, 2d

dition

.a/an m. portis, university of california. berkeley hug h d. y u ng, carnegie-mellon un,v r It y microwave produc tion a nd r fle ction Interference

nd

MO-'

Iffra c t/on

M O-2

t he k lystro micro

m

ave pro ag tlon

graw-hill b

n ew york

M

Sf lo uis

kco

san francisco

-3

MO-

pan y fu

seldort

microwave optics Copyright © 1971 by McGraw-HiJI, Inc.

All rights reserved. Printed in the United States of America.

No part of this publication may be reproduced, stored in a

retrieval system, or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher.

Library of Congress Catalog Card Number 79-125108

•

07-050488-1 1234567890 BABA 79876543210 The first edition of the Berkeley Physics Laboratory copyright © 1963, 1964, 1965 by Education Development Center was supported by a grant from the National Science Foundation to EDC. This material is available to publishers and authors on a royalty-free basis by applying to the Education Development Center.

This book was set in Times New Roman, printed on

permanent paper, and bound by George Banta Com­

pany, Inc. The drawings were done by Felix Cooper;

the designer was Elliot Epstein. The editors were Bradford

Bayne and Joan A. DeMattia. Sally Ellyson

supervised production.

rnicrow-ave optics

INTRODUCTION In this series of experiments you will study the properties and behavior of

electromagnetic waves with a wavelength of about 3 cm. The term microwave ordinarily denotes electromagnetic waves in the wavelength range of about 0.1 to 10 cm, located in the electromagnetic spectrum intermediate between uhf radio and television waves at one extreme and the far infrared at the other. The term optics is appropriate because many phenomena observed with visible light can be duplicated with microwaves; the scale is different, of course, because of the difference of wavelength. The first essentials for an experimental study of microwaves are a source and a detector. Ordinary radio and TV transmitters usually incorporate oscillators using LC resonant circuits along with transistor or vacuum-tube amplifiers. There are several reasons why it is not practical to build such oscillators for frequencies greater than a few hundred megahertz (l MHz 106 Hz 106 cycles/sec). A major difficulty exists in the small, but unavoid­ able, capacitances between electrodes and their associated connections in vacuum tubes and transistors. Since the ac impedance of a capacitor varies inversely with frequency, such stray capacitances act as short circuits at sufficiently high frequency. A second kind of difficulty arises when the transit time for electrons in the device becomes comparable to the period of the high-frequency signal. Present-day schemes for avoiding these difficulties actually take advantage of interelectrode capacitances and electron transit times as essential parts of the devices. The device used as an oscillator in these experiments is called a reflex klystron, shown in Fig. 1. In this device the conventional LC resonant circuit is replaced by a doughnut-shaped structure called a cavity. The two grids act as the plates of a capacitor, and the surrounding torus as a one-turn inductor. In operation, charge flows back and forth from one grid to the FIGURE 1

Grids

+

L

r.:~............ .... d····~-::

Q D

1

microwave optics

other through the outside surface of the cavity, and the device acts as an LC circuit. This oscillating current flow is accompanied by oscillating electric and magnetic fields between the grids and inside the cavity. These consist principally of an electric field between the grids and a magnetic field inside the doughnut, as shown in Fig. 2. If the charge and current oscillate sinu­ FIGURE 2

t=O

·c·

++++

j... ·l·· ·t· ~O·· ...•........

- - --



®•• ~.~. ®

® ~.~•••~•••••

.....•....••.

T

4"

t=

f

t -- 3T 4

®

®

····t···t···t·~O···

c

............

++++

·~·B··®·····®··~~ ®

~ ®

......•••....

®

soidally, then so do the fields. Since the charge and current are a quarter cycle out of phase, we also expect the electric and magnetic fields to be a quarter cycle out of phase. Figure 2 shows the approximate field configura­ tions and the charges and currents during various parts of the cycle. Oscillations in an LC circuit do not continue forever, but are damped by the energy loss due to resistance which is always present in the circuit. This phenomenon was studied in detail in Experiment EC-3. Similarly, oscillations in the klystron are damped by energy loss through radiation and currents in the cavity wall. The problem is to feed back energy into the oscillating field to compensate for these losses and permit sustained oscillations. This is accomplished by means of an electron beam passing through the grids. Electrons are emitted continuously by the cathode and are accelerated by the field between cathode and cavity. As they pass through the alternating field between grids, some electrons are accelerated, others slowed down, depend­ ing on the phase of this oscillation when they arrive. As a result, some elec­ trons drift forward and some backward in the beam, relative to the average beam velocity. The result is a bunching of electrons at certain positions along the beam. After passing through the screens, the entire beam is stopped and reversed by the negative potential on the reflector electrode (hence, the term reflex klystron). The bunched beam returns to the grids in the form of a series of 2

.rter

be a

ura­

introduction

FIGURE 3 (a)

A

V= -Vc

M

c·· · ··/.· t· ·. · ··,

= +VB

h I

V=o

A . . . . ......o. .

c

~v

..... J............ .. I I

I

M

lC

V

I

(c)

~v v

\

....l ............"-J

Electrons are emitted from a heated oxide cathode and accelerated to­ ward the anode.

f..~Y.

V=-Vc

I'~ - ---V=O

I

· · ·...................."-J

· · · · · · · · · . ·····, G,,

v=o

They pass through the anode grids into the drift region to be repelled back through the grids. (d)

~ l - r - --V=O I

\

..

·.·.·.·i·..·.. ....O C :

\

·~

~V

··....f····..·...... I

I

h

~coswt 2V B

If the resonant structure is in oscilla­ Electrons which cross the grids at t = 0 are accelerated, they go further tion, the electrons wiIl be velocity­ modulated when they emerge from before being reversed, and return later. the grids.

I

(e)

d by Th is jons ts in field lis is

( f)

~ . - - -­ V=o ,.,

. . . I ..k.0.. . C I

\

~V

......1' ............ I

h

~rids.

ythe field >end­ elec­ erage dong

ersed reflex 'ies of

(b)

Electrons which cross the grids at = ~ T are decelerated, they do not go so far before being reversed, and they return earlier. t

. . . ·. ............ ·......·, C •

I

I

tl ·~

.....

~

'

~V

..

~

n

Thus electron bunches form around the t = ~ T electrons. These bunches may reexcite the cavity on their re­ turn. 3

microwave optics

pulses ofelectrons, arriving with the same frequency as that ofthe oscillation. Now if the reflector voltage is properly adjusted, these pulses return again at a time in the cycle such that they are retarded by the alternating field between grids; the energy lost by the electrons goes into the oscillating field, and hence provides the mechanism for feedback of energy into the oscillations with the correct frequency and phase. The operation is shown schematically in Fig. 3. The crucial factor is the time that the electron bunches require for the round trip; this, in turn, is determined by the beam and reflector voltages, and so it is not surprising that the klystron will oscillate only with certain combina­ tions of these two voltages. However, for a given beam voltage there are several possible values of reflector voltage which give the correct phase relation. The operation of the klystron will be studied in more detail in Experiment MO-3. To draw microwave power out of the klystron, a small coupling loop is inserted in the cavity. The oscillating magnetic field in the cavity induces a voltage in this loop; the resulting microwave current flows down a coaxial output lead to an output antenna, which radiates into a waveguide and finally into the output horn. The output from the horn, at sufficient distances, is approximately a plane wave. Microwave detectors use a principle very similar to that of detectors in ordinary am radio receivers. We use a semiconductor diode which conducts preferentially in one direction. At microwave frequencies the junction capacitance acts to filter the rectified signal, so that the terminal voltage of the diode is a dc voltage which depends on the amplitude of the microwave signaL The response characteristic of the IN23, a typical microwave diode used in these experiments, is shown in Fig. 4. At small amplitudes the diode FIGURE 4

1V

lOOmV

v

IOmV

lO,uW

ImW

lOO,uW

lOmW

p

voltage is approximately proportional to microwave intensity (power per unit area); this, in turn, is proportional to the square of the E field amplitude. At higher levels the diode voltage becomes more nearly proportional to the E amplitude itself. This voltage is measured with an ordinary high-impedance 4

introduction

voltmeter such as a vacuum-tube voltmeter or an oscilloscope. By the nature of the device, the diode is sensitive only to the component of E along its axis; thus it can also be used as an indicator of the polarization of the wave. The diode is mounted in the waveguide of a receiver hom, which restricts the solid angle of acceptance.

5

experiment

M 0-1

microwave production and reflec tion

introdu c tion The 723 AlB reflex klystron is a commonly available low-voltage klystron, operating between 8.5 and 9.6 GHz, corresponding to wavelengths of 3.5 to 3.1 cm. The necessary voltages are supplied by connecting the klystron to the regulated power supply as shown in Fig. 5.

723 Al B Antenna Violet

Standing waves provide a convenient way to measure the wavelength of microwaves. A standing wave can be produced by the setup shown in Fig. 6. The wave is partially transmitted and partially reflected by each grid, so that the wave in the region between grids is a superposition of waves in both directions, a standing wave. To analyze this situation in more detail, we represent the action of each grid by means of a transmission coefficient t and a reflection coefficient r, defined as foll ows: If a wave of amplitude E j is incident on the screen, the transmitted wave has amplitude IEj and the reflected wave rEj. If the screen does not absorb power (which is the case provided it is a good conductor), 7

microwave optics

FIGURE 6

Incident wave

Grids

z=L

z=O

then the total power in transmitted and reflected waves must equal that of the incident wave, which gives the relation (1)

Now we consider the multiple reflections which occur when two screens are present; the situation is shown in Fig: 7. The wave transmitted through

FIGURE 7

Eo

.. ..

III'

rE!,l rt 2 E

.. rtEo

0

2

r tEo

- II

.. r r

3

tEo

-

4

tEo

II

Grids

z=O

z=L

the second screen is a superposition of waves which have undergone various numbers of "internal reflections." In general, these partial waves will not all be in phase, since their total path lengths are all different, and partial destructive interference occurs. However, if the distance between screens is an integer multiple of 1../2, the path lengths differ by multiples of A, and an the transmitted waves are in phase. In this case we can find the amplitude of the total transmitted wave by simply adding the partial amplitudes. We obtain Et 8

t 2 Eo

+ ,2 t 2 Eo + ,4t 2 Eo + ... +

(1

+ ,2 +

,4 + ... )t2Eo

(2)

microwave production and reflection

MO-1

This sum is easily evaluated by means of the formula for the sum of terms in an infinite geometric series: I

+

!X

+

!X

2

+ ... +

I -

!X

(3)

which can be verified by division. We find I 2 E t = -1--2 tEo - r

(4)

and (5)

When the spacing between grids is an integer multiple of i../2, the wave is completely transmitted through the second grid. Otherwise part of the energy will be reflected back to the transmitter horn. Thus when the second screen

is moved back and forth, the transmitted intensity will reach a maximum whenever the grid spacing is approximately ni../2, where n is an integer. An alternative procedure for determining the wavelength is to investigate the standing wave between the screens, using a fixed screen spacing. Because of the reflections, the region between grids has waves in both directions, as already pointed out. If these waves have the same amplitude, there will be nodes in the standing-wave pattern, spaced a half-wavelength apart. At these points the amplitude is zero; midway between nodes will be antinodes at which the amplitude is twice that of either individual wave. Ifthe reflected wave has smaller amplitude than the incident wave, the total amplitude reaches a minimum, but not zero, at the nodes. If the "forward" amplitude is A and the "backward" amplitude B, then the amplitude at the B). The spacing of nodes antinodes is (A + B), and at the nodes it is (A and antinodes is the same as when the two waves have equal amplitudes.

I)

1S ~h

experiment 1

)Us

all :ial an the the ain (2)

klystron operation

To put the klystron into operation, connect the power supply as in Fig. 5. First, energize the filament by turning the power supply switch to "standby" and wait about a minute. With the + Band - C controls in about the middle of their range, apply operating voltages. Adjust the + B supply to 300 V and the -C supply to about 100 V. The beam current should be about 25 rnA. Connect the VTVM to the detector, and place the detector horn facing the output horn of the klystron. Should the VTVM be set on ac or dc? Now while watching the voltmeter, vary the reflector voltage ( - C) and observe how the microwave output varies. By varying -C from 100 to 200 V you should observe several maxima and minima of output, corresponding to conditions that the bunched electron beam returns to the cavity with the proper phase to reinforce oscillations or to cancel them. Adjust - C for a maximum output in the vicinity of 150 V. 9

microwave optics

By moving the detector hom around, explore the radiation pattern of the output hom. How does the intensity vary with distance from the output hom?

2 polarization By changing the orientation of the detector hom, show that the microwaves are plane-polarized. By observing the orientation of the diode in the detector, determine the plane of polarization. Measure the detected signal as a function of detector hom angle. Can you predict what the functional relationship should be? The metal grids which are provided act very much like polarizing filters with a light beam. Transmission through the grid is almost 100% when the E vector is perpendicular to the slots, but much less than 100% when parallel. Can you understand this behavior on the basis ofcurrent induced in the grids? Verify this behavior of the grids experimentally. Now "cross" the transmitter and detector horns (at 90° to each other) so no signal is received. Under what conditions will inserting a grid between the horns produce a signal at the detector? What will be the polarization of this signal? Test your predictions experimentally. Is it possible to modify the polarization plane of the microwaves by inserting anything between source and detector? Try a few things. 3

standing waves The standing-wave pattern can be observed as follows: Insert a thin sheet of plywood between the two horns. Part of the wave is absorbed by the wood, part transmitted, and part reflected. Absorption will be greatest when the wood is near an antinode in the standing-wave pattern, and least when it is near a node. Thus when the wood is moved along the axis, the intensity measured by the detector will vary in a cyclic manner. Measure the positions of several consecutive maxima and minima. From these determine the wavelength, and from it calculate the frequency of the klystron.

4

frequency variation The frequency of the klystron varies slightly (the order of 1%) with reflector voltage, but larger frequency variations can be achieved by mechanically changing the klystron grid spacing by turning the screw on the klystron, provided for that purpose. Determine the maximum and minimum wave­ lengths your klystron can produce at large enough power levels to permit convenient measurements.

questions

10

1

What becomes of the electrons in the beam after they are reflected and pass through the grids for the second time?

2

Why will the klystron oscillate only with certain combinations of beam voltage and reflector voltage?

3

What electrical effect is achieved by changing the grid spacing in the klystron? Why does this change the frequency?

microwave production and reflection

le 11

es Ir,

MO-1

4 Is it possible to produce circularly polarized microwaves?

5 Prove that in a standing-wave pattern produced by waves of amplitudes A and B respectively, propagating in opposite directions, the maximum and minimum amplitudes in the standing-wave pattern are (A + B) and (A - B), respectively.

In

ip rs le

:1. ;1 ;0

le lIS

'y

of d, Ie is ty m Ie

or ly

s !is

m

11 11

experiment

M 0 -2

interference and diffraction

I

I introduction The terms interference and diffraction were originally used to refer to optical phenomena which show departures from the straight-line propagation of light predicted by the ray picture. The most famous example (and historically one of the most significant) is the two-slit experiment performed by Thomas Young in 1802; this experiment provided important support for the wave theory oflight and also made possible the first determination of wavelengths of light. Because of the small wavelengths of visible light (4 to 7 x 10- 7 m) interference effects are seen only in correspondingly small-scale ways, such as light passing through very small apertures or in the details of the edge of a shadow. The longer wavelengths of microwaves makes it possible to observe interference and diffraction easily with apparatus ofmacroscopic dimensions. In this experiment you will be able to duplicate Young's two-slit experiment with microwaves; and a number of related phenomena can also be observed. Suppose a metal plate with two thin slits is placed perpendicular to the microwave beam, as shown in Fig. 8. The slits then act as secondary sources FIGURE 8

d

of radiation, according to Huygens' principle, and the intensity of radiation at any point beyond the plate is determined by the relative phase of the waves arriving from the two sources. This, in turn, is determined by the difference in path length; if the difference is zero or an integral number of wavelengths, the two waves reinforce each other; if the difference is a half-integral number of wavelengths, cancellation occurs. In terms of the angle in the figure, the conditions for constructive and destructive interference, respectively, are: Constructive: d sin = n).. n = 0, ± 1, ±2, .. . (6) n 0, ± I, ±2, .. . Destructive: d sin = (n + t))..

e

e e

13

microwave optics

The intensity of radiation is proportional to the square of the maximum E field; hence, the intensity at the points of constructive interference is four times as great (not twice) as that due to one of the sources by itself. This relation may be checked experimentally. If the aperture width is not small compared with the wavelength, there is a phase difference between radiation emerging from different parts of each aperture. In this case the resulting radiation pattern is computed by using Huygens' principle, adding the contributions from the separate elements of aperture, with appropriate consideration ofthe phase differences, to find the total radiation field. This usually involves integrating over the width of the apertures. Such calculations are discussed in most standard texts and will not be repeated here. For a single long slit of width a, the radiation intensity is given by sin 2 (<1>/2) 1 = 10 (<1>/2)2 (7) where <1> = (2nj).)a sin () and 10 is the intensity in the direction ()

O.

experiment 1

klystron operation Connect the klystron to the power supply and the detector to the VTVM just as in Experiment MO-l, and put the klystron into operation. The angular selectivity of the detector can be improved, at the price of a decrease in sensitivity, by mounting a baffle in the form of a narrow slit across the opening of the detector hom.

2

two-slit diffraction With the two-slit sheet in place, locate the central maximum and as many maxima and minima at each side as possible. Measure the positions, and from them calculate the wavelength of the radiation. Compare your result with that of Experiment MO-l.

3

detector response

Block one slit with a metal sheet, and measure the intensity directly in front of the open slit. Compare this with the intensity of the central maximum in the two-slit pattern. What can you conclude from this about the response of the detector in this intensity range; that is, is the voltage proportional to E, to E2, or to something else? 4

14

single-slit diffraction Mount a wide (a > )./2) aperture; locate and measure as many maxima and minima as you can. Measure the intensity as a function of angle for at least two or three points between the central maximum and the first minimum on one side, and between the first and second minima. Is the diffraction pattern symmetric? Compare the positions of the maxima and minima with the predictions of Eq. (7), using the wavelength determined from the two-slit experiment. Plot a graph of intensity versus angle. You may wish to explore the diffraction patterns formed by other com­ binations of slits with various widths and spacings.

interference and diffraction

5

MO-2

axiaUy symmetric apertures An interesting variation is to use an axially symmetric aperture arrangement such as those shown in Fig. 9, and observe the variation in intensity at points

FIGURE 9

~--~--=-----------------~~-----------p

along the axis. If a plane wave is incident from the left, then all points in the plane of the apertures radiate in phase. The condition for constructive interference at a point on the axis is

(r 2

+ [2) 1/ 2

-

I = nA

n

= 0,1,2, ...

(8)

and for destructive interference it is n

= 0,1,2, ...

(9)

In practice, the intensi ty from the outer ring may be different from that of the inner ring, and t he destructive interference may appear as an intensity minimum rather than zero in tensity. 6 Fresnel zone plate An elaboration of this idea is the Fresnel zone plate. To illustrate, we consider a circular aperture, as in F ig. 10. Considering points at increasing distances r from the center, the waves from these points arrive at point P on the axis with a phase lag, which increases with, relative to the radiation from the center. Now suppose we draw circles to represent successive phase changes of multi­ pIes of n (one-half cycle) relative to the phase at the center. The radius of such a circle clearly must satisfy the requirement

'n

or

(10)

These circles divide the aperture into a set of rings. On the average, the radiation at point P from one ring is one-half cycle out of phase with that from either ring adjacent to it, and so partial destructive interference occurs. 15

micro wave optics

FIGURE 10

p

Now suppose we block out alternate rin gs, as shown in the figure. Interfer­ ence from the alternate rings which remain will then be constructive, on the average, and the result is an increase in intensity at point P. Since this con­ dition of constructive interference is established with a particular value of I, we shou ld not expect the interference to be constructive for different values of I. Hence, a the detector is moved along the axis, a sharp maximum should be seen at a distance I from t he zone plate. Thus the action of such an arrange­ ment, called a Fresnel zon plate, is similar to that of a converging lens with a light beam.

questions 1 In the diffraction and interference experiments using slits, should the E field

of the incident wave be parallel or perpendicular to the long dimension of the slits? Explain. 2 Show that the intensity of the central maximum of the two-slit pattern is

four times as great as the intensity from a single slit of the same dimensions. 3 With visible light it is customary to distinguish between Fraunhofer and

Fresnel diffraction. Is this distinction relevant for microwaves? Explain. 4 Suppose the plane containing the rectangular slits is not perpendicular to the

axis of the apparatus, but is tilted at some angle. Qualitatively, how will this affect the resulting interference pattern ? 5 In the experiment with circular ring apertures, suppose the apertures are

close enough to the output horn so the incident wave cannot be considered 16

interference and diffraction

MO-2

to be a plane wave, but instead is a diverging spherical wave. Qualitatively, how will the interference pattern along the axis be affected? 6

Suppose a Fresnel zone plate is to be used not with an incident plane wave, but with a spherical wave from a point source a distance d from the zone plate. How should the rings be designed?

17

experiment

M 0-3

the klystron

Introduction In this experiment you will investigate the operation of the klystron in more detail than in Experiment MO-I, and can observe its various modes of opera­ tion. As explained in the INTRODUCTION, the heart of the klystron is a doughnut­ shaped cavity which functions as an LC resonant circuit; the resonant frequency of this cavity determines the frequency of the oscillations. Cal­ culating this frequency from the geometry of the cavity is difficult to do with any great precision, but a rough approximation can be obtained by regarding the grids as a capacitor and the doughnut wall as a single-tum inductor, and using the familiar relation (j) = (l/LC)l!2. If the area of the grids is Ac and their separation d, as is Fig. 1, then the capacitance is given by (11)

The inductance of a toroidal solenoid with mean radius Rand cross­ sectional area A L with n turns (here n = 1) is (12)

Thus, the resonant frequency of the cavity is given approximately by (j)

= (BoAc fJ.O A L)-112 =

d 21tR

1 2= 1 1 (fJ.oBO) 12 ALA~)

(21tRt!\1

c(21tRd\ 12 1

ALA~)

(13)

In the klystron used in these experiments A Land Ac are each the order of

1 cm 2 , R is the order of 1 cm, and d is about 0.1 cm. Inserting these values in Eq. (13), we obtain a rough estimate of the frequency of the klystron:

f= 8 GHz

(14)

The corresponding radiation has a wavelength A in free space of

.1.=c

f

3 cm

(15)

The frequency of the resonant structure is adjusted by varying the spacing d between grids, using the external screw adjustment. Bringing the grids closer together increases the capacitance and decreases the frequency. The essential condition for the operation of the klystron is that the electron beam must feed in energy to replace losses due to cavity wall resistance and radiation. To accomplish this, the electron beam, which is "bunched" by the oscillating field during its first passage through the grids, must return to 19

microwave optics

the grids after reflection in the proper phase to reinforce the oscillations. We now examine in more detail the conditions under which this can happen. For simplicity we assume that all the electrodes can be treated as planes whose dimensions are much larger than the distances between electrodes. In this case the fields between adjacent electrodes are nearly uniform. Electrons are emitted from the cathode and accelerated by the potential VB' arriving at the first grid with speed Vo and corresponding kinetic energy imvo 2 given by (16)

Between the two grids is an oscillating voltage AV given by AV = Vo sin rut

(17)

which changes the kinetic energy of the electrons by an amount e A V. If L\ V is small compared to VB' then the corresponding change in energy is given approximately by . (IS)

Thus, the electrons emerge from the second grid with velocities depending on the time, according to e.

A

+ t..>v = Vo +

v = Vo

mvo

Vo sm rut

(19)

Those electrons which pass between the grids when AV is positive are accelerated and acquire energy from the rf field. Those which pass through when AV is negative are decelerated and give up energy to the rf field. Since the electron current is constant, these energy changes nearly balance out. In the region between the second grid and reflector the electrons move under a decelerating field E given by E = (VB + Vd/L and experience an accelera­ tion

a=

eE m

e m

(20)

An electron emerging from the second grid at a time to with a velocity Vo + Av will have its velocity and position described by the usual formulas for constant acceleration: v

z

(vo

(vo

+

+

Av) (I

Av) 10 )

+ Vc (1 - to )

e VB

m

-"2I

L

e VB

m

+

L

Vc (

t

(21)

(22)

where we have taken z 0 to be the position of the second grid. To determine the time at which this electron returns to the grid, we set z = 0 in Eq. (22) and solve for t - to. We find t - to = 2(vo

+

Av)

m

L

e VB

+ Vc

(23)

That is, electrons with a positive Av, corresponding to positive AV, return later than those with negative Av. This difference in round-trip time is responsible for the bunching of electrons in the beam. To understand how this comes about, consider the electrons passing through the grids when 20

the klystron

MO-3

a v is decreasing through zero. Electrons passing through slightly earlier are accelerated, so they take longer to return and hence fall behind. Those passing through slightly later are decelerated, take a shorter time to return and therefore catch up. Thus the electron density in the beam is enhanced in the vicinity of those which pass through the grids when a v is zero and falling. A similar argument shows that the density is correspondingly depleted during the opposite point in the cycle, when a v is zero and rising. If the bunched electron beam is to feed maximum energy into the cavity oscillations, the bunches should pass back through the grids when aVhas its maximum positive value so that they may be decelerated by the rf field and hence give as much energy as possible to it. Clearly, one possibility is for the total transit time to equal three-quarters of a period T of the oscillation. But the same effect is achieved if the transit time is greater than this by an integer number of periods; the phase relation is the same. Thus, the energy regenera­ tion to the cavity oscillations will be a maximum when the transit time given by Eq. (23) with is equal to (n + ~) times the period, T, where n is an integer. That is, the condition for maximum energy feedback is

°

av

2mvo

e

L

- - - = (n

VB

+

Vc

+ ~JT

n

0, 1, 2, . "

(24)

Finally, expressing Vo in terms of VB by means of Eq. (16), we obtain the necessary relation between VB and Vc:

(VAVB)1/2 4 ----'-'----''---- = n VB + Vc

~

+4

n = 0, 1, ...

(25)

where VA is an abbreviation for the quantity mL2 VA = 2eT 2

(26)

which is a characteristic of the klystron. For each value of VB there are several values of Ve , depending on the value of n. Thus Eq. (25) gives a family of curves, one for each value of n, which are called the modes of the klystron. Typical curves are shown in Fig. 11. Equation (25) gives the condition for maximum feedback of energy into the cavity oscillations. But even if this condition is not satisfied exactly, some FIGURE 11

150

100

50

21

microwave optics

feedback will occur, provided the bunches of electrons arrive some time during the positive half of the d V cycle. The difference is that less energy is fed back, and so the amplitude of the oscillations is reduced relative to its value when Vc has the optimum value. In other words, for each mode and for a given value of VB there is a range of values of Vc for which oscillations occur; the maximum power output occurs when Vc is near the center of this range. Figure 12 shows a graph of FIGURE 12

Reflector voltage

power output as a function of reflector voltage for a particular mode and for a fixed value of VB' The points at which the power reaches zero are the points where the electron bunches are nearly one-quarter cycle out of phase with d V and can no longer feed back enough energy to replace the cavity and radiation losses. Understanding of the electron motion in the klystron is facilitated by a diagram called an Appelgate diagram, shown in Fig. 13. This is a graph of position versus time for electrons passing through the grids at various times FIGURE 13

R

-----------------------------------------------------------vc

~'-------------------------------------------------------o

22

the klvstron

MO-3

during the ~ V cycle, which is also shown along the time axis. As the curves show, the total transit time is greatest for an electron starting out from the grids at a time when ~ V has its maximum positive value, and least when ~ V has its maximum negative value. The diagram also shows the bunching of electrons in the vicinity of those which pass through the grids when ~ V is zero and decreasing, as already discussed. The characteristics of the klystron for a particular value of VB are shown in more detail in Fig. 14.

experIment 1 klystron output

The dependence ofthe klystron power output on VB and Vc can be observed conveniently by holding VB constant and varying Vc periodically, observing how the power output varies with Vc. This variation is easily displayed on an oscilloscope by connecting the detector to the vertical input, and using the "sawtooth" horizontal sweep voltage to vary Vc. Thus the periodic variation in Vc is automatically synchronized with the scope sweep. In Fig. II, if VB is set at 100 V and Vc is swept from - 50 to 150 V, we should pass through the n = 8, 7, 6, and 5 modes. For each of these we expect to observe a maxi­ mum in power output near the mode center, as shown in Fig. 12. Thus if Vc sweeps through several modes, the scope curve should be similar to that shown in Fig. 14. To provide the sweep of Vc, the circuit of Fig. 15 may be used. The heater, cathode, and grids are connected just as in Experiment MO-I, but the reflector is not connected directly to the C supply. Instead, it is connected through a I-MO resistor, as shown, across which appears the voltage from the external sweep output of the scope. Thus, the instantaneous reflector voltage is the sum of the C supply and sweep voltages. The capacitor serves to isolate the sweep generator from the dc voltage of the power supply, but offers negligible impedance at the sweep frequencies used. 2 klystron modes

By reducing the horizontal gain of the scope, the amplitude of the sweep voltage to the klystron may be reduced. By adjusting the reflector voltage from the C supply, one of the modes may be centered in the scope sweep. Since the center corresponds to an instantaneous sweep voltage of zero, the values of VB and Vc under this condition are those for the center of the corresponding mode. Now it is possible to vary VB in increments and make corresponding changes in Vc to keep the mode centered in the sweep. In this way the data required to plot one mode may be obtained. Repeat this process for as many modes as you can locate and plot a family of curves similar to those in Fig. 1l. 3 equivalent trajectories

Because of the approximations made in the derivation of Eq. (25), especial­ ly the assumption that the retarding field between grids and reflector is uniform, one should not expect too precise agreement of the experimental curves with this equation. However, certain features of the modes are inde­ pendent of the details of the field configuration. This point is illustrated by 23

miclQwave optics

FIGURE 14

8530

=8510~-4++--~--~~~-4~~---4--~~ 8520

:

g

8~~~~--+----4-+~-4--4----+-,~~


g. 8490 -4-1+___+_--+----I+---'-~'---­ i: 8480 -4-++-+---~___jK-+~r__--+_- -ft__---tt--i 8470~++~--~~~···············1----4--~+---~·-i

40~~r=~~+=~+===r==+~~~ 30~~-r--~~~~~----+_---H~-4~

20 10~~-+--+_~~~___j----4---~----~

Reflector voltage

9690-4--~---+----+_--+_--___+_----t__--_i_--~

=9680r---~--_4----~--~----t__--~--_4--~ 9670~--_IH_--­

:

g
9660~-+-t'---­

g.9mror-+--i+--_4----~--___+_----t__--·-~--_4~~ Q ~ 9640~~~--~----+_--_T----t__--_r--_4_i_~ 9000~---tt----+--~~--~--___+_--~----++~

•

Reflector voltage

the technique used to determine the value of n corresponding to each mode. The simplest way to make this determination is to draw a line of convenient slope on the graph of VB versus V c , as shown in Fig. 16. The equation ofthis line is (27)

where k is the slope of the line. Substituting this equation into Eq. (25), we obtain 3

n

+"4 =

+k

Vj/2

(28)

This equation states that if Vc and VB are increased in the same ratio, then the values corresponding to modes (the intercepts of the straight line with the 24

the klystron

MO-3

FIGURE 15

Regulated power 'upp\y

1:1 Sia .J..

Wh it ,

723 AlB Antenna Violet

Sweep

various mode curves) occur at values of VB such that Vi 1/2 is proportional to (n + ~). Thus if these values are plotted at unit intervals along the hori­ zontal axis, as in Fig. 17, the result should be a straight line with intercept at - 1. Thus the value of n for each mode can be read off from this graph. It may tum out that the linearity of this graph is much better than the agree­ ment of the mode curves with Eq. (25). The reason is that if we change VB and Vc in the same proportion, the electron trajectory will be unchanged' the electrons will turn around at exactly the same place. This statement is FIG URE 16

25

microwave optics

FIGURE 17

1

~ B

independent of the details of the geometry. Thus the total transit time will be proportional to the speed Vo when leaving the grids; this in turn is propor­ tional to V.J/2. Since the transit times for various modes must be in the ratio n + ;i, we conclude that the linear variation shown in Fig. 17 is independent of the details of field shape.

4 determination of L Use the data obtained from your graph similar to Fig. 17 to compute the constant VA' From this calculate the distance L from grids to reflector. Compare with measurements on a disassembled klystron if one is available.

questions 1 The frequency of the klystron varies slightly (the order of 50 MHz) as Vc is varied from one side ofa mode to the other. Why should this variation occur? 2 What minimum sweep frequency must be used if the effect of the capacitor coupling the sweep generator to the reflector is to be negligible? 3 If all the dimensions of the resonant cavity were doubled, how would the frequency change? The wavelength? 4 What is the relative phase of the magnetic field in the cavity with respect to the passage of bunches of electrons back through the grids? 5 If the maximum value of A V between the grids is 10 V, what is the order of

magnitude of the maximum current flowing around the cavity walls between grids? Of the maximum value of B in the cavity?

26

expe riment

microwave propagation

introduc tion In this experiment you will investigate several additional aspects of micro­ wave propagation. These include wave propagation in a waveguide, circul r and elliptical polarization, and related phenomena. The same basic micro­ wave oscillator and detector equipment are used as in the previous microwave experiments.

vill or­ tio ent

the :or. ble.

MO-4

WAVEGUID E

In previous experiments we have considered the microwaves emitted by the klystron horn to be approximately plane waves. We now investigate the natu re of the waves when the hom is placed betwee two parallel conducting sheets forming a channel or waveguide, as in Fig. 18. These heets change the

FIGURE 18

ns E

;itor B

I the a

ct to

er of ween

nature of the wave because of the boundary conditions imposed by their presence. If the conductors are ideal (L e., have zero resistivity), the com­ ponent ofE parallel to the surface, usually denoted as E il' must vanish at the su face. The component E .l perpendicular to the surface need no t vanish, but is proportional to the surface charge density (J on the conductor surface. Thus any wave which propagates in the channel must have a field configura­ tion which satisfi es the conditions Ell = 0

(29)

on the boundary surfaces. There is an important practical motivation for studying such an arrange­ ment. The problem of conveying alternating currents from one place to 27

microwave optics

FIGURE 19

another by use of wires becomes increasingly difficult as the frequency increases, because of power losses due to radiation and because of the skin effect (the tendency of high-frequency currents to flow only on the surface of conductors). The radiation problem can be solved in some frequency ranges by using a pair of conductors in the form of coaxial cylinders, so the fields are confined to the region between conductors and cannot escape into space. But at extremely high frequencies the dielectric which is included for mechanical support of the center conductor becomes an absorber of energy. At microwave frequencies, corresponding to wavelengths of the order of 10 cm and less, the use of hollow waveguides becomes practical. The wave is contained in a hollow conducting pipe, usually of rectangular cross section. In general, many possible field configurations or modes are possible, but if the dimensions are chosen correctly in relation to the wavelength, the wave­ guide can be designed so that only one mode is possible. Examination of wave propagation between parallel conducting sheets exhibits the most important features of wave propagation in a waveguide. We consider two possibilities; the wave may have its E field perpendicular to the channel sides in the region between them, or it may have E parallel to the sides. In the first case there is nothing to prevent the propagation of plane waves just as though the channel were not present. The E field induces time­ varying surface charges on the side planes, but in the region between them the wave is still an ordinary plane wave, propagating with the same speed and wavelength as in free space. z

r

L

+ +++

++

+ +++

++ + i

+

+ +++ +

++

-I-

+ + +

++++

+++

++++

+++

7 "J" P'

++ ++

~~------a--~---+--~----------~~

28

microwave propagation

MO-4

When E is parallel to the planes, the situation is quite different. Since a plane wave has the property that at any instant E is uniform over any plane perpendicular to the direction of propagation, a plane wave would not satisfy the requirement that Ell be zero at the boundary surfaces. Hence a plane wave with this polarization cannot propagate in the channel. We are led to ask what sort ofwave can propagate in the channel; that is, what sort of wave can be constructed which will satisfy the requirement that Ell = 0 at the boundary surfaces? The nature of the wave that can propagate in the channel is suggested by recalling that the requirement Ell 0 is also responsible for the reflection of waves at the conducting surface. Suppose a plane wave enters the channel at an angle 0 and is successively reflected from the two sides. Such a wave can be represented as the superposition of two plane waves with the same wave­ length and polarization, propagating in directions which are horizontal but which make angles 0 and 0, respectively, with the channel. Such a super­ position is shown schematically in Fig.. 19. The figure shows wavefronts, with successive "crests" indicated by + + + + and successive valleys by - - - -. The superposition of the two waves gives a wave propagating along the channel. Along the center line of the figure the E fields add to give an amplitude twice as great as that of either plane wave. But because of the tilt in the direction of one wave with respect to the other, as we move away from the center line the two waves become more and more out of phase, until finally we reach a point where they are exactly a half-cycle out of phase, at which point their sum is zero at every instant. If such points lie on the bounding surfaces, the superposed wave then satisfies the boundary condition that Ell is zero on the boundary. As time progresses, the whole pattern moves down the channel, and this con­ dition is always satisfied. As the figure shows, the necessary value of 0 is determined by the width a of the channel and by . 1. 0, which, in tum, is determined by the frequency of the wave in the usual way: .

C

)\.0

=-

f

(30)

Referring to Fig. 19, we obtain the relation

2a = sin 0

(31)

Since sin 0 cannot be greater than unity, this relation shows that propagation of a wave, such as the one just described, is possible only when the free-space wavelength ;"0 given by Eq. (30) is less than twice the width of the channel. This imposes a lower limit on the frequency of the wave which can propagate in the channel. Alternatively, if the frequency is fixed but a can be adjusted, which will be the case in the present experiment, we expect that wave propa­ gation is possible only when

a> 2

(32)

It is interesting to note that the wavelength of the superposed pattern shown in Fig. 19, that is, the distance between successive maxima of E, is not equal to ..1.0 but rather is

A 9

=-~ cos 0

(33) 29

microwave optics

where the symbol Ag denotes the wavelength of the composite wave. Com­ bining Eqs. (31) and (33), eliminating (J by means of the identity sin z () + cos 2 (J = 1, and solving for Ag , we obtain the further relation (34)

which may also be written in the form =

1 -;::-z Ag

1

+ 4---Z a

(35)

The velocity of propagation of the composite wave is also of interest. Since each component plane wave is a solution of Maxwell's equations in vacuum, each component wave propagates with speed c = AO! Thus the propagation speed u along the channel for the composite wave is given by

u

c Agf = [1 _ (Ao/2a)Zr/2

(36)

This may seem alarming, inasmuch as this velocity is greater than c and may appear to violate the basic requirements of relativity. But we recall that the speed with which information is conveyed is not the phase velocity (which we have just calculated) but the group velocity, which, when velocity depends on frequency as in the present case, is quite different. In this respect the wave­ guide acts as a dispersive medium, in which the phase velocity is different for different frequencies. More specifically, the phase velocity u is given in general by (37)

whereas the group velocity v is df v = d(l/Ag)

(38)

Differentiating Eq. (35) with respect to I/Ag and using the above relations, we find that

A

v=c [ l - ( 2:

)ZJ1/2

(39)

which shows that although the phase velocity of the wave in the waveguide . is always greater than c, the group velocity v is always less than c. We also see that uv = c 2 , independently of AO' .

POLARIZATION

30

If the output horn is turned about its axis so that E makes an angle

microwave propagation

MO-4

ditions are arranged so the phase difference is n/2 and if the amplitudes are equal, the result is a circularly polarized wave, characterized by the fact that the total E field is constant in magnitude and rotates with angular velocity 0 ill = 2nf abou t the direction of propagation. If the phase difference is 180 , the emerging wa ve will be a plane wave polarized at right angles to the incident wave. Intermediate cases yield elliptical polarization. T hese various states of polarization can be observed experimentally.

1­

t­

4)

:5)

st. in :he by

experiment 1

36)

lay the lich nds

lve­

determination of gu ide wavelength Connect the klystron to the power supply and the detector to the VTVM, just as in Experiment MO-l , and put the klystron into operation. To investi­ gate wave propagation with the E field perpendicular to the guide plate , tum both the transmitting and receiving horns on their sides. To measure the wavelength in the channel we use the arrangement shown in F ig. 20, consist­

FIGURE 20

for

(37)

(38)

ions,

(39) ~uide

: also ing of the same pair of slotted plates used in Experiment M O-l with the addition of a p air of side plates. The wavelength determination proceeds just as in that experimen t: Maximum transmitted intensity occurs when the distance between grids is an integer multiple of }.9/2. Find various combinations of grid separation Ag/2 and baffle separation a that give maximum transmitted intensity. You may use baffles of different lengths to obtain a range of values. The grids need not be in actual contact with the baffles, so you may obtain several transmission conditions with one set of baffles.

with ntally md a

I

~ristic ~ perty ~perty

lariza2 COffi­

meral, .f con­

cutoff To check your data agamst Eq. (35), plot a graph of 1/A9 as a fu nction of 1/2a; the p oints should all lie on a circle of radius l /}.o. Observe that as the channel width a becomes smaller and smaller, the 31

channel wavelength becomes longer and longer; as a approaches Ao/2, it becomes infinite, and when a is less than this critical value the wave does not propagate, as discussed previously. 3 elliptical polarization

To produce an elliptically polarized wave, orient the output horn at 45° to the vertical. Adjust the separation a between bafHes until the intensity of the received signal is independent ofdetector horn orientation. This is character­ istic of circular polarization. Now reduce a further until there is no signal when the two horns are parallel, but a maximum signal when they are perpendicular. Under these conditions the phase shift is 180°. By further reducing the separation, you may be able to produce circular polarization with a phase shift of 270 and so on. 0

,

questions , Suppose the conductors forming the side bafHes of the waveguide are not ideal but have some resistivity. What effect will this have on the boundary conditions, Eqs. (29)? 2 In the microwave polarization experiments, what experimental setup is the analog of an optical quarter-wave plate? 3 How can you determine whether a circularly polarized wave is right- or left­ circularly polarized? 4 How can you detect the presence of an elliptically polarized wave? In this

experiment, how are the axes of the ellipse oriented relative to the output horn? 5 Is ther~ such a thing as an unpo/arized microwave beam? How does this

situation differ from the case of visible light? 6 Show that the product of phase and group velocities for the wave in the waveguide is equal to c 2 • 7 Note that Eq. (39) may be written

v = c cos (J How do you interpret this relation?

32

ber e/ y p

/abo~

r, 2d

dition

a l an . portis, university of californ Ia . berkoley h ugh d . y u ng, ca rneglc mellon un iversIty reflection and refra c tion

flight

LO-1

p olarization of light

LO-2

diffraction o f /I ht

LO-3

Interference of light

LOR4

holography

LO-5

t ~

e 1­

is ut

is rle

mcgraw-hill n evv y ork london

st lo u is mexico

0

k co

stl n franCISc o panama

pan y dusseldorf

sy ney

toronto

,

II

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E

laser optics Copyright~) 1971 by McGraw-Hill, Inc.

All rights reserved. Printed in the United States of America.

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Library a/Congress Catalog Card Number 79-125108 07--oS0489-x 1234567890 BABA 79876543210

E E E. E E

"

F­ F­ F­ F­ F­

The first edition of the Berkeley Physics Laboratory

copyright © 1963, 1964, 1965 by Education Development

Center was supported by a grant from the National Science

Foundation to EDC. This material is available to publishers

and authors on a royalty-free basis by applying to the

Education Development Center.

This book was set in Times New Roman, printed on

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pany, Inc. The drawings were done by Felix Cooper;

the designer was Elliot Epstein. The editors were Bradford

Bayne and Joan A. DeMattia. Sally Ellyson

supervised production.

e. EC EC

EC EC EC

s

rk tal its :nt ns )Ie lre

laser optics

INTRODUCTION

ber CAUTION

In this series of experiments we shall study several basic properties of light, its propagation, and its interaction with matter. The light source used is a helium-neon gas laser. Light from this source is electromagnetic radiation, like all light, but it differs from light from most ordinary sources (such as the sun or an electric light bulb) in several important respects. Ordinary light is a mixture ofseveral wavelengths, with a corresponding variety offrequencies, whereas laser light is very nearly monochromatic, or single-frequency, and has a very definite, precisely known wavelength [6328 A for the helium-neon (He-Ne) laser]. Furthermore, ordinary light exhibits rapid and random variations in the phase of the emitted waves, whereas laser light preserves definite phase relations over much longer time intervals, and over much larger distances; this property is called coherence, and it is essential in observing various interference effects to be studied in these experiments. The physical basis of laser operation is investigated in Experiment AP-I and need not concern us in detail here. Very briefly, in an ordinary light source individual atoms act as the basic sources; an atom is given excess energy by thermal agitation or electrical excitation, and then radiates for a time (typical­ ly the order of 10- 8 sec) until it has lost this energy. Various atoms attain various amounts of excess energy, depending on their energy-level scheme, and the resulting radiation is a random mixture of the emissions of individual atoms. In the laser the situation is arranged so that many atoms are excited to exactly the same excess energy states, and the radiation from them is synchronized in frequency and phase by a process called stimulated emission. Thus laser light is much more coherent and more nearly monochromatic than ordinary light, and so provides a much better approximation to the simple sinusoidal plane electromagnetic waves often discussed in textbooks. In Experiment LO-I we investigate reflection and refraction of laser light at an interface between media of different indexes of refraction, using the ray representation of geometrical optics. Experiment LO-2 deals with the production and properties of linearly and circularly polarized light. Experi­ ments LO-3 and LO-4 deal explicitly with the wave nature of light; several examples of interference and diffraction of coherent light are studied. Finally, in Experiment LO-5 we see a particularly interesting application of diffraction of coherent light, the use of holograms to store and reconstruct three-dimensional images. The He-Ne gas laser used in these experiments produces about 0.6 mW (milliwatts) of radiant power over an area of about 2 mm 2 , corresponding to an energy flux of 0.03 W/cm 2 . For comparison, the energy flux from the sun is about 0.135 W/cm 2 • Thus, just as looking directly at the sun can cause eye damage, a laser beam may also be sufficiently intense to produce permanent damage to the retina. Therefore, NEVER look directly into a focused laser beam or its mirror reflection! 1

experiment

L 0 -1

reflection and refraction of light

introduction The phenomena of reflection and refraction are most conveniently dis­ cussed in the language of geometrical optics. This provides a very useful approximate representation of the behavior of lenses and mirrors in situa­ tions in which the dimensions are sufficiently large compared to wavelengths of light so that interference and diffraction effects can be neglected. In this approximation the propagation of light is represented as a strictly straight­ line phenomenon. In geometrical optics the propagation oflight is represented by rays, which travel in straight lines in a homogeneous optical medium, and whose behavior at an interface between two media is governed by two simple rules, the law of reflection and Snell's law of refraction. The law of reflection states simply that the reflected ray makes an angle with the normal to the surface equal to that of the incident ray, as shown in Fig. 1, and that the two rays are coplanar FIGURE 7

with the normal to the surface. The law of refraction states that the angles of the incident and refracted rays are related by n 1 sin rx = n 2 sin f3

(1)

where n 1 and n 2 are the indexes of refraction of the two optical media. In general the ray is partly transmitted and partly reflected at the interface between two transparent media, but if the surface is a mirror, then, of course, only the reflected ray occurs. These laws of geometrical optics are sufficient to analyze the behavior of all lenses and mirrors when diffraction and interference effects can be neglected. They can be regarded as empirical laws, forming what is perhaps the simplest of all physical theories. Alternately, they can be derived from electromagnetic theory, using Maxwell's field equations to analyze the behavior of a plane electromagnetic wave incident on a boundary surface. The laser beam, being a small pencil oflight about 1 mm in diameter, provides a physical situation approximating the abstract concept of a ray of light. 3

laser optics

FIGURE 2

By using a thick glass plate we may observe multiple reflections, as shown in Fig. 2. If d is the thickness ofthe plate, then the lateral separation a between adjacent rays, observed either in transmission or in reflection, is given by a

=

2d sin cx cos cx ~~--~--~

(n 2

-

sin 2

CX)1 / 2

(2)

Derivation of this formula is left as a pro blem. This formula can be solved for n; then by measuring d and a, one can determine the refractive index n of the material. The refractive index of a material always varies somewhat with wavelength. Following are a few typical values of n for the red region of the visible spectrum : Material air water fu sed quartz plexiglass crown gl ss flint glass diamond

n

1.000276 1.333 1. 458 1.49 1. 515 1.57 to 1.88 2.417

For vacuum, n = 1 by definition. Equation (1) can be used when the fi rst medium is either more or less dense than the second. When the first medium is less dense, the ray is refracted toward the normal ; when more dense, away from the normal. An interesting limiting case is the situation in which Eq. (1) gives an angle of 90° for the refracted ray; this occurs when the incident angle is given by (3)

When the incident angle is greater than this critical value, there is no real value of fJ which can satisfy Eq. (1). In this case there is no refracted ray, and the incident ray is totally reflected; not very surprisingly, this phenomenon is called total internal reflec tion. Total internal reflection can be observed easily using a 45 °-45°- 90° triangular prism, as shown in Fig. 3. Note that after two internal reflections, the ray emerges parallel to its original direction, even if that direction is not

reflection and refraction of light

L0 - 1

FIG URE 3

normal to the prism face. In that case, however, the internal reflection may not be total. The laws of reflection and refraction also provide the basis for analysis of lenses. Although we shall not investigate the behavior of lenses in detail here a few simple examples which are useful in later experiments in this series will be discussed. First, we consider the planoconcave lens shown in Fig. 4. Rays

tl

~

l) (I

n

FIGURE 4

~.

le

L --~

:ss is

\.n

0°

}) ~al

nd on

ns,

which are parallel to the optic axis initially are divergent after leaving the len , as tho ugh they had originated at the point F, called the fo cus of the lens. The distancef from F to the lens is called the focal length of the lens. We can show that the focal length of a planoconcave lens is given by

f=_R_ n - 1

(4)

where R is the radius of curvature of the spherical concave surface, and n is the refractive index of the material. Derivation of this formula is discussed in most introductory textbooks. Referring to Fig. 4, we see that the tangent of the angle f3 - a can be ex­ pressed either as r0/f or as (r - r0)/L (if we neglect the thickness of the lens relative to the other dimensions) which leads to the relation (5)

lot 5

laser optics

Thus the radius of the laser beam increases linearly with L, just as though it had originated as a point source at the focus F. Furthermore, since r is proportional to '0, the beam is spread uniformly. That is, if the initial beam has uniform intensity over its cross section, the diverging beam retains this property. By measuring the diameter of the diverging beam at several distances, one can determine accurately both the focallengthJ of the lens and the original diameter,o of the beam. The value ofJcan be compared with that computed from Eq. (4), or this equation can be used to determine the value ofn for the lens. For this, the value of R is required. A simple way to measure R is by front-surface reflection of the laser beam, as shown in Fig. 5. From the figure, we have

. SIll

8

a

=R

(6)

so R can be determined from the easily measured quantities a and 28. FIGURE 5

a

The addition of a planoconvex lens is just opposite to that of a plano­ concave lens. The convex lens converges a parallel beam to a point. A divergent beam is made less divergent, parallel, or convergent, depending on the position of the lens. When the distance from the virtual source of a divergent beam (i.e., the point from which the beam appears to diverge) is equal to the focal length of the convex lens, the final beam is parallel, with the same diameter the divergent beam had when it entered the lens. Thus a pair of lenses, one concave and one convex, can be used to obtain a laser beam which is larger in diameter than the original one but still parallel. This magnification also illustrates (although in a somewhat unfamiliar context) the basic principle of the Galilean telescope.

experime nt 1

m ult iple reflection

Place a thick slab of glass in the laser beam, and observe the multiple rays emerging from both sides. Measure the separation between adjacent rays, and the angle between the beam and the normal to the surface, and use Eq. (2) to determine the refractive index of the material. Repeat for at least two values 6

reflection and refraction of light

L0 - 1

of the angle. Replace the glass by a Plexiglas plate, and repeat the measure­ ments, again determining the refractive index. 2 total internal reflection Place the 45°-45°-90° prism in the beam as shown in Fig. 3 and determine whether total internal reflection occurs with the beam at normal incidence to the prism face. Change the incident angle until the total reflection situation changes, and measure the critical angle. Can this information be used to determine the refractive index of the prism?

3 diverging beam Place the planoconcave lens in the beam, and measure beam diameter r as a function of distance L from the lens, for several values of L. Plot r as a function of L; from the graph determine both ro and/, using Eq. (5). 4 determination of n Measure the radius of curvature R of the concave surface by the method illustrated in Fig. 5. Use this value of R and the value offobtained previously, together with Eq. (6), to determine the refractive index of the lens. 5 Galilean telescope With the planoconcave lens still in place, measure the focal length of the planoconvex lens by finding a position for it such that the emergent beam is parallel (neither diverging nor converging). The focal length is then given by the sum of the distance between the lenses and the focal length of the concave lens. Can you prove this? 6 determination of n Measure the radius of curvature of the convex surface by the same method used previously; from this value of R and the value of f just obtained, determine the refractive index of this lens.

questions 1 Solve Eq. (2) to obtain n as a function of a. In the limiting case n value of a do you expect? Does the formula give this result?

= I, what

2 Derive Eq. (2). 3 Can total internal reflection be observed in the situation of Fig. 2? Why? 4 Find the minimum value of n needed for total internal reflection in the

situation of Fig. 3, with the incident beam normal to the face. 5 The inside of a hollow cube is lined with mirrors. Show that a ray of light aimed into a corner (but not precisely at the corner) experiences three succes­ sive reflections and returns parallel to its incident direction, no matter what that direction was. An array of one hundred small corner reflectors was placed on the moon during the Apollo 11 landing. Laser pulses reflected from this array have been observed with a large earth-based telescope. 6 Suppose the prism you used to observe total internal reflection were im­ mersed in water. Would total internal reflection still occur? 7

experiment

L0 -2

polarization of light

Introduction Light, like all other electromagnetic radiation, is characterized by propaga­ tion in vacuum with electric and magnetic fields which at each point in space are perpendicular to the direction of propagation and to each other. If in addition each field lies always parallel to a single plane, the wave is said to be plane-polarized. The plane of the E field is designated as the plane of polariza­ tion, since in most interactions of electromagnetic radiation with matter, the E field rather than the B field provides the dominant interaction. It is convenient to represent the electric field in terms of two transverse components Ex and Ey as in Fig. 6, in which the direction of propagation is FIGURE 6

y

--------------~-------.~-----x

out of the page, toward the reader. If only one of the components is different from zero, then the wave is linearly polarized or plane-polarized in the corresponding plane. Conversely, if the x and y components have equal amplitudes but are completely uncorrelated in phase, then the wave is said to be unpolarized. Unpolarized light is inherently a statistical concept, implying complete randomness of the relative phase of the two components so that no prediction about the direction of the resultant is possible. However, if the x and y components have equal amplitude and are in phase, the result is a plane-polarized wave with polarization at 45° to the x andy axes. Linearly polarized light is most easily produced by selective absorption using H-type Polaroid. This material, which is available in large sheets, is prepared by absorbing iodine into stretched sheets of polyvinyl alcohol. In this process polymeric iodine is formed in a configuration which is strongly 9

laser optics

dichroic, which means that the absorption of light is different for different planes of polarization, in this case over 100 times as great for the component parallel to the direction of stretching as for the perpendicular component. For a sufficiently thick sheet, the emerging light is for all practical purposes linearly polarized. An alternative way to produce plane-polarized light makes use of a principle known as Brewster's law. When a wave is incident obliquely on an interface between two media, in general it is partly transmitted and partly reflected. According to Brewster's law, in the particular case in which the transmitted and reflected directions are perpendicular, as in Fig. 7, the FI GURE 7

n

Vacuum

component wave with its E vector parallel to the page is transmitted com­ pletely, without any reflection. In this case the component which is reflected is plane-polarized perpendicular to the page, no matter what the initial polarization of the incident wave. The angle of incidence satisfying the condition of Brewster's law, called Brewster's angle, is easily obtained from Fig. 7 by noting that in this case fJ = n/2 - Ci. and Snell's law becomes

n=

sin ex sin Ci. = - -= tan Ci. sin [(n/2) - ex] cos Ci.

(7)

Similarly, it is easy to show that for internal reflection the condition for com­ plete polarization of the reflected wave (Brewster's angle) is

n = cot ex

(8)

Brewster's law can be derived from electromagnetic theory, and is in fact a special case of a set of relations called the Fresnel equations, which give the amplitudes of the transmitted and reflected waves in general, as functions of the angle of incidence and the state of polarization of the incident wave. The derivation, although straightforward, is somewhat involved, and it will not be discussed in detail here. Instead, we present a brief argument to show why it is reasonable that there should be an angle at which a component wave polarized in the plane of incidence should be completely transmitted. This argument is given in Fig. 8. When a wave at normal incidence is reflected from an interface with a denser medium, the reflected wave has its E reversed, as in Fig. 80, as a result of the requirement that the parallel component of E must be continuous across the boundary. Similarly, a wave polarized in the plane of the page at grazing incidence results in a reflected wave with its E 10

polarization of light

tan

FIGURE 8

(a)

(b)

a

L 0 -2

=n

(c)

vector as shown in Fig. 8b. Now suppose the angle of incidence is varied con­ tinuously from 0 to 90°. For small angles the E of the reflected wave is away from the normal, for large angles toward the normal. Thus, there must be a transition point at some angle where this component goes through zero, and at this critical angle the wave is totally transmitted. Now suppose the dielectric in the Brewster's-law discussion is a glass plate of finite thickness, and suppose that light is incident on it at Brewster's angle, as in Fig. 9. The component with E in the plane of the figure is transmitted FIGURE 9

completely by the first surface. But since at Brewster's angle tan ft = nand f3 = (n/2) - ft, the angle of incidence at the second surface also satisfies Brewster's law, Eq. (8). Thus this component of the wave is transmitted completely, without any reflection at either surface. Such an arrangement is called a Brewster window. A common arrangement for a gas laser is to have windows at the Brewster angle at both ends as shown in Fig. 10. A wave propagating along the axis of the laser with its E in the plane of the figure is transmitted completely by the windows, and the external reflectors establish the standing waves neces­ sary for laser operation. A wave with E normal to the plane of the figure is partially reflected out by the Brewster windows, and so a standing wave with this polarization cannot be established. Thus the emerging light is plane­ polarized in the plane of the figure. Some lasers use mirrors inside the gas­ 11

laser optics

FIGURE 10

t Reflector

Gas tube

Reflector and collimator

filled tube, eliminating the need for Brewster windows. Such lasers oscillate simultaneously in both polarization modes, producing unpolarized light. With circularly polarized light the E vector is not confined to a single plane, hut instead it has constant magnitude and rotates in a plane perpendicular to the direction of propagation. Conventionally a wave is called right circularly polarized if the E vector rotates counterclockwise as seen in Fig. 6, that, is. as viewed in a direction opposite to the direction of propagation. Circular polarization can be represented in terms of the x and y compo­ nents of E; these components always have equal amplitude but a phase difference of 90°, just as circular motion of a particle can be represented as a superposition of two simple harmonic motions in perpendicular directions, and with a phase difference of 90°. In Fig. 6, for a right circularly polarized wave the y component of E lags the x component in phase by a quarter-cycle; for left circular polarization it leads by the same amount. Circular polarization can be produced by use ofa material whose refractive index is different for the two components of E. If a plane wave polarized at 45° to the x and y axes is incident on such a material, the x and y components of E are in phase as the wave enters the material. Because these components propagate with different velocities in the material, however, a progressive phase difference develops as the wave propagates through. If the plate is of the proper thickness, the two components emerge with a phase difference of 90°, just what is needed for circular polarization. Such a plate is called a quarter-wave plate (often abbreviated Aj4 plate). When the phase difference between x and y components of E is other than 90°, or when they have different amplitudes, the E vector as it rotates traces out not a circle but an ellipse, and the result is called elliptically polarized light. This case also can be regarded as a coherent superposition of plane- and circular-polarized waves. The process of depolarization, or making polarized light into un polarized light, is slightly more subtle. As explained above, it is necessary to introduce random phase differences between the two components of E; this is best accomplished by using a material which is inhomogeneous across the wave­ front, as well as anisotropic, in order to provide the necessary random phase shifts. In these experiments a sheet of ordinary waxed paper serves as an effective depolarizer.

experiment 1 polarization Project the laser beam onto a screen. Insert a piece of H-type Polaroid in the laser beam and observe the variation in intensity as the polarizer is rotated in 12

polarization of light

L 0 -2

the plane perpendicular to the beam. If you find an orientation for which the beam is extinguished, then the light emerging from the laser is already linearly polarized. For an unpolarized beam, inserting the polarizer should drop the intensity by a factor of 2, but the intensity will be independent of the orientation of the polarizer. Why? 2 extinction

To demonstrate that the beam emerging from the polarizer is indeed linearly polarized, insert a second polarizer between the first polarizer and the screen. With the orientation of the first polarizer fixed, rotate the second polarizer. You should be able to find an orientation of the second polarizer which extinguishes the beam. 3 three-polarizer experiment

With the two polarizers in this "crossed" position, insert a third polarizer between the original two. Note that the beam is now partly transmitted. For what orientation of the intermediate polarizer is the transmitted light most intense? Can you explain why inserting the third polarizer permits light to strike the screen? 4 Brewster's law

To observe Brewster's law, mount in the linearly polarized laser beam a glass plate which can be rotated about a vertical axis, as shown in Fig. 11, FIGURE 11 /

/

/

/

/

/

/

/

/

/

Beam

which is a top view of the setup. Rotate the glass and look for an orientation at which the reflected beam vanishes. Try various orientations ofthe polarizer until you find such a condition. Is the incident light then polarized in the plane of the figure? Measure the Brewster angle with a protractor and from Eq. (7) compute the refractive index of the glass. 5 polarization of reflected light

With the glass plate at the Brewster angle, but without the initial polarizing filter in place, insert a polarizer in the reflected beam, and show that it is linearly polarized. Repeat this observation for the transmitted beam. Why is the transmitted beam not completely polarized? 13

laser optics

FIGURE 12 (b)

To analyze an unknown photon beam

first find the orientati-on of a polarizer for maximum transmission.

(d)

Insert a quarter-wave plate at this orientation. . .

and with an analyzer, find the new orientation for maximum transmis­ sion.

(f)

Measure the transmitted flux with the analyzer in this position ... 14

and rotated through 90°.

polarization of light

6

circular polarization

Circularly polarized light can be made from linearly polarized light by using a quarter-wave plate, as discussed above. A very simple quarter-wave plate is provided by an appropriate thickness of stretched polyvinyl alcohol. The phase difference on transmission for the component of E parallel to the direction of stretching will be different from that for the perpendicular com­ ponent. Most sheet plastics which are rolled show such an anisotropy. Circularly polarized light can be identified by using a second quarter-wave plate, which introduces an additional phase shift of ±90" (depending on its orientation) and brings the two components back in phase, or 180" out of phase. When the axes of the two quarter-wave plates are parallel, you should find that the transmitted light is plane-polarized perpendicular to the original plane of polarization. When the axes of the quarter-wave plates are perpen­ dicular, the transmitted light should be plane-polarized parallel to the initial plane of polarization. Can you explain these observations?

er

:W

L 0 -2

7

depolarization

Insert a sheet of waxed paper between a pair of polarizers, and observe how the transmitted intensity changes with the angle of the second polarizer. What do you conclude from this observation about the polarization state of the light transmitted by the waxed paper? 8 analysis of polarization

A light beam may always be represented as the superposition of a circularly polarized component, a linearly polarized component, and an unpolarized component. A systematic method for analyzing a beam of unknown com­ position is shown schematically in Fig. 12. How are the angles e and

questions

is­ 1 Are polarization effects observed with sound waves in air? In solid materials? 2 Discuss the relation between Brewster's law and the effectiveness of Polaroid

sun glasses in reducing reflected glare from water or highway surfaces. 3 Does the axis of a Polaroid filter have a direction, like an arrow? That is, is its action changed by rotating it by 180" around the beam axis? 4 What would be the action of a half-wave plate (like a quarter-wave plate but twice as thick) on a linearly polarized beam? Circularly polarized? 5 When unpolarized light is passed through an ideal polarizer, the transmitted intensity is found to be exactly half that of the incident beam. Why? 6 Suppose light is incident at Brewster's angle on a stack of glass plates con­ taining a large number of individual sheets. It is found that both the trans­ mitted and reflected beams are plane-polarized. How does this result differ from the case when only a single sheet is used? Why? 7 How could a right circularly polarized beam be transformed into a left circularly polarized beam? Would the same technique transform a left circularly polarized beam to right circularly polarized? 15

experiment

L0-3

diffraction of light

introduction The term diffraction refers to phenomena in which light or other radiation exhibits departures from the straight-line propagation predicted by the simplified model of geometrical optics. Analysis of diffraction phenomena always requires the more complete description given by the wave picture of light. Typically, we are concerned with situations where a beam of light strikes a barrier with openings or edges, producing a wave beyond the barrier which can be projected on a screen as a diffraction pattern. Calculations of the principal characteristics of simple diffraction patterns are discussed in most textbooks and need not be discussed in detail here. These calculations are usually based on Huygens' principle, which states that when light emerges from an aperture, the various points in the aperture can be treated as secondary sources of radiation. If the incident light is a plane wave (a parallel beam), then these sources are all in phase, provided the plane of the aperture is perpendicular to the beam direction. Radiation from these secondary sources then travels to the viewing screen, arriving with phases which depend on the distances from the various points in the aperture to the point on the screen under consideration. The simplest arrangement for observing diffraction is to use a long thin aperture or slit of width a, as shown in Fig. 13. If such an aperture is illuminatFIGURE 13

Slit

-----if ----I----Ll.

a~---~--~-~-4

----I-----fT

I

Parallel beam ~--------L--------~

Screen

ed by a plane wave, the screen will show an intensity maximum in the straight-ahead direction, since in this direction all the secondary sources in the slit are equidistant from the screen (assuming the distance to the screen is much larger than the aperture width) and the corresponding waves all arrive at the screen in phase. As we move away from this point, the distance from one side of the slit becomes longer than from the other side, and cor­ responding phase differences develop. Complete destructive interference occurs at points for which a sin (J = nl.

n

= 1,2, ...

(9) 17

laser optics

Between these points ofzero intensity are other regions of maximum intensity, but not as bright as the central maximum. (Why?) As Eq. (9) shows, the distances between adjacent minima in the diffraction pattern are inversely proportional to the slit width a, but directly proportional to the wavelength A. Another situation, similar to the first but easier to analyze, is the case of two thin slits separated by a distance b, as in Fig. 14. If we may neglect the FIGURE 14

beam

width of each slit, then constructive interference between radiation from the two slits occurs when the path difference is zero or a whole number of wave­ lengths, that is, when

b sin 0

=

nA

n

=

0, 1,2, ...

(10)

Destructive interference occurs when the path difference is a half-integer number of wavelengths, or b sin 0 = (n

+ !)A

n = 0, 1, 2, ...

(11)

Note that again the spacings in the diffraction pattern are inversely propor­ tional to b, but directly proportienal to A. An elaboration of the two-slit idea is to use several slits, equally spaced. The phase relations are exactly the same as for the two-slit arrangement, and so the positions of the maxima and minima are the same as before. The difference is that the maxima are sharper than before; that is, the intensity drops off much more quickly on both sides of a maximum. The reason, briefly, is that as we go slightly to one side of a maximum, waves from two adjacent slits have only a slight phase difference, and the corresponding intensity decrease is rather small. But if there are many slits with the same spacing, the phase differences between slits at opposite ends of the array grow much more rapidly, and destructive interference develops. An array of many equally spaced slits is called a diffraction grating; the intensity pattern for a grating of N slits is calculated in many textbooks. An interesting variation is to use a square or rectangular array of apertures, as shown in Fig. 15a. The intensity maxima in the resulting diffraction patterns are most easily found by using a two-stage procedure. First, we represent the array as a series of parallel rows of apertures, as shown in Fig. 15c, which shows two different choices for the square array. Again assuming that tht; distance to the screen is much larger than the separation of the apertures, we note that at any point in the plane perpendicular to the rows, radiatiQn from all apertures in an individual row will be in phase. Second, we then ask under what conditions radiation from adjacent rows will also be in 18

diffraction of light

L0 -3

FIGURE 16

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0 0

0

0

0 0

0 0

0

0 0

0

0 0

0

(b)

(a)

(c)

phase, so that radiation from all the apertures is in phase. Clearly, this will occur when the path difference from adjacent rows is a whole number of wavelengths. Thus, for example, the diffraction pattern for the square array consists in part of a row of spots of maximum intensity oriented as shown in Fig. 16, with positions determined by the condition a sin

e=

n).

n = 0, 1,2, ...

(12)

FIGURE 16

f

a

t

A different choice of rows (e.g., the 45° diagonals) leads to different aperture row spacing and a different row of spots, and a little thOUght shows that in each case the row of spots is perpendicular to the corresponding rows of apertures. Thus the diffraction pattern is uniquely .determined by the structure of the aperture pattern. A precisely similar analysis can be made for the hexagonal array shown in Fig. 15b. The foregoing analysis is very similar in principle to that used for x-ray diffraction. A beam of x-rays whose wavelength is of the same order of magnitude as the atomic spacing in a crystal is used. The individual atoms 19

laser optics

scatter the radiation, performing the same function as the individual apertures in the arrays just discussed. The shape of the resulting diffraction pattern is characteristic of the structure of the crystal being used. X-ray diffraction is an extremely important experimental tool which has provided most of the information we now have about the crystal structure of solids, as well as much information about the structure of liquids, polymers, and other arrangements of atoms.

experiment 1

single-slit diffraction

Aim the laser beam at one of the single-slit apertures and measure the posi­ tions Xn of the intensity minima in the diffraction pattern. Compute the corresponding values of tan en> using the fact that tan en = xn/L. Using a table of trigonometric functions, obtain the corresponding values of sin On and plot a graph of sin en as a function of n, drawing the best straight line through the date points. Using this graph, together with Eq. (9), determine the value of a/A. Using the known value of A for the He-Ne laser, determine the slit width a. If the width of your slit is greater than about 0.1 mm you can determine the width directly by looking at the shadow of the slit edges produced when diverging laser light is projected through the slit. To do this place a short focal-length concave lens directly in front of the laser. Place the slit in the diverging beam as close to the lens as possible. You should now see on the scree1}a magnified image of the slit. From a measurement of the width of the image, the distance to the screen, and the focal length of the lens you should be able to compute the slit width. Alternatively, you may use the ratio of the diameter of the unobstructed laser beam at the screen to the diameter of the beam in the plane of the slit as the magnification. Compare the slit width determined in this way with the value computed from the diffraction pattern. Replace the single slit with another slit of different width and observe qualitatively how the pattern changes. You may wish to repeat the above measurements with this slit or with a different screen distance L. 2

two-slit diffraction Aim the laser beam at one of the double-slit patterns and repeat the above procedure. Use Eq. (10) to determine the value of b/A and from this find the slit spacing. How does the pattern depend on the distance b between slits? On the width a of the individual slits? Can you understand the effect of varying the width of the individual slits?

3

diffraction grating

Aim the laser beam at the various grating arrays to observe the effect of increasing the number of slits. For a sufficiently coarse grating you should be able to use a diverging laser beam to look at the magnified image of the grating. In this way you can determine the width of the individual slits, the separation between slits, and the number of slits. 20

diffraction of light

4

L 0 -3

arrays of apertures

Using one of the square arrays of apertures, observe the diffraction pattern. Measure the spacing in a row of maxima in the diffraction, and determine the aperture spacing. Replace this array by another with the same spacing but different sized apertures. You should find that the positions of the maxima do not change but that more of the pattern is visible in one case than the other. Why? You may wish to repeat the above observations with a hexagonal array of apertures. For sufficiently coarse arrays you should be able to use the diverging laser beam to produce a magnified image of the array.

questions 1 Suppose the single slit and the region between it and the screen are immersed

in water. How does the diffraction pattern change? 2 Is a two-slit diffraction pattern the same as two superimposed single-slit patterns with the same slit width and spacing? That is, could a two-slit pattern be recorded on film by making exposures with first one slit open and then the other? Explain. 3 Suppose a planoconcave lens is placed in the laser beam before the slit, so that the beam striking the slits is diverging. Qualitatively, what effect would this have on the diffraction pattern ? 4 In a two-slit diffraction experiment, suppose one slit were illuminated by one

laser beam, the other by a beam from a different laser. Would the diffraction pattern be the same as before? Suppose a single laser beam is split by mirrors and the two parts of the beam illuminate the two slits. Is this the same as using two separate lasers? Explain. 5 In the diffraction pattern formed by a grating, suppose the laser were replaced by an ordinary light bulb with lenses to form a parallel beam. How would the resulting diffraction pattern look? 6 A slit is placed directly in front of a diverging laser beam produced by a concave lens with a focal length of - 1.5 cm. How wide must the slit be in order that a reasonably well defined shadow is obtained? (Take as the criterion that the angle subtended by the slit must be greater than the diffraction angle.)

21

.

"-..,,­

experimen t

L 0 -4

interference of light

introductio n The diffraction effects observed in Experiment LO-3 resulted from inter­ ference between waves coming from different points on the same wavefronts of the initial waves. For example, in the two-slit experiment we made use of the fact that the waves emitted by the two slits are in phase when they leave the slits. This definite phase relationship, in turn, results from t he fact that the slits are equidistant from the source, and that various points on a wavefront from the source always have a definite phase relationship to each other. The existence of such a relationship is called spatial coheren ce. In this experiment we make use ofa similar but somewhat different property, that of a defini te phase relation between points that are separated along the direction ofpropagation of the wave. By using a combination of mirrors we split the laser beam into two beams and arrange the mirrors so that the beams travel different distances before they are brought back together. Interference between the two recombined beams requires a definite phase relation along the direction of travel, since the two waves have traveled different distances. Such a relation is called temporal coherence. A purely sinusoidal wave of definite frequency (an idealization which cannot be precisely realized in nature) would be completely coherent, both spatially and temporally. A convenient arrangement to demonstrate interference effects associated with temporal coherence is the Michelson interferometer, shown in Fig. 17. FIGURE 17

IL-_~--,I Mirror

-

I

Source / //

"­

Mirror

Beam splitter /

r....~____ I Screen It was invented by Albert Michelson, one of the pioneers in measuring the speed of light precisely and demonstrating that the speed is independent of the direction of propagation, one of the cornerstones of relativity. As a beam splitter, Michelson used a half-silvered mirror, made so that at 45° incidence the reflected and transmitted waves had about equal amplitude. Interference 23

laser optics

effects were observed by looking into the interferometer at an extended source. We expect constructive interference when the difference in the total path lengths of the two beams is zero or an integer number of wavelengths, and destructive when it is a half-integer number of wavelengths. Because of the high degree of collimation, intensity, and monochromatic­ ity of the laser beam it is possible to project the interference pattern onto a screen rather than having to view it by looking into the interferometer. Two advantages of this arrangement are that a number of observers can view the interference pattern simultaneously and direct measurements of the inter­ ference pattern may be made on the screen. In addition, it is probably easier to discuss the real interference pattern produced on a screen than to discuss the virtual interference pattern seen when one looks into an interferometer.

CAUTION

We remind you again that you should not look into the interferometer at a focused laser beam. You may safely look at a diverging beam, however.

The interference pattern appears on the screen as a series of circular bright and dark fringes. To understand the origin of these fringes, it is useful to represent the functions of the mirrors in terms of images. The mirrors form two virtual images of an initial point source, in the positions shown in Fig. 18. FIGURE 18

r2d ~~+~E--------------L I --------------~

() r

S1 Virtual sources Screen

That is, the light from each partial beam striking the screen looks just as it would if it had originated in the corresponding virtual source. The positions of the virtual sources are determined by the total path lengths from source to screen for the two beams. Clearly, the two virtual sources are coherent. Suppose the mirrors are positioned so that the beams reaching the screen are exactly parallel and the optical path lengths are exactly equal. Then if one mirror is moved forward or backward a distance d, the virtual sources separate by a distance 2d. Now we consider light from the two virtual sources 24

interference of light

L 0-4

reaching the screen (at a total distance L from the virtual sources) at a point a distance' from its center, as shown in Fig. 18. Constructive interference occurs when the path difference, or cosine error, is given by 2d cos () = n2

n

=

0, 1,2, ...

(13)

2

where cos () may be approximated as 1 - ,2 /2L if the source and screen are small compared to L. In this case we expect the interference pattern to form a series of concentric bright rings whose radii 'n are given by 2d(1

If 'n and Eq. (14)

'n+ 1

-~) 2L2 = n2

(14)

are the radii of adjacent bright fringes, we may write from (15)

If one or both mirrors are tilted slightly, not the center of the ring pattern, but a section to one side of the center, will appear. Now if either mirror is moved forward or backward, moving the mirror a distance 2/2 changes the optical path by an amount 2 and causes the pattern to shift over so that each fringe is in the position of an adjacent fringe. By moving the mirror a measured distance and counting fringes, we can obtain a direct measurement of the wavelength. This also gives an indication of the extent of the temporal coherence. By how great a distance can the path lengths differ without losing the coherence needed for the formation of fringes? Several interesting variations of the Michelson interferometer can be obtained by using polarizers in various locations. Analysis of these variations is left for the student.

experiment 1

Michelson interferometer

Project the laser beam into the Michelson interferometer and onto a screen. Normally, you will observe two intense spots on the screen. If your beam splitter has more than one reflecting surface you may also see a number of subsidiary spots. Adjust the orientation of the adjustable mirror until the two intense spots exactly overlap. You may notice a flickering ofthe combined spot. What causes this? Now insert a diverging lens immediately in front of the laser. (You may need to adjust the position of the lens so as to illuminate the central region of the interference pattern.) You should now observe a set of circular fringes whose center is on the screen. If the center of the fringe pattern is not visible, you should by careful adjustment of the mirror be able to move the center of the interference fringes into the center of the screen. With a centimeter scale measure and record the radii of successive bright or dark fringes, starting from the center of the pattern. Compute the squares of the radii and determine by subtractlon the difference in the squares of successive radii. This number should be a constant as given by Eq. (15). Carefully measure the distances from the beam splitter to the two mirrors 25

laser optics

of the interferometer and compute the distance d. You will have to make these measurements very carefully. You will also have to determine just what are the light paths through the beam splitter. It makes a difference whether the beams reflect off the same surface or off opposite surfaces of the beam splitter. Also, is the path in glass equal for the two rays? If not, you will need to know the refractive index of the glass or else insert a compensating plate identical to the beam splitter. Finally, determine the total light path L from the virtual source behind the diverging lens to the screen. Substitute all this into Eq. (15) and compute rn 2 r:+l' Compare your computation with the observed fringe separation. If your two values do not agree reasonably well, you have probably made a mistake in determining d and should recheck the light paths through the beam splitter.

2

interference of polarized light If your laser has internal reflecting mirrors, the beam will be unpolarized as discussed in Experiment LO-2. Ifthe beam leaves the discharge tube through Brewster windows, it will be linearly polarized. For an un polarized laser beam place a sheet polarizer in front of each reflecting mirror. When the polarizer axes are parallel, we should expect a reduction in intensity by a factor of 2, but no change in the fringe pattern. Why? What happens when the axis of one polarizer is rotated until the two axes are perpendicular? Explain. With the two polarizers crossed, insert a third polarizer directly in front of the laser with its axis of polarization at 45° to that of the other two polarizers. Finally, place a fourth polarizer directly in front of the screen. What happens as you rotate the axis of this polarizer? Can you explain all this? If your laser beam is linearly polarized, the experiment described above will have to be done differently. First, find the plane of polarization of the beam by placing a polarizer in front of the laser and rotating the axis of polarization until the beam is extinguished. Now, place crossed polarizers in front of the two mirrors with their axes at45° to the plane of the laser beam. You should find that the interference pattern has disappeared. Why? Now, place a third polarizer in front of the screen and rotate its axis of polarization. What do you observe? Why?

3

circular polarization By using eighth- and quarter-wave plates in addition to linear polarizers, you may produce circularly polarized light and study the interference be· tween left and right circularly polarized light. Design of such experiments is left to your ingenuity.

questions

26

1

Suppose that the "beam-splitter" mirror does not split the beam equally, but that one beam is more intense than the other. What effect does this have on the resulting interference pattern ?

2

Explain why displacing the diverging lens alters the part of the fringe pattern which is illuminated.

3

Why is the center of the fringe pattern unaffected by a displacement of the diverging lens? Note that the position of the center remains fixed and alw that there is little, if any, shift in the fringes.

interference of light

L 0-4

4 When a right circularly polarized beam is reflected by a mirror, what is the state of polarization of the reflected beam? 5 A plane-polarized beam passes through a quarter-wave plate with its axis at

45 0 to the plane of polarization. It is then reflected normally by a mirror and passes back through the quarter-wave plate again. What is the state of polar­ ization of the final beam? 6 In Question 5, suppose the quarter-wave plate is replaced by an eighth-wave plate; what is the final polarization state? 7 Suppose a transparent box which can be evacuated or filled with a gas is

placed in front of one of the mirrors. Describe how such an arrangement could be used to measure precisely the index of refraction of a gas whose index differs only very slightly from unity. 8 Instead of projecting a real image, a Michelson interferometer may be used with an extended source to produce a virtual image. Explain why it is necessary to employ an extended source if virtual interference fringes are to be observed.

27

experiment

L0-5

holography

introduction Holography is a technique for storing and reproducing a three-dimensional image of a three-dimensional object. Ordinary photography, by comparison, always records two-dimensional images. The lens in a camera forms an image of the object being photographed. If the object is plane and the lens free from aberration, the image is also plane and can be recorded by placing a photographic film in the image plane. But, in general, the image is three­ dimensional; portions of the image which happen to lie in the film plane are recorded without distortion, but those which are in front of or behind this plane simply appear out of focus. For example, an image of a point is formed by a converging cone of light; if the image is "focused" on the film the cone converges at the film plane, otherwise it intersects the film plane in a circle, known to lens designers as the "circle of confusion." In any case, the three-dimensional character of the image is not recorded on the film; what is recorded is distinctly two-dimensional, as is the image formed on a screen by a slide projector. Holography, however, does produce true three-dimensional images with all the attributes such images should have. Such an image can be viewed from different directions to reveal different sides of the image, and from different distances to reveal changing perspective. In fact, to anyone who has not seen a hologram image, and to many who have, the whole phenomenon seems quite impossible. First, we describe the simplest possible setup to prod~ce a hologram and reproduce its image; then we give a semiquantitative and not completely rigorous discussion of how and why the images are formed. As we shall see, in general, both a real and a virtual image are formed. The basic setup to make a hologram is shown in Fig. 19a. The beam from a laser, spread out by means of suitable lenses, is split, part of it striking a photographic film directly, the other part illuminating the object to be holographed. Scattered light from the object also strikes the film; since the light is highly coherent, both spatially and temporally, the direct and scattered light have a definite phase relation at each point on the film, and an interference pattern is formed and recorded. The images are formed by simply projecting laser light through the devel­ oped film, as in Fig. 19b. A virtual image is formed in the same position relative to the film as that of the original object, and a real image is formed on the opposite side in the mirror-image position. The virtual image may be viewed directly, or a converging lens may be used to form a real image on a screen of this virtual image. The real image formed is not easily viewed directly, but again it may be projected on a screen or a second image formed with a converging lens. In an effort to understand how hologram images are formed, we first 29

laser optics

FIGURE 19

Film Mirror

Laser beam

Laser beam

Virtual image

Real image

(b)

(a)

consider a simpler situation. Suppose two coherent parallel beahls (plane waves) are incident on a photographic film, one normally and one at an angle ct to the normal, as shown in Fig. 20. Clearly, an interference pattern is formed on the film. The broken lines perpendicular to the beam at angle ct represent successive wavefronts, spaced one wavelength apart. The intersections of these with the line representing the film plane are points where the two waves are exactly in phase, so maximum constructive interference occurs. Midway between these are points of maximum destructive interference. A little thought shows that the intensity of the resultant wave is in fact a sinusoidal function of position along the film, with successive maxima separated by a distance d given by

d sin

ct

=

.A.

(16)

After the film is exposed and developed, the blackest regions are those of maximum intensity in the interference pattern, the most nearly transparent are those of minimum intensity. Now we make a positive print on transparent

FIGURE 20

30

Film

holography

LO-5

film, which reverses light and dark. (As we shall see later, this step is not really necessary, but it helps clarify the basic ideas.) The resulting record of the interference pattern is a series of stripes of maximum transmittance corresponding to the interference maxima, separat­ ed by equal distances d A/sin IX, as given by Eq. (16). We now project the same laser beam used before through this pattern on the film. What is the result? We can guess its general nature by noting the similarity of this arrangement to an ordinary diffraction grating. If a parallel beam is incident normally on a grating with spacing d between adjacent slits, and if the individual slits are very narrow compared to d, the result is a series of maxima in the resulting interference pattern, at angles 0 to the normal given by the usual grating formula dsinO=nA

n

= 0, ± I, ±2, ±3, ...

(17)

as also illustrated in Fig. 21. The difference in our situation is that the trans­ mission of the grating does not change abruptly from zero to maximum and FIGURE 21

le

\

~e

\ \

d

~d

nt of

es

Ily

tIe lal

'a

6)

of m.t m.t

I~O

\

~ d sin (J

Screen

back as we go across the lines; instead it changes smoothly, and in fact sinusoidally. It is perhaps not completely obvious what effect this difference should have on the interference pattern. We state dogmatically and without proof that the effect is simply that only the n 0 and n = ± I maxima pre­ dicted by Eq. (17) actually appear; all higher-order maxima are completely absent. Complete proof of this statement is beyond our scope, but we can draw an analogy for the reader with a little knowledge of Fourier series. Any periodic wave can be represented as a sum of sinusoidal waves whose fre­ quencies are integer multiples ofthe fundamental periodicity frequency of the wave. A "square wave" contains the fundamental frequency together with all possible harmonics or multiples of the fundamental, while a sine wave, of course, contains only the fundamental-frequency component. To make a long story short, the diffraction pattern represents a Fourier analysis of the grating, which is, after all, a periodic function. The intensities of the various "orders" (values of n) correspond to the amplitudes of the various sine functions in the series. The "square-wave" grating has all orders, correspond­ ing to all possible harmonics, whereas the "sine-wave" grating has only the first order, corresponding to the fundamental frequency. Thus the diffraction pattern from our sinusoidal grating, formed initially by the interference of two parallel beams, one normal to the grating and one at angle IX, is a pair of parallel beams, both at angle IX, one on either side of the normal, as shown in Fig. 22. Now we consider a different situation, the interference of two coherent waves, one a plane wave at normal incidence as before, the other this time a spherical wave emitted from a point source at a distance from the film which 31

laser optics

FIGURE 22

d=~ sma

we shall assume is much larger than a wavelength. Again, we develop the film and project laser light through the resulting transparency. What does the interference pattern look like? Consider a small region of the transparency near the point a in Fig. 23. The corresponding small segment ofthe spherical wave is nearly a plane wave, FIGURE 23

Film

if we look at a sufficiently small solid angle, and so the contribution from this portion ofthe grating can be predicted immediately from our previous result. It is simply a pair ofsegments of waves in the directions shown, which we have represented in the figure as rays. The same argument can be made for other points in the transparency, such as point b. Taken all together, these sets of rays clearly show the formation of two images as shown in Fig. 24. One is a real image, where the rays actually do converge, at a point to the right of the film the same distance as the original source point was to the left. The other is a virtual image, corresponding to the 32

holography

LO-5

FIGURE 24

\a I

-----~---------------------------~a~----------TL-------------

----....._ ______--:::--"'"' . . :\~/ _/-,,""­

\a

fa

b

Real image point Virtual image point

diverging rays whose directions are the same as though they had originated at a point in the same location as the original source point. Thus, at least for single points, the images are indeed formed as advertised. Now suppose the original object consists not just of one point source but of many. Since each interference pattern results from adding amplitudes of the component waves, it should be clear that the principle of linear super­ position holds both for the formation of the pattern on the film and for formation of the final image. That is, the final image for many point sources is just the sum of the images of the individual sources. This completes our demonstration of the assertion that the arrangement described at the begin­ ning actually does form three-dimensional images. The basic idea of holography was developed in 1947 by Gabor, but it attracted relatively little attention at that time because of the great difficulty of obtaining a sufficiently coherent light source. With the invention of the laser in 1960, light of great coherence became readily available, and holog­ raphy became a practical reality. Although our discussion here has been confined to monochromatic light, an elaboration of the same basic ideas can now produce full-color images which reproduce the color as well as the spatial characteristics of the object. Three-dimensional television and many other interesting applications are now under development.

he :he

23.

ve,

experiment his lIt. lve

1

observation of images Using a laser source with the beam spread by a diverging lens, observe the virtual image formed by a hologram. Using a ruler and a flashlight, devise a method for measuring its position and size. Can you develop a technique for observing the real image and measuring its size?

2

wavelength variation Replace the laser source by a point source of white light. Observe the virtual image. Why does the image appear colored? Place a red filter over the

tier

wo do nal the

33

laser optics

light source and again observe the image. Can you distinguish between the image produced with the laser and that produced with the incoherent source? What differences do you observe? Replace the red filter with a blue filter and repeat the observations. What differences do you observe? 3 hologram size

Make an aperture about a centimeter on a side and view the virtual image through this aperture. Move the aperture over the surface of the hologram. What differences, if any, do you observe in the virtual image? What do you conclude from this? With a pair of cards, make a narrow slit and view the hologram through this slit. What happens to the image as the slit width is made narrower and narrower? At what width does the quality of the image begin to deteriorate? What is the character of the deterioration? To what is this slit width related?

questions 1 Suppose a hologram is made with a laser whose wavelength is 6000 A (orange) and then viewed with a laser whose wavelength is 5000 A (green). How will the images differ from those formed when the orange light is used for viewing? 2 Can the real image be viewed directly? Why or why not? Is this difficulty related to the side ofthe object that is illuminated when the hologram is made? 3 How does the image formed by a hologram compare with the results of stereoscopic "three-dimensional" photography, in which two cameras take pictures from two slightly different positions and the images are then viewed separately by the two eyes of the viewer? What are the most significant differences? 4 Do a hologram "negative" and the corresponding "positive" print (with

black and white reversed) make the same image? Note that changing the phase of the "reference beam" by 1800 would interchange black and white on the hologram.

34

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atomic physics Copyright 1971 by McGraw-Hill, Inc.

All rights reserved. Printed in the United States of America.

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retrieval system, or transmitted, in any form or by any means,

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without the prior written permission of the publisher.

Library ()f Congress Catalog Card Number 79-125108

07-050490-3

1234567890 BABA 79876543210

The first edition of the Berkeley Physics Laboratory

copyright \963,1964,1965 by Education Development

Center was supported by a grant from the National Science

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Education Development Center.

This book was set in Times New Roman, printed on

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supervised production.

atomic physics

INTR 0

DUCTIO N

Several basic concepts ofquantum mechanics which are ofcentral importance in the understanding of the structure and properties of atoms are explored experimentally in this series of experiments. These include the existence of atomic energy levels, the particle nature of electromagnetic radiation, and the wave nature of particles. We begin in Experiment AP-I with the study of atomic spectra, which provided much of the original motivation for the development of quantum mechanics. The analysis and interpretation of spectra give strong support to the existence of discrete energy levels in atoms and to the photon concept. In Experiment AP-4 we measure the ionization energy of atoms in a very direct way. Experiments AP-2 and AP-3 involve the photoelectric effect, which gives even more direct evidence of the particle nature of radiation than does the study of spectra. Finally, in Experiment AP-5, we examine a straightforward experiment to demonstrate the wave properties of electrons by observing their diffraction from a crystal lattice.

1

experiment

A P-l

atomic spectra

introduction In this experiment we study the relationship of atomic spectra to energy levels of atoms. Throughout the development of quantum mechanics, atomic spectra have played a central role. The problem of the origin of atomic spectra was one of the original motives for the beginning of quantum mechanics, and in later years the analysis of spectra has been the most important analytical tool in the investigation of atomic and molecular structure. The central idea in the relation of atomic structure to spectra is the existence of discrete energy levels in atoms. An atom in a state with energy E1 can make a transition to a state with lower energy E2 by emitting a photon E 2). The energy of the photon is in turn related to its whose energy is (E1 frequency f and wavelength A. by the familiar Planck relation hf= he A.

(1)

where c is the speed of light and h is Planck's constant. Conversely, an atom can be raised or exited from a lower energy state to a higher energy state by absorption of a photon whose energy equals the difference of energy of the two states. The hydrogen atom is the simplest of all atoms, consisting of a single elec­ tron and a single proton. Thus, it is not surprising that the spectrum of hydrogen has a corresponding simplicity. Elementary quantum mechanics show that the energy levels En of the hydrogen atom are given (in MKS units) by a simple formula:

En

(

4::J

2

2n":h2

(2)

where e is the electron charge, eo is the permittivity of free space, h = hj2n, and n is a positive integer called the principal quantum number. The lowest energy state or ground state is the state with n = 1. The energy E = 0 (with n = CI) corresponds to a state in which the electron has been completely separated from the proton and is at rest. If the proton were infinitely massive, the mass m in Eq. (2) would be simply the electron mass. It is possible to show that the appropriate correction for the finite mass M of the proton is obtained by simply replacing the electron mass m by the reduced mass f1 of the system, defined as

1 f1

=

I m

+

I M

(3)

This change has the effect of decreasing the magnitudes of all the energy JeveJs by about 0.05% compared to the vaJues they wouJd have with M = W. 3

atomic physics

The lines in the hydrogen spectrum correspond to all possible transitions between energy levels. The wavelengths of these lines are given according to Eqs. (1) and (2), by

I A

(e 2)2 41rh c (In/

E2 ---'--:-hc--=- = 4m::o

11

3

(4)

n:2)

where n I and n 2 denote the quantum numbers of the initial and final states, respectively. The combination of constants outside the parentheses in Eq. (4) is often called the Rydberg constant (abbreviated R H ), after J. R. Rydberg, one of the pioneers in atomic spectroscopy. The sUbscript H (for hydrogen) distinguishes it from the corresponding constant containing m rather than Jl, which is denoted by Roo. The numerical value of the Rydberg constant, which has been measured very precisely by spectroscopic measurements, is RH =

(4::J2

4';3

C

=

1.0967757 x 10-

7

m-

I

Thus, the wavelengths of the hydrogen spectrum are given by

I A

(5)

In particular, transitions from higher initial states to the final state n2 = 2 yield wavelengths in the visible region of the spectrum, with wavelengths given by

:2)

RH

n2 - 4) (4T

n = 3,4,5, ...

(6)

This series of lines is called the Balmer series after Johann Balmer, who discovered Eq. (6) empirically (by trial and error) in 1885, long before its relation to the structure of the hydrogen atom was understood. Table I shows the remarkable agreement between observed wavelengths and those com­ puted from the Balmer formula. This table also shows the usual spectroscopic notation used to designate these lines. TABLE 1

Line

n

Ha Hp Hr

3 4 5 6 7 8

H,;

H, H{

A. (observed),

6562.79 4861.327 4340.47 4104.74 3970.07 3889.06

A

(red) (blue-green) (blue) (violet) (ultraviolet) (ultraviolet)

A. (computed),

A

6562.80 4861.33 4340.48 4101.75 3970.08 3889.06

For more complex atoms it is not practical to calculate the energy levels, but they can still be deduced from spectra. In every case there is a direct correspondence between a spectrum line and a transition from one energy state to another. Helium-Neon Laser The relations between energy levels and spectra form part of the basis of operation ofthe He-Ne laser, which you may have used in the Laser Optics experiments, and the laser is an interesting and very useful application of 4

atomic spectra

)fiS

to

(4)

es, (4) rg, ~n)

I

p.,

ich

(5)

; 2 ths

(6)

rho its ~ws

'm-

pic

:ls, ect 'gy

of ics of

A P- 1

these ideas. To illustrate, we first discuss briefly the operation of the laser. The importance of the laser stems from the fact that many atoms which are initially excited to the same excited state can decay to their ground states in a synchronized manner, that is, with a definite phase relationship between the radiations emitted by individual atoms. As a result, light is emitted in a definite phase relation which is preserved for a long time, or over a corre­ spondingly long distance along the light path. This is in contrast with the usual situation in which atoms radiate independently, each for a time the order of 10 - 8 sec, with no correlation in the phases of the various atoms. In the more usual case the phase of the radiation varies rapidly and randomly. The existence of a long-lasting phase relation is called coherence. The synchronizing of radiation from a large number of atoms is possible because of a phenomenon called stimulated emission. An atom in an excited state has a certain probability (per unit time) to emit a photon spontaneously and make a transition to a state with correspondingly lower energy. But if there is some radiation already present with the same frequency as that of the emitted photon, the transition probability is increased. Photons "like" to be with other photons of the same energy; a photon of a given energy is more likely to be emitted if there are already some photons of this energy in the vicinity. Photons are gregarious! In practice, the gas is placed between two parallel or concave mirrors so that light is reflected back and forth between the mirrors, forming a standing wave pattern. It is this standing wave that stimulates the emission from atoms in the region between the mirrors. By making one of the mirrors only partially silvered or by some other means, part of the energy can be extracted as an external beam. The energy in the standing wave is constantly replenished by emission from atoms, provided of course that a continuous supply of atoms in excited states can be provided. In the He-Ne laser the initial excitation is provided by imposing an electric field across the gas to set up a glow discharge. A few atoms are ionized, and the ions and electrons carry the discharge current. The electrons in turn can collide with gas atoms, exciting them to various higher-energy levels. This in itself does not guarantee a sufficient population of atoms in the desired excited state to maintain the standing wave at sufficient amplitude for the required stimulated emission. Usually some means must be found to channel the electron-collision excitation into the particular levels needed. In the He-Ne laser this is made possible by a fortunate coincidence in the energy­ level schemes, which are shown in Fig. 1. Typically a helium-neon laser contains the order of 1.0 mm pressure of He and 0.1 mm pressure of Ne. Electron impact excites some of the He atoms to the 21 S state as shown, and to higher states from which some atoms cascade down to the 2l S state. Ordinarily, an atom in such an excited state would quickly emit a photon with energy 20.61 eV and decay back to the ground state. This is not possible here because of a selection rule forbidding this particular transition. The basis for the selection rule is conservation ofangular momentum; both the excited state and the ground states have total orbital angular momentum of zero, so it is not possible to emit a photon, which must have at least one unit of angular momentum. Such a state, in which radiative decay is forbidden by a selection rule, is called a metastable state. The He atoms can lose energy by another mechanism, however. Since Ne has an energy level almost exactly equal to that of the 2 I S level of He, an energy-exchange collision between an excited He atom and an unexcited Ne atom can leave the He atom in the ground state and the Ne atom in an 5

atomic physics

FIGURE 1

He

Metastable 20.61 eV

21S-~~-:--

Ke

------Collision

...

20.66 eV 't;"­ ___.l::!.3,912 Alaser

58

\

--­

\.

20.30eV

\

Metastable 19.82 eV

23 S - - - - ­

-------­ Collision

4

p

0

\--6328 A laser 1\ 19.78o eV ",, , \ \ 11,523 A laser~, \

4s~~=-~~

,

\

. '\

'\

3p 18.70 leV I I I

I I I

I

I

Electron im act

38 16.70 eV

Diffusion to walls

j

Ground state

excited state. This particular state is again metastable; the Ne atom cannot decay directly to the ground state because of selection rules, but it can decay to the 3p level shown, and this, in fact, is the transition involved in the laser action. Finally, from the 3p state the Ne atoms make another radiative transition to the metastable 3s state, and from there decay to the 2p ground state, usually by collision with the walls of the container. We note that it is important that the 3p -+ 3s transition occur rapidly; ifthere were an appreci­ able buildup of atoms in 3p state, their probability for absorption of laser radiation would be appreciable, and this would inhibit the laser action.

experiment Wavelengths of spectrum lines can be measured quite directly by using a spectrometer containing a diffraction grating. If light of wavelength ;. is incident normally on a diffraction grating with line spacing a, the resulting 6

atomic spectra

A P- 1

diffraction has strong intensity maxima at angles 0 with respect to the direction of the incident beam determined by the relation asinO=nl

n = 1,2,3, ...

(7)

1

hydrogen spectrum By measuring the angles for the various spectrum lines and using the known value of the grating constant a, one can calculate the wavelengths in the hydrogen spectrum, which can then be compared with values in Table 1. Alternatively, if the grating constant is not known, the spectrometer can be calibrated by regarding the Balmer wavelengths as known and using them to determine the grating constant. Measurements should be repeated several times in order to get some idea of how reproducible the results are. Identify as many possible sources of error as you can. What sources of error are most important? Is the spectrum of molecular hydrogen present in addition to that of atomic hydrogen?

2

helium spectrum Similar observations can be carried out with the helium and neon glow tubes. Since these are inert gases, there is no molecule; there is, however, the pos­ sibility of observing lines associated with ionized atoms. In the helium spectrum you will observe about six strong lines. Determine their wavelengths, being especially careful with the strong line in the yellow and the line in the blue. The blue line is from singly ionized helium. The energy levels for singly ionized helium are given by Eq. (2) except that e 2 must be replaced by 2e 2 (Why?), and in Eq. (3) the mass of the hydrogen nucleus is replaced by that of the helium nucleus (only a very small correction), Determine the initial and final states corresponding to the blue helium line.

3

neon spectrum Observe the neon spectrum and pick out the most intense lines. Determine the wavelengths of these lines. Can you find the lines corresponding to the 5s -+ 3p laser transition and the 3p -+ 3s transition? If a helium-neon laser is available, observe the radiation coming out the sides of the laser tube. Can you understand the spectrum observed? Can you find the blue ionized helium line?

.t

y

:r

e

d

$

i­

questions

~

1

How can you be sure the lines in the hydrogen spectrum you observe are from atomic hydrogen and not molecular hydrogen? What differences would you expect to find in the molecular hydrogen spectrum?

2

How does the energy required to dissociate a hydrogen molecule compare with typical energies of photons of visible light? Look up the value of the dissociation energy in a handbook. Can you predict a region of the spectrum in which hydrogen would be opaque?

3

What would the spectrum of doubly ionized lithium look like? What transitions would be in the visible spectrum?

t

a s g

7

atomic physics

4 Why is the emission corresponding to the 3p ~ 3s transition not present in

the laser beam with the same intensity as that of the "laser transition" 5s~ 3p? 5 What advantage might there be in using higher orders in the diffraction pattern, rather than the first order (n = I)?

6 Is there any evidence that the "known" value of the grating parameter a is incorrect? If so, compute the correct value, and state the probable percent error in the given value.

8

experiment

A P -2

photoelectric effect

s

t

introduction The photoelectric effect is the liberation of electrons from the surface of a material by absorption of energy from light striking the surface. The simplest experimental arrangement for observing this effect is shown in Fig. 2. The I

FIGURE 2

.. Incident light Anode

V (Variable)

Photocathode

cathode is illuminated with monochromatic light, and the current (resulting from photoemission of electrons from the cathode) is measured as a function of voltage. The most important experimental observations are the following: • The kinetic energy of the photoelectrons (as determined by the reverse voltage needed to completely stop the flow of electrons from cathode to anode) is independent of the intensity of the light, but is a linear function of the frequency of the radiation. • There is a maximum wavelength beyond which photoemission does not occur; the value of this maximum depends on the composition of the surface. • The saturation photocurrent is directly proportional to the light intensity.

These experimental facts were first understood in 1905 by Einstein, who interpreted them in terms of quantization of electromagnetic energy, first postulated by Planck five years earlier. A photon of energy E hI, cor­ responding to radiation of frequency I, is absorbed by an electron in the cathode. To remove the electron from the surface requires an amount of work W (called the work function for the material), The electron thus emerges with a kinetic energy !mv 2 given by

!mv2

hf - W

(8)

Increasing the intensity increases the number of photons and thus the number of photoelectrons, but not the energy of each electron. 9

atomic physics

Correspondingly, the negative potential Vo needed to stop the electron flow is determined by the fact that the corresponding potential energy eVo must equal the electron's initial kinetic energy. Hence the "stopping poten­ tial" Vo is given by eVo

= hf -

W

(9)

That is, Vo is a linear function of frequency, as observed experimentally. Thus measuring Vo as a function offrequency should provide values for both hand W, as shown in Fig. 3. FIGURE 3

Vo

L---~~----------------f

The earliest experiments on the photoelectric effect were performed with alkali-metal cathode surfaces because of their low work function and relatively high photoelectric efficiency (ratio of electrons emitted to photons absorbed). Even so, the efficiency was only about 0.1 %, and photocurrents produced by moderate light intensities were in the microampere range. Modern phototubes often use a cesium antimonide photocathode surface, and have efficiencies as high as 20% together with a low work function. The disadvantage of these surfaces for the present experiment is that they have a rather nonuniform work function, so there is no sharp cutoff of the photo­ current for negative potentials. Even so, hand W can be determined with a precision of 10 to 20%. Figure 4 shows photoelectric efficiency as a function of wavelength for three commercially available photosurfaces. In actual practice the l-versus- V graph will not look exactly like Fig. 5, but will be more like Fig. 6. Instead of a sharp cutoff of current at a certain voltage Vo, the current for negative values of V approaches a small negative saturation value. This is a result of photoemission from the anode. Small quantities of photocathode material evaporate and can be deposited on the anode surface. Then when light strikes the anode, it acts as a photocathode. This effect can be reduced by masking the tube so that no direct light strikes the anode wire, but still some scattered light inevitably strikes it. Thus the measured current 1(V) is the sum of cathode and anode currents, whereas what is needed is the cathode current lo(V) by itself. This separation can be made if we make the simplifying assumption that the anode current has the same dependence on Vas the cathode current except for a constant factor a and ofcourse a change in sign of V because of the interchange ofroles of the two electrodes. That is, we assume that the total current l( V) can be represented in terms of the cathode current lo(V) as 1(V) = lo(V) - alo( - V)

(10)

where a is a constant much smaller than unity, and equal to the ratio of 10

photoelectric effect

A P-2

FIGURE 4

5-5

0.1

.01~~--------~--------~1--------~1--------~1~----

2500

5000

7500

10,000

12,500

Wavelength. angstroms FIGURE 5

I

--------~----T---------------v

Vo

FIGURE 6

I

--------~----+---------------V

anode-to-cathode saturation currents (i.e., "forward" and "backward" saturation currents) with a fixed intensity and frequency. Equation (10) gives 1(V) in terms of lo(V). Since 1(V) is the directly meas­ ured quantity and lo(V) the quantity we wish to determine, we should like to 11

atomic physics

solve this equation for Io(V). To do this, we first change Vto - V and multiply through by IX, obtaining (11)

Now we add Eqs. (10) and (II) and divide by the factor (l I(V)

+ IXI( -

V)

0(2)

to obtain (12)

1

experiment A mercury arc lamp serves as a convenient source of a wide range of spectral lines for studying the photoelectric effect. Although the relative intensity of the lines depends on the design of the source, the strongest lines are those listed iR Table 2.

TABLE 2

Wavelength, 5790.65 5769.59 5460.74 4358.35 4046.56 3650.15 2536.52

A

Wavenumber, em-I

Color

17,269 17,332 18,313 22,950 24,713 27,397 39,424

Yellow Yellow Green Blue Violet Long wavelength ultraviolet Short wavelength ultraviolet

There are many types of mercury arc lamps operating at various pressures of mercury vapor and at various power levels, from a few watts for a small, low-pressure lamp to several kilowatts for the high-pressure mercury lamps used for highway illumination. One convenient mercury lamp is the 4-W germicidal lamp. This lamp contains a few millimeters pressure of argon in addition to mercury, which has very low vapor pressure at room temperature. The discharge starts in the argon, but as the bulb warms up the vapor pressure of mercury rises to about 0.9 atm. Electrons to initiate the argon discharge are emitted thermionically from a tungsten filament stretched between the electrodes. Because of the negative-resistance characteristic of the discharge, the lamp must be used in series with a high-reactance ballast (as with ordinary fluorescent lamps). The bulb is made ofa glass which is transparent down to a wavelength of about 1850 A, and the 2537- and 3650-A lines are transmitted with very little absorption. Another commonly available mercury lamp is the high-pressure mercury arc, used primarily for illumination. These lamps contain a quartz arc tube surrounded by a glass bulb. Typically the outer glass bulb cuts off radiation of wavelength shorter than 3000 A so that although these lamps do produce the mercury 3650-A line, their output in the short wavelength ultraviolet is quite limited. Typically, at each end of the quartz arc tube is a main electrode 12

photoelectric effect

A P-2

and a starting electrode. When voltage is applied the discharge first takes place between the main electrode and the adjacent starting electrode. The current in this discharge is limited to a reasonable value by series resistors enclosed within the lamp bulb. Once this auxiliary discharge has generated a sufficient quantity of ions and electrons, the arc jumps to the operating electrodes. Lamps of this type are usually operated on alternating current provided through a high-reactance transformer. These transformers supply an initial high voltage for starting and a ballast reactance to compensate for the negative resistance of the arc.

CAUTION

Because of the high intensity of ultraviolet light produced by a mercury arc, never look directly at the arc. Permanent damage to vision could result.

To isolate individual spectrum lines, various filters are used. For example, an ordinary glass filter blocks the uv lines but transmits the lines from 4046 A upward. A green filter blocks the blue and violet lines but transmits the green and yellow. The green filter by itself may not sufficiently block the uv lines, so a glass filter may be needed in addition. Suitable clear sheet plastic will block radiation below 2800 A but will pass the 3650-A line. This may be checked by looking at a U.S. postage stamp which is "tagged" with fluores­ cent ink, which fluoresces under short wavelength ultraviolet but not under the 3650-A line. Some foreign stamps, notably from Denmark, also fluoresce under the 3650-A line. 1 determination of h

To provide the necessary variable voltage, along with provision for measuring the phototube voltage and current, the circuit shown in Fig. 7 is suggested. FIGURE 7

8 25kQ

The meter measuring V 2 must be a high-impedance voltmeter, with internal resistance of at least 11 Mil; this meter measures the current through the phototube. For VI an ordinary low-impedance voltmeter may be used, 73

atomic physics

provided it is left in the circuit at all times (since the potentiometer current will change appreciably if this meter is disconnected). If only the high­ impedance meter is available, it may be used for both VI and V 2, by switching one lead back and forth from one side to the other of the phototube. In this case VI will not change appreciably when the meter connection is changed, since the meter currents will then be very small compared to the potentiometer current. Note that the phototube voltage is given by V VI - V2 , and the phototube current! is given by 1 V z/ R, where R is the internal resistance of the voltmeter. Attach a thin strip of opaque tape on the tube so as to cast a shadow on the anode to shield it from direct light from the mercury lamp. Measure carefully the characteristic for the green 5461-A line. Vary VI in steps of 0.1 V from 3 to +3 V, measuring V z for each value of VI' Compute the values of V and I. From the ratio of reverse to forward saturation current, determine the value of IX. Using Eq. (12), compute values of 10 for the various values of V, and plot a graph. From this, determine the "cutoff" voltage Vo for this spectrum line. Replacing the green filter with a dark blue filter, repeat the measurements and analysis for the blue 4358-A line. Using results from this line and the green line, together with Eq. (9), determine the value of the quantity he/e, and also the value of W. The accepted value of he/e is

he = 1.23986 + 0.00001 x 10- 6 e ­

V-m

Using the accepted values of e and e, compute your experimental value ofh, and compare it with the accepted value. 2

yellow lines You may wish to make similar measurements for the yellow 5770- and 5790-A lines, which may be isolated with an amber filter. However, because these lines are not very strong and because the efficiency of the S-4 surface is rather low in this wavelength range, you may find that the dominant photocurrent still comes from the small amount of 5461-A line which gets through the amber filter. A better way to compare the green and yellow mercury lines is to use a tube such as the 925, which has an S-I photosurface.

3

ultraviolet lines Because of the very considerable wavenumber difference between the 2536­ and 3650-A lines, it should be possible to obtain an improved value of hele by using these lines. You may wish to try this measurement using an S-5 tube such as the 935. At these shorter wavelengths, corresponding to high photon energies, the 7t-V battery will not be sufficient, and you will want to replace it with a 22t-V battery. Unfiltered light from the mercury germicidal lamp will contain the 3650-A line and the 2536-A line; clear sheet plastic will block the 2536-A line but pass the 3650-A line, as noted earlier.

questions 1 14

If a material has a work function of2.0 eV, what is the longest wavelength to which a photocathode made of this material will be sensitive?

photoelectric effect

t

.­

g

s

2

What are some advantages and disadvantages of the various yield character­ istics (efficiencies) shown in Fig. 4?

3

Estimate the error which would result in your values of Vo if the reverse current (anode-emission current) were neglected completely, assuming instead that I(V) = Io(V). How would this affect the measured value of h?

4

Is the value of Cl: the same or different for various spectrum lines? What might be some possible reasons for variation of Cl: with wavelength?

5

From the observed saturation photocurrent with the 929 phototube illumi­ nated by the green line, compute the number of photoelectrons emitted per second. Assuming the photo yield efficiency of 5%, how many photons strike the cathode per second? Estimating the solid angle subtended by the cathode, about how many photons are emitted by the lamp in the green line, per unit time? What fraction of the total input power is emitted in the green line?

6

When the anode is positive with respect to the cathode, why does the current not rise immediately to its saturation value? What happens to the electrons which do not reach the anode?

7

How valid is the assumption that cathode and anode emission currents have a similar dependence on V. What are some considerations that might make the two currents different in form ?

l,

r

e e

.e

y

n V

le

,

I'

is

ts

le e,

A P ­2

h,

A

se er nt

ne

to

16­ by be on ~ce

np

ck

t5 to 15

experiment

A P -3

the photomultiplier and photon noise

introduction In Experiment AP-2 we studied the photoemission ofelectrons from a surface by absorption of energy from incident photons. Each photoelectron results from absorption of a single photon, but the quantity actually observed is the average emission current, which is the electron charge e multiplied by the average number of electrons r emitted per unit time. Because of the small magnitude of the electron charge, a macroscopically observable current corresponds to a very large number of electrons per second. This current, although it may have a constant average value, actually has an ac component corresponding to random fluctuations in the rate at which electrons are emitted from the photocathode. These current fluctuations, called photon noise because of this origin, can be observed directly with suitable equipment. The photomultiplier is a device which, compared with a conventional phototube, has greatly increased sensitivity to incident photons. Correspond­ ingly, photon noise is much easier to observe using a photomultiplier. The basic principle of operation is as follows. After photoelectrons are liberated from the cathode they are accelerated toward an intermediate electrode called a dynode. If the potential difference between cathode and dynode is great enough, each electrode reaches the dynode with enough kinetic energy to knock off several electrons. These secondary electrons are then accelerated toward the anode by an additional potential difference. The arrangement is shown schematically in Fig. 8. The current in the resistor R, corresponding FIGURES

Incident light

1

Dynode

R

to the flow of secondary electrons from dynode to anode, is clearly greater than that corresponding to the primary photoelectrons leaving the cathode. Depending on dynode materials and accelerating potentials, multiplication factors of 10 or more are possible. Furthermore, the yield of secondary 17

atomic physics

electrons for each primary is not constant. If for example the average multiplication factor is 4, the yield may be 3 or 5 a substantial fraction of the time. The number of secondaries per primary follows a statistical distribution law determined by the characteristics of the particular situation. It is often convenient to assume that this is a Poisson distribution, and, in fact, if certain special assumptions can be justified, we can show that the distribution function is indeed a Poisson distribution. The number of primary electrons emitted during a given time interval also follows a Poisson distribution, analogous to the statistical description of radioactive decay. The electron multiplication produced by the dynode can be repeated in several stages by using several dynodes at successively higher potentials. If the multiplication factor is 4, for example, a single primary electron yields 4 at the first dynode, 42 or 16 at the second, 43 or 64 at the third, and so on, so that with several dynodes the overall current multiplication may 'be very large. With n dynodes and a factor f> per dynode, the overall mUltiplication is bn. The arrangement of a common photomultiplier, the type 931 A, is shown in Fig. 9. This tube has a cesium antimonide photocathode with an S-4 response FIGURE 9

D9

U

~----------~

Photocathode Re~onof

best collection

Anode

Direction of incident radiation

similar to that of the 929 phototube. There are nine dynodes, each of which is also coated with cesium antimonide. It is of interest to study the matter of random current fluctuations in more detail. We consider first the situation of Fig. 10, which uses an ordinary phototube. If photoelectrons are emitted at an average rate r (per unit time), with a charge of e for each electron, the average current through R is I = er, the voltage V across Rand C is V = erR, and the average charge on the capacitor is (13) Q = erRC 18

the photomultiplier and photon noise

rage fthe Ltion >ften ;t, if ltion :rons tion,

AP-3

FIGURE 10

2.9MU

21pF

+

;xl in

Is. If rields In, so arge.

is 5/1.

linin )onse

In a time intervall1t, the average number of electrons emitted is r At, corre­ sponding to a charge er At. But since the number of electrons emitted in this time interval follows the Poisson distribution, as suggested above, the standard deviation of this number (i.e., the root-mean-square deviation from the average) is given by the square root of the average number, that is, (r At)1I2. The corresponding standard deviation of the charge is given by

AQ

=

(14)

e(r At)1!2

Now an excess charge AQ placed on the capacitor at time t ' decays ex­ ponentially through the resistor, so the excess charge at a later time t is given by AQ(t)

= AQ(t')e-(t-n!RC

(15)

Thus, the mean-square value [AQ(t)J2 at time t due to an excess charge accumulated in a particular interval At at an earlier time t' is given by [AQ(t)J2 = e 2 r At e- 2 (I-t')!RC

(16)

le Finally, to find the total mean-square excess charge we must integrate over all times t' prior to t, obtaining [AQ(t)J2

e2r f~-2(t-t')!RCdt' = le 2rRC

(17)

Thus, the root-mean-square charge fluctuation is given by rRC)1!2

A Q( = 2

(RC)1!2

e=2r

I

(18)

where I = er is the average current. The corresponding rms voltage across the RC network is given by

AV= AQ C

(~)1!21 2rC

(19)

which 1 more

dinary time), r = er, on the

(13)

For typical values, such as I = lO pA, R = 2.9 MQ, C 20 pF, we ex­ pect an rms noise voltage of the order of 0.3 m V. Since deflection sensitivities of typical oscilloscopes are of the order of 25 m Vlin., we would need an addi­ tional amplification of the order of a factor of lOO to see this noise directly. But the electron multiplication occurring in the photomultiplier makes photon noise much more readily visible. In this case the noise associated with random photoemission of electrons is augmented by additional noise associated with fluctuations in yield at the various dynodes. The end result 19

atomic physics

is that the total noise is given by expressions essentially the same as Eqs. (18) and (19), with two differences: I is now interpreted as the final anode current rather than the primary cathode current, and there is an additional factor bj (b - 1), where b is the average multiplication per stage. The derivation of this factor need not be discussed in detail; in any case the most interesting features of the relations are the dependence on R, C, and the counting rate.

experiment 1 determination of ~Q An arrangement for applying suitable potentials to the 931A is shown in Fig. 11. A string of I-MO resistors mounted on the base of the tube provides FIGURE 11

Oscilloscope

Regulated power supply

-1250 V

2.7 M!2

lMQ

5.6M!1

-a voltage-divider network. The voltage for the photocathode may be taken from the accelerating voltage supply of the oscilloscope. The voltage between the last dynode and the anode should be taken from a regulated power supply whose current capacity is greater than that of the oscilloscope. As an exercise you may wish to calculate the current through each resistor in the voltage divider, assuming a cathode current 10 and a gain per stage of b. Remember that the 10-kO resistor shown is shunted by the input capacitance 20

the photomultiplier and photon noise

AP-3

of the oscilloscope. With a 21-pF input capacitance, the computed time constant is given by r

RC = IOkn x 21 pf = 210nsec

The photomultiplier may be excited with the 5461-A line of mercury as in Experiment AP-2. The amount oflight permitted to fall on the photocathode should be restricted by a slit or a small iris. The light intensity should be sufficiently low so that the anode current is kept below 10 /lA. If the anode current exceeds this value, the loading of the voltage-divider network will become excessive and the voltages at the final dynodes will drop. As a result their mUltiplication will drop and the photomultiplier will respond non­ linearly. From the voltage drop across the I-Mn resistor in series with the anode, compute the current I = rbne. Also measure the rms noise voltage at the oscilloscope. This requires some estimation and judgment; experience has shown that the apparent "peak-to-peak" voltage is usually about four times the rms value. Using this estimate, compute AQ. 2 determination of rand (j

Finally, use Eq. (18) or (19) with the factor b/(b - 1) to compute the counting rate r and the average multiplication b. What is the overall gain, as deter­ mined from bn ? How many photons are detected during a time interval equal toRC? 3 changing (j

Repeat your measurements for a number of different voltages across the dynode network. Determine the counting rate r and the average mUltiplica­ tion b for each value of voltage. Plot b as a function of the voltage between dynodes. 4 direct determination of (j

By comparing the anode current with that at the last dynode, you may make a direct determination of b. Compare your result with the value of b deter­ mined from noise.

questions 1 The quantum efficiency of the S4 surface in the vicinity of the mercury green

line is about 5%. Thus your computed value of r is only 5% of the rate at which photons strike the cathode. Would you expect the reduced yield to introduce additional noise, or is the noise entirely accounted for in the above discussion. Explain. 2 How large a voltage pulse would be produced across the input capacitance

of the oscilloscope by a single photon? 3 How much additional gain would be required in order to observe individual photons? Would such an observation be feasible?

21

experiment

A P -4

~n~atlonbye~c~ons

introduction In Experiment AP-2 we studied the voltage-current characteristic of a photo­ tube with the photo surface illuminated by light of various wavelengths. We paid particular attention to the characteristic for small negative voltages, which decelerate the electrons ejected from the photo surface. By determining the cutoff voltage we found the relation between the peak energy of the ejected electrons and the wavelength of the light. In the present experiment we shall study the characteristic of the phototube for moderate accelerating voltages. We shall find that for accelerating voltages of a few volts or more the current is independent of voltage and depends simply on the light level. Next, we shall examine the current characteristic of a phototube containing a small amount of a rare gas. We shall see that for positive voltages across the tube a similar plateau is reached but that it is followed by several abrupt increases in current. These increases result from the ionization of atoms of gas in the tube by impact of sufficiently energetic electrons. The resulting increase in density of ions and free electrons provides more current carriers, hence the increase in current. By measuring the voltage-current characteristic, we may determine the ionization energy of the gas atoms. Finally, we shall study a more completely ionized state that occurs at even higher voltages. Commercial gas phototubes are designed to operate with the anode about 90 V positive with respect to cathode. For voltages in excess of this value the ion multiplication increases very rapidly, finally resulting in an electrical discharge across the tube. To understand the reasons for the development of a discharge we consider two processes: (I) the growth of tube current as a result of gas ionization and (2) the production of secondary electrons as a result of positive ion impact on the photocathode. In Fig. 12 we show schematically the several electron currents. The current io is the initial current produced as the result of photons incident on the cathode. The current is, is the secondary current due to ad­ ditional electrons liberated from the cathode by bombardment by positive FIGURE 12

23



atomic physics

gas ions, and i is the total electron current reaching the anode. If we let IX be the electron multiplication due to ionization in the electron stream, we may write (20) where IX depends on the voltage across the tube. Below the first threshold is equal to unity. At normal operating voltages it is between 5 and 10 and may be still higher above normal voltages. Under steady-state conditions the cathode current and anode currents must be equal if there is to be no charge buildup in the tube. Thus, there must be positive ion current moving to the cathode and equal to i (i o + is)' just the difference between the anode and cathode electron currents. Now if y is the probability that a positive ion reaching the anode ejects a secondary electron, we may write for the secondary current IX

(21)

Eliminating the expression for the secondary current is between Eqs. (20) and (21), we obtain for the anode current IX

i = - - - - - io Y(IX - 1)

(22)

For low voltages across the tube IX is close to unity, y is small, and the denominator is only slightly less than unity. But as we increase the voltage, IX and y both increase. At a sufficiently high voltage we may approach the condition 1

1X=1+-

(23)

Y

for which the denominator goes to zero, implying the possibility of current flow even though io is zero. Once a discharge has been established a space charge builds up, stabilizing the tube current even in the presence of light. As you will observe, once a discharge has been established, light is self­ generated so that the discharge generates its own photoelectrons. This process is not important before discharge, and it has been ignored in the above dis­ cussion.

experiment 1

24

vacuum phototube characteristics To determine the characteristic of a vacuum phototube assemble the circuit shown in Fig. 13. The voltmeter Vi may either be an external voltmeter or the voltmeter across the power supply. Use a type 925 photo tube. As a source of illumination you may use either a flashlight or daylight. Fluorescent light should be avoided as the variation in output at 120 Hz (twice the line frequency) will confuse the characteristic as observed on the oscilloscope. The advantage of the circuit shown in Fig. 13 is that by varying the ampli­ tude of the sine-wave signal and the output voltage of the regulated power supply one may examine various portions of the phototube characteristic. Then by turning the sine-wave amplitude to zero one may measure with the

ionization by electrons

be lay

AP-4

FIGURE 13 4

20)

DId

Regulated ~ power supply

lnd

)ns no lng the

t

phototube

+

a

for Sine-wave generator

n) W)

lOOkU

~2)

he

voltmeters the output voltage of the power supply and the current through the phototube (from the drop across the 100-kn resistor). The voltage across the phototube may be computed from the difference between Vi and V 2 . Observe the voltage-current characteristic of the photo tube and determine the value of the voltage across the phototube for the current to rise to half its peak value. Can you explain why it is necessary to apply a moderate forward voltage in order to saturate the photocurrent? For smaller forward voltages, what happens to the electrons that do not reach the anode? From the magnitude of the saturated photocurrent, compute the number of electrons per second reaching the anode. Assuming a photoyield of 0.1 % for the 925 (that is, 0.1% of the incident photons actually eject electrons), compute the number of photons reaching the photo surface per second. Estimate the light flux in watts per square centimeter.

~e,

he ~3)

:nt ce It. If­ ss s-

2

it

e

FIGURE 14

gas phototube characteristics Replace the 925 phototube by a 930 phototube, which contains a small amount of rare gas. You should be able to distinguish three regions in the characteris­ tic, as shown in Fig. 14. At low voltages the characteristic resembles that of a vacuum phototube. Above a threshold voltage VA there is an additional contribution to the current which, if permitted to saturate, would just double the current. One may detect a second threshold voltage, above which the I

f

t

r

25

atomic physics

current rises more rapidly. Attempt to locate the two voltage thresholds; reduce the amplitude of the sine-wave signal, keeping the threshold region centered on the oscilloscope screen, and measure the voltages Vl and V2 • It will be necessary to subtract the voltages in order to determine the voltage across the tube at each of the two thresholds. The interpretation of the characteristic shown in Fig. 14 is the following: For voltages below threshold the gas molecules slightly impede the flow of the electrons to the anode as a result of collisions but otherwise have very little effect on their behavior. But when the energy of the electrons is sufficient to ionize gas atoms on collision (that is, to eject an electron from the gas atoms), we obtain a multiplication of the electron current, since for every initial photoelectron there is now the possibility of two electrons reaching the anode. The positive ion formed in the collision drifts toward the photocathode where it is neutralized. If this ion strikes the cathode with sufficient energy, it may release secondary electrons, producing further multiplication of the current. Both the initial electron and the electron produced by ionization are now accelerated toward the anode. If the anode voltage is sufficiently large, both electrons have the possibility of exciting other gas atoms, producing a second threshold. Still further thresholds may be produced, although it is rather difficult to identify them on the characteristic. In order to compute the ionization energy from the first threshold, it is necessary to have a way of extrapolating back to the beginning of the rise; this is difficult. It is generally easier to determine the difference in voltage for the two thresholds, where one takes similar points on the two thresholds. By doing this you should be able to determine the ionization energy and thus identify the gas. Table 3 gives the ionization energies of the rare gases. From this table, which gas do you think is contained in the 930?

TABLE 3

Gas

Helium (He) Neon (Ne) Argon (Ar) Krypton (K) Xenon (Xe) Radon (Rn)

Ionization energy 24.46 eV 21.47 15.68 13.93 12.08 10.698

The electron energy in electron volts is the energy acquired by an electron when accelerated through a potential of the same number of volts. Thus, for elec­ trons the energies in electron volts are numerically equal to the measured potentials.

3 glow discharge Raise the voltage across the gas-filled phototube until you can just detect a faint glow in the dark. About 250 V is needed. Note that now, even though no outside light is permitted to strike the photosurface, current continues to flow. If you have a spectrometer available, measure the wavelengths of the emitted light. Table 4 lists the wavelengths of the lines most strongly emitted by a glow discharge of each of the rare gases. What does this suggest con­ cerning the gas in the phototube? Does this identification agree with that which you made from the determination of the ionization energy? 26

ionization by electrons

n

TABLE 4

[t

,.r·

)f

y

Characteristic glow-discharge lines in Angstroms

Gas Helium (He) Neon (Ne) Argon (Ar) Krypton (K) Xenon (Xe) Radon (Rn)

~e

AP-4

4686 5400 5570 4501

5876 5852 6402 6965 7067 5871

7504 8115

4624 4671 4682 4826

1t

lS

y

You may find that the discharge is too faint to determine a line spectrum. You should be able to identify the gas present by comparing the color with discharge tubes containing the rare gases.

Ie

Ie it

Ie

w

:h

questions

ld er

1

In Experiment AP-2, the wavelength of the radiation incident on the photo­ cathode was an important consideration. Why is this of less importance in the present experiment?

2

Why is the potential at which the first jump in current occurs not simply equal to the ionization potential of the gas?

3

Suppose the gas pressure in the phototube is increased until the mean free path of an electron is much smaller than the distance from cathode to anode. What effect would this have on the voltage-current characteristic?

4

In a self-sustaining glow discharge, what is the source of energy which creates the initial ionization?

5

Suppose the gas-filled phototube contained a mixture of two gases having different ionization potentials. How would this affect the voltage-current characteristic?

6

In a self-sustaining glow discharge, why does the current not continue to increase indefinitely?

is e;

)r ~y

IlS

:s.

en ~c-

ed

:a

[10

to he ed

ID­

tat

27

experiment

A P -5

electron diffraction

introduction In 1924, when Louis de Broglie advanced his hypothesis concerning possible wavelike behavior of particles, the dual wave.particle nature of electro­ magnetic radiation was well established. De Broglie proposed that this duality also extended to matter, and that the entities commonly called particles may in some circumstances exhibit wavelike behavior. He proposed that the characteristic wavelength A should be related to the momentum p in the same way as for photons, namely,

A=~

(24)

P

whereh is Planck's constant, already known from analysis ofthe photoelectric effect and other experiments. The presently accepted value of h is 6.6256 X 10- 34 J·sec. The first direct experimental evidence for the wave nature of particles was obtained in 1927 by Davisson and Germer, who reflected a beam of slow electrons from a single crystal of nickel and found that their results could be understood on the basis of diffraction of the electron waves by the crystal lattice. The experiments of G. P. Thomson with fast electrons, supplied additional confirmation (reported in 1928) of the correctness of de Broglie's suggestion. Thomson passed an electron beam through thin films of alumi· num, gold, and platinum, and observed ring-shaped diffraction patterns from these polycrystalline targets. From measurements of the sizes of these rings, which were observed on a fluorescent screen, he computed the electron wave­ length needed to account for such a pattern. This wavelength can also be predicted from the de Broglie hypothesis; the electron momentum can be obtained from the accelerating potential V, since p2 2m

(25)

The wavelength computed from Eq. (24) was found to agree to within 1% with the value observed from the diffraction pattern. The present experiment uses Thomson's method of passing a beam of electrons through a thin film of randomly oriented crystals. The condition for an intensity maximum in the diffraction pattern is most simply stated in terms of the various sets of crystal lattice planes. Constructive interference from a given set of planes occurs when the angles of inciden~ and reflection are equal, and when the path difference for adjacent planes is an integer number of wavelengths. Reference to Fig. 15 shows that this condition is satisfied when 2d sin

e=

nA

(26) 29

atomic physics

d sin (J

FIGURE 15

L\

r-~

~\(Ji(J/~ . n-..; d

L

\

I / \11

~ where A is the wavelength, d the distance between adjacent crystal planes, and n an integer called the order of the reflection. Equation (26) is called Bragg's law; it is identical to the condition for constructive interference in the diffraction of x rays by a crystal lattice. The angle e is shown in Fig. 15; we note that the angle between incident and diffracted beams is 2e. Equation (26) can be given a useful reinterpretation by dividing both sides by n to obtain 2(dln) sin e = ), (27) That is, the condition for an nth-order reflection for planes with separation d is the same as that for afirst-order reflection for planes with separation din, which would be the situation if there were n 1 additional intermediate planes between the actual lattice planes. For a given set of reflection planes, there is only one angle for which the Bragg condition is met. Now if a crystal which is properly oriented is rotated about the direction of the incident beam, the diffracted beam will also rotate. Thus for a powder or polycrystalline specimen with random crystal orientation we expect a ring pattern with half-angle 2e. If the structure of a crystal is known, there is a systematic way of finding the various values of d, the separation between adjacent crystal planes. We shall examine this question in detail for two-dimensional lattices, and then quote the corresponding relations for real three-dimensional lattices. Figure 16 shows a square lattice with spacing a. The row of lattice sites labeled A has the property that each site is 3 units to the right and 2 units up from the previous one. The row labeled B is parallel, but displaced vertically. The perpendicular distance between these rows is easily shown to be 3a (22 + 32)1/2 FIGURE 16

B

TO

0

lo

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

A

a:

0

30



electron diffraction

A P -5

However, between these there are two other parallel rows, equally spaced, as shown by the broken lines in the figure. Hence, the spacing ofadjacent rows is actually a (22 + 3 2)1/2 It is easy to generalize this result, and we sketch the derivation. If in a given row each site is h units to the right and k units up from the previous site, where hand k are integers, the corresponding row displaced one unit vertically is a distance d from the first row, given by d

99'S

the ; we

+

(29)

k 2 )1/2

The set of numbers (h,k) are called Miller indices; each such set uniquely specifies a set of rows, and in each case the spacing of adjacent rows is given by Eq. (29). Several examples are shown in Fig. 17.

(27)

ding .We then ;ices. >eled nthe The

(28)

e)1/2

a d = (h 2

ides

Hate mes, if a . the ~r or ring

+

But between these rows there are h - 1 intermediate rows, equally spaced, so the distance between adjacent rows is given by

and

ond din,

ha (h 2

FIGURE 17 0--0-------0--­ 0--0---0-0---0

0--0---0-0---0

0----0-------0­

d

= alVI (a)

~ ~ d

0--0--------0-­

0---0-0-----0-­ 0--0--------0-­ 0--0--------0-­

= alV2

d

(b)

= alV4

d

= alVS (d)

(e)

We note that the first-order reflection for the planes (h,k) = (2,0) is identical with the second-order reflection for (h,k) = (1,0). We will find it simpler always to imagine we are dealing with a first-order reflection, even though this may lead to fictitious intermediate planes. Suppose we place an additional atom in the center of each of the squares in Fig. 17; such a lattice might be called a "square-centered" lattice. For the sets of planes shown in Figs. 17a and 17d, these additional atoms would be midway between planes and would just cancel the reflections from those atoms lying in the planes as drawn. But for the sets of planes shown in Figs. 17band 17c, the additional atoms lie in the planes and reinforce the reflections. Generalizing this result, the reflections for a square-centered lattice corre­ spond to the values of d given by Eq. (29), provided that hand k are either both even (as in the third set ofplanes) or both odd (as in the second set). Otherwise, constructive interference does not occur. The generalization to three dimensions is straightforward and will not be discussed in detail. For a set of crystal planes specified by the set of Miller indices (h,k,l), the distance between adjacent planes is given by d

a

(30) 31

atomic physics

where h, k, and I are integers. If we place an atom in the center of each cube face, as in Fig. 18, the resulting lattice is called/ace-centered cubic, abbreviatFIGURE 18

ed fcc. The condition for constructive interference is that h, k, and I must be either all even or all odd. The allowed reflections for the face-centered-cubic lattice are given in Table 5.

TABLE 5

h, k, I

1 I 1 200 220 3 1 1 222 400 331 420 422 5 I 1, 3 3 3 440

3 4 8 11 12 16 19 20 24 27 32

1.732 2.000 2.828 3.316 3.464 4.000 4.358 4.472 4.898 5.196 5.656

Finally, the diffraction angles may be expressed in terms of Miller indices by combining Eqs. (27) and (30) to obtain . A. J./2a sm () = - - = --"-"-'~'2dln (h 2 + k 2 + [2)1/2

(31)

experiment In this experiment we use a permanently evacuated multiple-target, electron­ diffraction cathode-ray tube. Typical target materials are a polycrystalline aluminum film and single-crystal hexagonal pyrolytic graphite. This latter material grows by deposition from the vapor phase and forms quite perfect small single crystals. In order to obtain adequate electron transmission 32

electron diffraction

AP-5

through these films, electron voltages approaching 10 kV must he used. As a result certain precautions must he observed:

CAUTION

1

Do not touch connecting wires at the rear of the power supply or tube. The voltages employed are sufficiently high to he lethal. In order to preserve the internal targets, the beam current should he kept below 10 1lA. At higher currents heating may he sufficient to destroy the films.

electron diffraction in aluminum Before turning on the power supply he sure that the high-voltage control and

bias control are set fully counterclockwise to the OFF position. The power supply is energized by turning the ac line switch to the ON position. After the power supply has been on for a few minutes the high voltage may safely be set to the desired value. With the electron beam defocused the intensity may be raised and the target scanned for a section containing aluminum. The beam may now be focused to a fine spot and measurements made of the ring diameters. In order to make measurements at another voltage setting the beam should be defocused, the voltage adjusted to its new value followed by a resetting of the beam position. After the set of measurements is completed the voltage and bias controls should he set to the OFF position before the ac line switch is turned off. 2

determination of h

From the observed ring diameters and the known value of L (the distance from the target to the screen) compute the observed values of sin 0 for each voltage. Compute the quantity

o

=

2,a (h2

+ k2 +

12)

(32)

j.

for each value of voltage. Determine the Miller indices for each set of rings. Given the lattice constant of aluminum a

= 4.041 A

determine the de Broglie wavelength A at each voltage and compute the product Ap. Compare your set of values with the accepted value for h.

questions ,

What is the energy of a photon having the same wavelength as a 100-eV electron? In what region of the electromagnetic spectrum does such a photon lie?

2

What is the wavelength of a "thermal neutron"; that is, a neutron whose kinetic energy is 3kT/2, where k is Boltzmann's constant, and T is the temperature of the material through which the neutron is moving? 33

atomic physics

34

3

What does the diffraction pattern from pyrolytic graphite tell you about its crystal structure? How would the pattern differ if ordinary powdered graphite were used?

4

How is the energy of the electrons in the beam affected by the work function ofthe cathode material? Does the work function of the target have any effect? Explain.

5

Could protons be used for crystal diffraction experiments? What energy would be appropriate 1 What difficulties might arise which are not present with electron diffraction 1

berkeley p h ysi

lab rato y, d edition

alan m. p ortis, university of ca lifo rnia. berkeley hugh d . y ung, carnegie-mellon university geiger- mueller tube

N P-f

ra dioactive deca y

N P-2

the SCintillat ion c oun ter

NP-3

b eta and gamma absorption

NP-4

neutron activation

NP-5

m gr w-hill boo c omp a ny n ew york

Sf

louis

san francisco

diisseldorf

nuclear physics Copyright 1971 by McGraw-Hill, Inc.

All rights reserved. Printed in the United States of America.

No part of this publication may be reproduced, stored in a

retrieval system, or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher.

Library of Congress Catalog Card Number 79-125108

07-050491-1 1234567890 BABA 79876543210 The first edition of the Berkeley Physics Laboratory copyright 1963,1964,1965 by Education Development Center was supported by a grant from the National Science Foundation to EDC. This material is available to publishers and authors on a royalty-free basis by applying to the Education Development Center.

This book was set in Times New Roman, printed on permanent paper, and bound by George Banta Com­ pany, Inc. The drawings were done by Felix Cooper; the designer was Elliot Epstein. The editors were Bradford Bayne and Joan A. DeMattia. Sally Ellyson . supervised production.

nuclear physics

INTRODUCTION In these experiments we study some of the properties of the atomic nucleus,

with particular attention to the decay of unstable nuclei. We study severa] instruments used to detect the emission of particles from nuclei during their decay, particularly the Geiger-Mueller tuhe and the scintillation counter. Using these instruments, we can make a detailed study of rates of radioactive decay and of the properties of the emitted particles, including the particles referred to as IX, /3, and y. Finally, we study an analytic technique in which stable nuclei are made radioactive (unstable) by bombardment with neutrons. When a nucleus absorbs a neutron and subsequently decays, the type and energy bf the decay products can he used to identify the original nucleus.

r

experiment

N P-1

Geiger-Mueller tube

introduction Experiments in this series involve the detection of two common radioactive decay products, fJ particles and 1 rays. The Geiger-Mueller (G-M) tube, which is a commonly used detector of these products, is the object of investi­ gation in this experiment. The radioactive sources and their decay products will be studied in some detail in later experiments, but a brief description of the decay modes is in order here. The thallium-204 (atomic mass number) source emits fJ particles (high-energy electrons) with a maximum energy of 0.77 MeV. In the decay of cesium-l 37 a fJ with maximum energy 0.52 MeV is emitted, leaving the barium-137 daughter nucleus in an excited state. The barium nucleus decays to the ground state by emitting a 0.66-MeV 1 ray (high-energy photon). There are a great number of different particle detectors, each with its own particular limitations and advantages. Charged particles are usually detected by the ionization they produce in an ionization chamber, proportional counter, G-M tube, crystal counter, or electron multiplier tube, or by the excitation ofa phosphor which subsequently emits photons. Neutral particles are usually detected indirectly by the ionizing particles they produce under proper conditions. Photons in the visible (and the ultraviolet if the phototube envelope is transparent at the desired wavelength) may be detected with a photomultiplier: Nuclear 1 rays are detected by their conversion into elec­ trons by the photoelectric effect (in high-Z materials for 1'S up to several hundred kilovolts), the Compton effect (0.1 to 3 MeV), or pair production (> 1 MeV). Slow neutrons are detected by a boron trifluoride ionization chamber or proportional counter. Other detectors include cloud, bubble, and spark chambers and photographic emulsions. One of the simplest detectors, the Geiger-Mueller tube, will be studied in this experiment.

THE GEIGER-MUElLER TUBE

A G-M tube is simply an inert-gas-filled conducting cylinder with a fine tungsten wire down the center as shown in Fig. 1. When an ionizing particle passes through the gas, it liberates electrons. The electrons are attracted to the central wire, which is at a positive potentiaL As these electrons approach the central anode they may gain sufficient energy to produce additional ionization; the ionization potential of argon is 15.68 eV. The secondary electrons produced in this way are themselves accelerated and may be able to produce additional ionization. The positive ions, being quite heavy, move much more slowly than the electrons, and have the effect of reducing the field intensity around the anode. One might expect the discharge to "quench" itself if the potential across the tube is not too high. However, photons emitted from the excited gas around 3

nuclear physics

FIGURE 1 ( b)

(a)

L Rarc "as

'"

A Geiger-Mueller tube is a conducting cylinder filled with a rare gas and with a fine wire on the axis. (c)

A positive potential is applied to the central wire or anode.

(d)

,,

,, "

,

,/ +

+

,

o = Positive inn ( + ) • = Negativc ckdron (-

)

. When a beta particle or gamma ray passes through the tube, it ionizes atoms of the rare gas.

(e)

The electrons move rapidly toward the anode. The heavy positive ions move more slowly toward the outside cylinder. (f)

+ /

00/

o

o

/

~o 00

00

The initial electrons ionize other atoms, producing an "avalanche" of electrons.

4

The "avalanche" is stopped by the accumulation of positive ions around the anode, reducing the magnitude of the electric field.

Geiger·Mueller tube

NP-1

the anode may strike the surrounding cathode, ejecting additional electrons and possibly sustaining the discharge. In order to absorb these photons, most tubes contain about 10% of a quenching gas. An organic compound such as ethyl alcohol is used with argon. When neon is used as the inert gas a halogen may be used for quenching. The quenching gas absorbs the photons and dissociates. In each quenching process some 10 10 molecules are dissociated. Since ethyl alcohol will not reform, an organic-quenched G-M tribe has a limited life, usually about lOB counts. The halogens can recombine, so a halogen-quenched tube has an indefinitely long life. Typically, a G-M tube is used in a circuit which counts the number of discharges in some definite time interval. The counting rate depends on the applied voltage in the way shown in Fig. 2. Below a minimum voltage, the FIGURE 2

I

, I

I

150

I I

I

50

600

700 Applied voltage

starting voltage, no counts will be registered. This minimum voltage is a function of the gas pressure and the anode diameter, and may be between 500 and 900 V. As the voltage is increased, more and more counts are registered. Over a range of voltages, called the plateau range, the counting rate is relatively insensitive to applied voltage. The change in counting rate over a 100-V range of applied voltage may be as little as 5%. Organic­ quenched tubes usually have a flatter plateau than halogen-quenched tubes. For still higher applied voltages the tube may go into continuous discharge. It is particularly important that an organic-quenched tube not be permitted to go into continuous discharge, as the quenching gas may be exhausted in this way.

HEALTH HAZARDS OF RADIATION The intensity ofa radioactive source may be characterized by the number of

decays per second. The conventional unit is the activity of I g of radium, 3.666 x 1010 disintegrations per second. This unit is called the curie (ab­ breviated Ci). The maximum activity that may be used without a special Atomic Energy Commission license is 50 /lCi of 204TI or 10 /lCi of 137CS. Thus, the total activity may approach 60 /lei or about 220,000 disintegrations per second. [n assessing the health and safety hazard of a specific level of radiation, the important consideration is the amount of ionization induced in body 5

nucle
tissue. As a steady-state rate for radiation workers, the Atomic Energy Com­ mission recommends an upper limit of 8 x 10 10 ion pairs per gram of tissue in an eight-hour day. This is roughly translated as an upper limit of 106 p particles or 108 I-MeV y absorbed per square centimeter of body surface in an eight-hour day. Although you should not handle the sources any more than necessary, you can acquire at most a small fraction of the limit recom­ mended by the AEC.

CAUTION

One precaution which you should exercise is not to bring a f3 source close to your eye, as the cornea can become damaged in this way.

experiment The characteristics of an inexpensive halogen-quenched G-M tube are investigated. In particular, a study is made of the pulse shape, the charge per pulse, and the dead time. 1

pulse shape Set up the circuit of Fig. 3. The high-voltage power supply of an oscilloscope, which provides the accelerating voltage for the CRT, is used to supply the

FIGURE 3

+

high voltage needed to operate the G-M tube. An internal series-limiting resistor is placed in the oscilloscope to prevent serious electrical shock, but care should be taken. Check all your connections before making the high­ voltage connection to the oscilloscope. One should also be careful to dis­ charge capacitors before removing them from the circuit to avoid annoying shocks. With the oscilloscope at its slowest sweep rate and with the voltage control set at the minimum voltage position, place one of your sources opposite the sensitive area of the G-M tube. Slowly increase the applied voltage until you 6

Geiger-Mueller tube

NP-l

are able to detect negative pulses on the face of the scope. Now increase the voltage by about one-tenth tum to provide an additional increase of about 50 V, placing the G-M tube in the plateau region. When the sweep rate of the oscilloscope is increased to about 1 kHz, the pulse shape should become visible. Be sure that the sweep is set to synchro­ nize on negative pulses. Calibrate the sweep duration with a square-wave generator, and make a careful sketch of the output pulse. Ideally, the pulse height should be equal to the difference between the starting voltage and operating voltage. However, because of the loading of the G-M tube by the I-Mil resistor and the input capacitance of the oscilloscope, a substantially reduced pulse is observed. How long a time is required for the pulse to drop from its maximum amplitude to half its maximum? 2 number of ion pairs per pulse

In order to determine the amount of charge which passes through the G-M tube per pulse, place a 0.002-IlF capacitor in parallel with the I-Mil resistor. You should now observe a small negative step v instead of a pulse. Since the RC decay time, which is about 2 msec, is much longer than the buildup time of the pulse, neglect the resistor during this period. Then the charge stored by the capacitor will be given by q

= Cv

(1)

where Eq. (1) is just the definition of capacitance. Calibrate the oscilloscope and determine v. A low counting rate should be used to avoid pulse distor­ tions introduced by dead-time effects (to be studied in Part 4). Compute q, the amount of charge that passes through the G-M tube per pulse. How many ion pairs are formed per pulse? 3 counting rate

A vacuum-tube voltmeter (VTVM) may be used as a counting-rate meter by replacing the oscilloscope of Fig. 3 by the VTVM. If V is the measured VTVM voltage, the counting rate is r

I q

V Rq

= - =-

(2)

where the loading by the meter (about 11 Mil) has been neglected. (Dis­ connect the oscilloscope to prevent further loading.) What is the maximum counting rate that you can obtain with the 137CS source? 4 dead time: direct measurement

Return to the circuit shown in Fig. 3 and establish as high a counting rate as possible. With a sweep duration of about 1.5 msec carefully examine the trace on the oscilloscope. You should observe a pattern similar to that shown in Fig. 4, where the synchronizing pulse is shown followed by subsequent smaller pulses. Note that for about 100 Ilsec after the first pulse, the G-M tube is inoperative. This period is called the dead time '1:. How does one explain the gradual increase in height which follows? Think in terms of two electrons which penetrate the G-M tube in succession. The first electron fires the tube, producing a pulse of full height and triggering the oscilloscope sweep. If the second electron enters in an interval shorter than the dead time '1:, there will be no second pulse. If it enters in the time just slightly longer than '1:, a second pulse is produced but with reduced height. The longer the time interval, the more fully the tube has recovered and the more nearly does the height of the 7

nuclear physics

FIGURE 4

Time

second pulse approach that of the first. A high counting rate is necessary in order that the pulses will occur during the recovery interval. But because of imperfect oscilloscope triggering, the recovery pulses are obscured by many background pulses. Adjusting the focus control sometimes makes the recovery pulses easier to see.

5 dead time: indirect measurement At high counting rates the tube will be dead a significant part of the time, and the apparent counting rate will be too low. Imagine that ro particles per second enter a Geiger tube with a dead time t, and that r pulses per second are registered. Now in anyone second the tube is dead for a time rt. This means that only a fraction 1 rt of the pulses is counted; thus if the rate at which particles enter the tube increases by dr o• the actual number dr of pulses counted is given by dr = (1 - rt)dr o

or

dr dro = - - ­

I - n::

(3)

Integrating, one finds

(4) Note that when r is much smaller than l/t, r = roo But when ro is much larger than l/t, l/t is the limiting counting rate. Determine an effective dead time for'the Geiger tube by counting a pair of samples together and then separately. Label the samples A and B; first count sample A using the voltmeter as in Part 3. Then bring up sample B and count A and B together. Finally, remove sample A and count B. The advantage of this procedure is that it is not necessary to place the sample in a reproducible position. Note thatthe counting rate r(A + B) is c1earlyless than r(A) + r(B). The following analysis will permit a determination of the dead time t. Let ro(A) and ro(B) be the actual activities. For sample A I - r(A}r

e-ro(A)T

(5)

e-[ro(A)+ro(B)]T

(6)

For samples A and B together 1 - r(A 8

+ B)1:

Geiger-Mueller tube

NP-1

and finally, for sample B alone 1 - r(B}r =

(7)

e-ro(B)t

With these three determinations we may expect to be able to determine ro(A), ro(B), and ,. Since the product of Eqs. (5) and (7) equals Eq. (6), [1 - r(A),] [1 - r(B),] = 1 - r(A

+ B),

(8)

Solving for " - r(A + B) ,= r(A) + r(B) r(A) r(B)

(9)

Determine, in this way and compare with your direct observation from the oscilloscope. You may expect only qualitative agreement using the VTVM as a counting-rate meter. The counting rate in the VTVM method depends on the pulse height as well as the number of pulses. Then for high counting rates the recovery time is important in the VTVM measurement. The effective dead time is then longer than for the direct measurement. If a scaler is available, repeat the indirect dead-time measurement. Since the pulse must be lalger than some minimum value to trigger the scaler, the scaler-measured, should be longer than the value found from the oscillo­ scope trace. But since the pulse need not be a full-size pulse (trigger pulse in the oscilloscope method) to be counted as a complete pulse, the scaler­ dete/mined , should be shorter than " as determined with the VTVM. Estimate how large a variation in the measured, can be expected from this mechanism. How do your experimental results compare?

r in : of my the

me, per )nd ~his

eat lses

(3)

(4)

rger

rof mnt mnt :e of :ible

Questions 1

Explain the role of a quenching gas in a Geiger-Mueller tube. Explain why for excessively high voltages across a Geiger tube the quenching gas is unable to prevent continuous breakdown.

2

Explain why the starting voltage depends on anode diameter.

3

Explain why for an unloaded G-M tube the output pulses should equal the difference between the starting and operating voltages. Explain why resistive and capacitive loading of the tube reduce this voltage.

4

Explain Eq. (3). How would the following discussion be modified if there is a radioactive background?

5

Solve Eq. (4) for roo Make a plot of ro, versus r-r.

(B).

Let (5)

(6) 9

experiment

N P -2

radioactive decay

introduction In this experiment we use the Geiger-Mueller tube as a detector of radioactive decay. We shall be interested in measuring the fluctuations in the number of events counted in some interval of time. If the interval of time is sufficiently long and the counting rate sufficiently low, we can count individual events. For higher counting rates we may use a scaler or, alternatively, a device for integrating the total charge from the G-M tube. In the present experiment we shall also introduce a simple variant on these techniques. In addition to counting for a definite period of time, we shall introduce the pulses into a filter which has a time constant r = RC. As we shall see, the fluctuations just correspond to those in a definite time interval A.t = tr. A source containing a radioactive element ordinarily has a very large number of unstable nuclei, and the probability for anyone of them to decay in a specified time interval is very smalL As discussed in Experiment MS-5, the probability for n decays to occur in a specified time interval is given by the Poisson distribution

n!

(10)

where a is equal to the average number of counts (n) in the intervaL The standard deviation (1 of this distribution is given by (11) As n becomes larger and larger, the Poisson distribution of Eq. (10) ap­ proaches the normal, or Gaussian, distribution given by Eq. (67) in the Mathematics and Statistics Unit.

Pen)

where (1

=

=

(211:(1)-1/2 e-(n-n)/2tT 2

(12)

(n) 1/2 is the standard deviation, in general, defined by (12

=

L (n

- n)2 Pen)

n

When the Gaussian distribution is used as an approximate representation of the Poisson, we have a = n = (12. We may now consider the matter of long-term charge fluctuations, as observed as voltage fluctuations detected by the oscilloscope, with input connected as in Fig. 5. Roughly speaking, we may expect that the observed fluctuations in voltage will correspond to the variation in charge accumula­ tion during one time constant of the circuit, r = RC. Under these circum­ stances, the charge fluctuation will be of the order (13) 11

nuclear physics

FIGURE 5

IMQ

v

O.lJ.LF

-where q is the charge per pulse and r is the counting rate. The fluctuation in voltage is given by

tiV = tiQ C

(14)

Since the mean voltage across the circuit is given by V= rqR

(15)

we obtain for the ratio of ti V over V _tiV V

~

(,,)-1/2

(16)

The above treatment has been rough in two respects. First, we are not really concerned with the fluctuations in a definite interval of time. Rather, we have a system which relaxes exponentially. In Eq. (26) we shall show that the appropriate standard deviation in charge is given by

tiQ

,,)1/2 q ( 2

=-

(17)

Second, if we attempt to judge the peak-to-peak amplitude of the fluctuations by eye, how many standard deviations do we take in? This is in large part a question of human response and memory, and a number of studies have suggested that about four standard deviations is appropriate. This would mean that a measure of the fractional fluctuations would be given by 4 ti V = 4 (~)112 V rr

(18)

Charge Fluctuations In this experiment we examine random events in two ways. In the first way

we count the number of events taking place in an interval of duration tit. We find that (1) where there is no limit to the number of possible events, and (2) where the number of events observed on the average is large, we obtain a normal, or Gaussian, distribution for the probability of observing n events in the interval tit. The root-mean-square displacement from the average, or standard deviation, is simply equal to the square root of the average number of events ii occurring in the interval tit. Where we are interested in an actual experiment, such as the collection of 12

radioactive decay

NP-2

charge from a G-M tube, the standard deviation in collected charge is given by AQ

= (r At)1/2 q

(19)

where r is the mean counting rate and q is the charge collected per event. We now ask the following slightly different question. Instead of counting events, let us inject the charge increments q into an RC circuit as shown in Fig. 6. Again, if r is the mean counting rate, what will be the root-mean-square fluctuation in charge on the capacitor? FIGURE 6

-

I

:-,.

tl

The current [ which flows into the filter will be a random function of the time of the form shown in Fig. 7. Here q is the charge per event and J rq is

i) FIGURE 7

I

;) ~t

)t

r,

It

7)

-

tS

~

t'

-

'-----''-----'

­

,--,---,WLlL

a

re

.d

the mean current. What is the mean charge on the capacitor C? Since the voltage across the capacitor and across the resistor must be equal we have

Q = JR

8)

C

rqR

(20)

Solving Eq. (20) for Q, we obtain

Q

ty .t. td

a

1s

)r

er

of

n:q

(21)

where 't = RC is the time constant of the filter. Let us next compute the mean-square deviation in charge AQ2 from the average value Q. In a short interval of time At at t' as shown in Fig. 7 we obtain a mean-square deviation AQ2(t') = rq2 At

(22)

Since charge placed on the capacitor decays according to the equation Q(t)

Q(t')e-{t-t'llr

(23)

we can expect the mean-square charge accumulated in At at t' to decay as (24) 13

nuclear physics

Finally, we must integrate over all increments of previous time to obtain

~Q2.

=

rq2.

foo

e-2.(I-I ' )/ t

dt'

= tn:q2.

(25)

Taking the square root of Eq. (25), we obtain for the root-mean-square charge deviation ~Q =

n:)1 /2. q ( -2

(26)

and we see that the standard deviation in charge for an exponential filter of time constant l' is the same as that for an interval ~t = 1'/ 2.

Output Spectrum of a Scaler

One problem which we can investigate experimentally is the output spec­ trum of a scaler. Let us imagine as in Fig. 8 that pulses enter a scaler at a mean rate r and leave at a mean rate r/S.

FIGURE 8 I

r

~

Scale of S

r

S

Are the output pulses randomly distributed? That is, may they be character­ ized by a Poisson distribution? If the pulses enter a filter of time constant l' = RC, what will be the rms charge variation ~Q? First the probability ofn input pulses in a time ~t is given from Eq. (10) by (n)ne - n P (n) = - (27) In n! where n = r ~t. Now the probability of m output pulses during the same interval must just be equal to the sum of the probabilities of mS to (m + l)S input pulses: (m+ 1)S-1 (n)ne-n (28) Pout(m) = n=mS n!

L --.-

If S is sufficiently small so that Pin(n) is slowly varying over the interval ~n = S, we may write Eq . (28) as _ (mS)mS mS Pout(m) = Se(mS)! (29) Finally, if ~t is sufficiently long so that mS is large, we obtain a Gaussian distribution (30)

where (J = (m/S)1 /2. = (n)1 /2./S. Note that the standard deviation in the output is not equal to (m)1 /2. as for random pulses, but is equal to (m/ S)1 /2. . Thus, the output pulses are more evenly spaced than random pulses would be. With a scaler available you may wish to verify that the charge fluctuations will now be given by n:)1 /2. q (31) ~Q = (2 S where 14

r

is the input pulse rate.

radioactive decay

experiment

;) 1

5)

)f

can

nt

2

longer intervals

By adding the counts in successive lO-sec intervals, you may obtain data for twenty-five 20-sec intervals. Plot this histogram and, for comparison, the expected frequency. Note that as nbecomes larger and larger, the distribution becomes more and more sharply peaked around n.

by

:7)

ne

Poisson distribution

Assemble the circuit shown in Fig. 3. With a 10-IlCi 137CS source in the vicinity of the G-M tube, increase the voltage across the tube until counts become visible on the oscilloscope. Now increase the voltage approximately 50 V by a one-tenth tum increase of the control knob. This should place you within the plateau of the tube. With the oscilloscope on its slowest sweep, remove the 137CS source sufficiently far from the G-M tube so that the average counting rate as observed on the oscilloscope screen is about one count every second. Determine the number of counts in successive 10-sec intervals, counting 50 such intervals. Plot a histogram of your data, with the frequency of n counts (that is, the number of intervals in which each number of counts is observed) plotted as a function of n. Use your experimental data to compute n. Compute values of the Poisson distribution using this n = a, for all values of n which you observed. Multiply each value of the distribution, which gives probabilities for various values of n, by 50 to obtain the corresponding expected frequen­ cies, and plot these on the histogram for comparison with the experimental results.

~e

~r-

NP-2

3

long-term fluctuations

As a convenient example of long-term fluctuations, assemble the circuit shown in Fig. 5. Bring the 137CS source sufficiently close to the G-M tube so that an appreciable bounce is observed in the oscilloscope trace. Determine the fractional amplitude of the fluctuations for a wide range of counting rates and time constants, and compare your observations with Eq. (18).

)S

tS)

tal

t9)

questions

an

10)

'Or )re

1

The parameter a in the Poisson distribution, Eq. (10), is estimated by calcu­ lating n from your data. How reliable is this result? That is, what is the standard deviation of a obtained in this way?

2

In judging the peak-to-peak amplitude of charge fluctuations as observed with an oscilloscope, why is it reasonable to take this as about four times the standard deviation of the charge fluctuation?

3

If the average number of decays counted in a certain time interval is n, what is the probability that in anyone interval at least 2n decays will be observed?

IDS

31)

15

experiment

N P -3

the scintillation counter

introduction In this experiment we shall examine the properties of a device which is a particularly effective detector of gamma rays. Although the Geiger-Mueller tube may be used as a detector ofgamma rays, its efficiency is quite low, being only 0.5 to 1.5%. This is because direct ionization of the gas within the G-M tube by a gamma ray is quite small. A gamma ray is principally detected as the result of ejection of a photoelectron from the surrounding metal or glass envelope. But since the envelope is intentionally made quite thin in order to permit low-energy electrons to enter the chamber, the probability of photo­ ejection is correspondingly small. A scintillation counter used for the detection of gamma rays makes use of a large sodium iodide crystal to which has been added a small amount of thallium. The gamma rayon passing through the crystal has a high probability of producing a photoelectron. This electron will in turn produce a high degree of ionization in the crystal. Most of the secondary electrons produced in this way recombine, generating radiation in the ultraviolet. This radiation is absorbed within the crystal. With thallium impurities present some of the radiation reaches the thallium centers, producing an excited atomic state of the thallium. When the thallium atoms decay, they produce blue light. Since NaI is transparent through the whole visible range, this blue light is able to escape from the crystal and be detected by a photodetector. In order to obtain fast response and high sensitivity the light pulses are usually introduced into a photomultiplier. An interesting feature ofthe scintillating crystal is that the number of thallium centers excited is proportional to the gamma-ray energy. Thus, from an analysis of the distribution in pulse height, one may determine the distribution in gamma-ray energies.

experiment 1

operation of a scintillator Assemble the circuit shown in Fig. 9. The 931A phototube was studied in Experiment AP-3, and you may refer to this experiment for a discussion of its operation. A thallium-activated sodium iodide crystal about one centimeter thick should be attached to the envelope of the 931A. Placing a thin layer of a clear grease (Vaseline is satisfactory) between the crystal and the envelope will reduce light scattering. The entire assembly should then be covered with a light-tight shield in order to avoid any background light pulses. Black photopaper fastened with black tape should be adequate for this purpose. 17

nuclear physics

FIGURE 9

Oscilloscope

Regulated power supply

-12.'50 V

2.7Mll

4MS1

~._-,

IMS1

5.6MS1 931A Red

With a light-tight cover in place, voltage may be applied to the phototube. Pulses may be observed on the oscilloscope and the anode current read with a vacuum-tube voltmeter. Place the 137CS source in front of the crystal. What range of pulse heights do you observe? What is the number of detected photons entering the photomultiplier per detected gamma ray? (For a dis­ cussion of the gain of the photomultiplier, see Experiment AP-3.) Replace the 13 7Cs source with 204Tl. Are you able to detect any counts from the thallium source? 2 observation of pulses Using a procedure similar to that described in Experiment NP-I for the G-M tube, attempt to observe single output pulses from the photomultiplier. What is the duration of these pulses? What determines this?

questions g/cc. The photoelectric absorption coefficient of the 0.66-MeV gamma ray from 137CS is about 0.04 cm 2/g. What is the probability that a photoelectron is produced by the gamma ray on passing through the crystal? Compare this with the probability that a photoelectron is produced in the 0.002-in .-thick stainless-steel jacket of a G-M tube.

1 Sodium iodide has a density of 3.67

2

18

You may have observed some counts from the 204TI source, which is a pure beta source. How were these counts produced? How can they be eliminated?

experiment

N P -4

beta and gatntna absorption

introduction In this experiment we shall determine the absorption of beta and gamma rays by various thicknesses of absorbing material. There are several reasons for making this determination. First, there may be occasions when we will want to shield out beta or gamma rays and we shall wish to know what thickness of absorber is necessary. Conversely, we may wish to know the effect of a cover (like the stainless-steel jacket on the G-M tube) in reducing counting rate. Or we may wish to know the probability that a gamma ray which we want to detect is absorbed in a given thickness of sodium iodide. The range of a beta particle or the absorption coefficient of a gamma ray is strongly dependent on initial energy. We may thus use the observed variation in counting rate to determine the initial energy.

)tube. i with What reeted a dis­ eplace m the

experiment 1

~G-M

absorption of beta particles Assemble a G-M tube as shown in Fig. 3. Bring a 204TI sample sufficiently close to the G-M tube to obtain a counting rate of about 1000 counts/sec. Insert varying thicknesses of aluminum between the source and the detector. The absorber should not be much larger than the detector so that additional beta particles are not scattered into the detector. The densities of various metals are given in Table 1. Also given is the thickness in inches correspond­ ing to 100 mg/cm 2.

What

rons

,rption cm 2 /g. maray that a et of a

a pure nated?

TABLE 1

Metal

Density, glee

Al

2.70 8.92 7.86 11.34

eu Fe Ph

Thickness for 100 mglem2, in. 0.0146 0.0044 0.0050 0.0035

Plot counting rate as a function of absorber thickness in milligrams per square centimeter (add the thickness of the G-M jacket in milligrams per square centimeter) on semilog paper. Is the decay exponential? Actually, the beta-particle counting rate is not expected to be exponential. A beta particle passing through matter loses energy continually by ionization 19

nuclear physics

of the medium. It is not difficult to show that the rate of loss of energy is inversely proportional to the square of the velocity, or the kinetic energy: dE

k E

dx

(32)

Integrating Eq. (32), we find for the particle energy

2kx

E2 = E02

(33)

where Eo is the initial energy. Note that when the electron has gone a distance

E/

Xo

(34)

= 2k

the energy has gone to zero and the electron has been stopped in the medium. In Fig. lOis shown the range-energy curve for beta particles. Note that FJGURE 10

lOS~m:H

2 1---+--+--+-1-+,...,

e

21---+--+--+-I-+-hH-+-++H--4--~-+-HH-+-I-+~--+--+-~~~

~


~

0.11---+--+­

1.0

100

10

Range, mg/cm

1000

2

5

10,000

2

although the range does increase as the square of the energy for energies below 0.1 MeV, it increases more slowly for higher energies. This is because as the energy increases' above 0.1 MeV, the particle velocity is approaching the velocity of light. At constant velocity the energy loss per unit length approaches a constant, yielding a range which increases more nearly linearly with increasing energy. By differentiating the experimental data we can obtain a plot of the number of particles stopped per incremental thickness of absorber. Using Fig. 10 one may then convert this to number of electrons per unit energy. Make such 20

beta and gamma absorption

r

N P-4

a plot and determine the energy spectrum of the beta particles from 204Tl. Find the endpoint energy and compare with the accepted value 0.77 MeV.

is

32) 2 33) nee 34) 1m.

hat

absorption of gamma rays Unlike beta particles, gamma rays do not produce a trail of ions along their path but instead interact strongly at some particular point on the path. Thus, we may expect that the gamma ray is either scattered out of the beam or is relatively unaffected. There are three principal scattering mechanisms for gamma rays. The first is the Compton scattering of the gamma ray by an atomic electron. In such a scattering process the gamma ray is scattered out of the forward beam while transferring recoil energy to the electron. A second process is photoelectric absorption in which all the gamma-ray energy is transferred to the ejected electron. Finally, pair production, the creation of electron pairs, is a process which becomes important above the energy threshold for this process, 1.02 MeV. At low energies photoproduction is the dominant scattering process. There is an intermediate region around 1 MeV where Compton scattering domi­ nates. At still higher gamma-ray energies, pair production is the dominant process. In order to measure the absorption coefficient of gamma rays, we shall use a 137CS source and the scintillation counter. Assemble the circuit shown in Fig. 9 and measure the counting rate as a function of the thickness of lead between source and scintillating crystal. Remember that the 137es is also a beta emitter. You may detect some beta particles at low absorber thicknesses. Ifwe have a gamma ray flux I, the decrease in 1 in passing through a distance dx may be written as

dl = -J1 dx

(35)

where J1 is called the linear absorption coefficient. Integrating Eq. (35) we obtain (36)

where 10 is the initial gamma-ray flux. Plot the observed counting rate as a function of thickness oflead on semilog paper. Determine the linear absorption coefficient. Compute the mass absorption coefficient J1m from the relation

0,000

(37)

where p 11.34 glcc is the density of lead. From Fig. 11 find the apparent gamma-ray energy and compare with the accepted value of 0.66 MeV.

:rgies ~use

~hing

mgth early

mber g. 10 'such

3

mass absorption coefficient You may wish to try other gamma-ray sources of known energy and in this way determine the mass absorption coefficient of lead as a function of energy. You may wish to use the G-M tube to measure a pure gamma source. You can also look at the 137CS source, but you will probably find that the counting rate is dominated by electrons at moderate counting rates. 21

nuclear physics

100

bIl "-

'"SCJ

10

....: >=:

Q)

.(:1

$Q) 0

CJ

>=:

.~

.....

...

0.. 0

'"

..0 CIl

'"CIl

'"

:::E

Energy, MeV

questions 1 Explain why the counting rate for gamma rays is expected to decrease

exponentially with thickness, but the counting rate for beta particles is not expected to decrease exponentially. 2 Invent an experiment to show that the beta particles leaving an absorber have lower energy than those particles entering. 3 How might you show that the un scattered gamma rays have the same energy

as the incident gammas? 4 On the same coordinates plot activity versus absorber thickness for the beta particles and gamma rays from 137CS. Now reduce the gamma activity by a factor of 100, corresponding to the approximately 1% efficiency of a G-M tube for the detection of gamma rays. At what absorber thickness are the beta and gamma activities equal? By what factor is the beta activity reduced at this thickness? 5 Design an experiment to measure the angular distribution of Compton­ scattered gamma rays. 6 Design an experiment to determine elm of beta particles. 7 Design an experiment to magnetically deflect beta particles and thus deter ..

mine their energies. 22

experiment

N P -5

neutron activation

Introduction The process of neutron activatit):1 involves the production of a radioactive source as a result of the absorption of neutrons. Since a radioactive source emits characteristic alpha, beta, or gamma rays, one may, from an analysis of the radiation, determine the source. Thus, neutron activation may be used for determining the composition ofa sample. Activation is more commonly used, however, in searching for trace amounts of certain elements that strongly absorb neutrons and result in the production of highly radioactive nuclei. As we shall see, if the initial neutron flux, the activation cross section, the life and the decay rate are all known, one may determine the concentration of trace nuclei. Neutron activation is used, for example, in analyzing for trace amounts of gunpowder. Plutonium-Beryllium Neutron Source

s

A commonly available source of neutrons is prepared by mixing 239Pu with finely ground beryllium powder. Plutonium-239 decays by spontaneous fission with the emission of an alpha particle (the nucleus of helium). Some fraction of the alpha particles are absorbed by the 9Be, nuclei resulting in the production of an excited state of 13C. Although 13C is normally stable, the excited state of 13C may decay to 12C with the emission of a neutron. The nuclear reactions that we have been describing may be written as 2~~pU ---> fission products

+ ~He

(38)

(39)

ve

(40) Note that the asterisk (*) symbol in Eqs. (39) and (40) indicates that the 13C is formed in an excited state. Note that the subscript indicates atomic number and the superscript indicates atomic mass. Both mass and charge must balance on the two sides of the equation. A source with an activity of 1 Ci has the same activity as I g of radium which is entirely 2~~Ra. The half-life is 1,622 years so that the activity of 1 g must be 1 x6023xlO 23 x 0.69135 =3.666 x 10 10 226.0254' 1622 x 3.156 x 107

sec- 1

where 1/226.0254 is the fraction ofa mole corresponding to 1 g; 6.023 X 10 23 is the number of nuclei in a mole; 3.156 x 107 is the number of seconds in a year; 0.69135 is In 2 which, when divided by the half-life, gives the decay rate per nucleus. Now, the half-life of 239Pu is 24,360 years or about 15 times as long as that of 226Ra. Since the atomic masses are about the same, this means that we shall need 15 g of plutonium to yield 1 Ci. One curie of plutonium in 23

nuclear physics

tum produces 1.5 x 106 neutrons/sec. Thus, only 1 neutron is produced for every 20,000 alpha particles. The plutonium-beryllium source is surrounded by a lead shield, which absorbs any alpha particles not captured by the beryllium metal. The lead also absorbs most of the emitted gamma rays. The neutrons are absorbed very little by the lead, however, and pass through the lead shield. The proba­ bility that a neutron is captured by a nucleus depends on the energy of the neutron, generally being larger the lower the neutron energy. In order to re­ duce the neutron energy as much as possible, the source is surrounded by either paraffin or water. As a result of inelastic collisions with protons, the neutrons slow down until they reach a mean energy of 4 x 10- 21 J, which is the mean thermal energy at room temperature. (At this energy the most probable neutron velocity is 2200 m/sec.) Slowing the neutrons to thermal energies is called moderation. Neutrons in paraffin are moderated in a distance of about 4 cm. In order to provide adequate moderation and shielding, a I-Ci source should be placed at the center of a moderator about 2 ft high and I! ft in diameter (just about the size of a garbage can). The sample to be activated is dropped into a tube at the same depth as the source. Activation of Silver

Naturally occurring silver is composed of two isotopes, 107Ag, which is 51.82% abundant, and 109Ag, which is 48.18% abundant. The extent to which a nucleus interacts with an incident particle may be described in terms of a capture cross section. That is, an incident particle coming within that area surrounding the nucleus will be captured. The thermal-neutron capture cross section of 107Ag has been measured to be 40 b (a bam is 10- 24 square centimeters); 109Ag has a thermal-neutron capture cross section of 82 b. When 107Ag captures a neutron, it is converted to I08Ag. Silver-108 has a half-life of 2.4 min, decaying principally to 108Cd by the emission of a beta particle. About 2 % of the 108Ag nuclei decay to I08Pd by the emission of a positron. Silver-lOO is converted to 110Ag by neutron capture. Silver-110 decays in 24 sec to 110Cd by the emission of a beta particle. These reactions are summarized below: (41) (42) In addition to the reactions described above, there is a small probability that isomeric states of 108Ag and 110Ag are formed. These are higher energy nuclear states that are relatively stable. They may also decay by beta emission, but the lifetimes are much longer so that they contribute very little to the observed activity. In the following discussion we shall ignore all but the re­ actions indicated in Eqs. (41) and (42).

Calculation of Activity

24

How long should the silver foil be activated in order to become reasonably active? Does this time depend on the strength of the neutron source? We can answer these questions from a solution of the equation describing the rate of production of radioactive nuclei. In what follows we consider a single nuclear species. Our results may later be combined to describe the production of several nuclear species. Let n be the number of source nuclei in the foiL

neutron activation

N P -5

(J the thermal·neutron capture cross section, and e/> the neutron flux in particles per unit area per second. Then if N is the number of radioactive

r

nuclei, the rate of production of these nuclei is given by

dN

dt

=

(43)

n(Je/>

But these nuclei decay to the ground state in a mean time r at a rate given by N

dN

(44)

dt

1

In order to describe both processes, we must add Eqs. (43) and (44) to obtain

t U.

dN dt

-

a

=

n(Je/> - ­

n

(45)

r

= 0 at t

The solution of Eq. (45), assuming that N

:e

N = 0 is given by

N = n(Je/>r(l - e- tl<)

is

(46)

Note that the number of radioactive nuclei produced does not increase indefinitely, but levels off at a value n(Je/>r. In a mean time r, we achieve lie 63.2% of this value. 1 Let us now ask the following question: Ifwe irradiate the foil for a time T and then remove it from the moderator, what is the expression for the activity A dNldt following the removal of the foil? At time T we have from Eq. (46)

is

to

ns

.at .re ,re

(47)

b.

With the sample removed from the moderator, the rate of change of N is given by Eq. (44), which may be solved to obtain

,a ~ta

fa 10 ms

(48)

The activity is given by

A

41)

=

dN / N

--

dt

=

~e-t/t

(49)

r

Substituting from Eq. (47), we have finally

A

42)

:rgy ion, the re-

ably :can te of ngle :tion foil,

n(Je/>O

e-T/~e-t/t

(50)

Note that from Eq. (50), as long as Tis much longer than r, the initial activity is independent of r. Finally, if we have two nuclear species corresponding to nl and nz, we may write the total activity as

hat

~

=

A = n 1(Jle/>O -

e-T/tl)e-t/tl

+ nz(Jze/>(l -

e-T/t')e- I/t ,

(51)

experiment 1

beta activity In this experiment the beta activity of the activated nuclei will be measured as a function of time following activation and as a function of activation 25

nuclear physics

time. A scaler may be used although a counting-rate meter is sufficient. The circuit shown in Fig. 3 with the cathode-ray oscilloscope replaced by a vacuum-tube voltmeter provides an adequate counting-rate meter. Insert the silver cylinder into one of the test holes in the moderator for 5 min. In this time, which is approximately two half-lives of 108 Ag, the activity of this nucleus will reach I (t? = i of its peak value. The activity of 110Ag will reach essentially its peak value. Remove the cylinder from the test hole and at the same time start a stop clock. Measure the counting rate every 5 sec until you are no longer able to detect any appreciable activity above background. About 10 min should be sufficient. Plot your date on semilog paper. Is a correction for dead time necessary? Explain. Determine the half-lives of the two radioactive nuclei. Find the ratio of their original activities. From Eq. (51) we may expect

Compare your observed result with this value. Can you suggest why your observed results might differ from this predicted result? You may wish to repeat the activation for longer and shorter times and note the way in which the two activities vary. Can you explain your results?

2

determination of neutron flux By making careful measurements, we should be able to compute the neutron flux from the observed activity. Alternatively, if we know the neutron flux, we should be able to use the observed activity to compute the capture cross section. Let us assume here that the capture cross section is known and we are attempting to measure the neutron flux. What value might we expect for the flux? Taking the initial production rate of neutrons to be 1.5 x 106 sec-I, what ultimately happens to these neutrons? They diffuse through the paraffin, ultimately escaping into the air. The flux of neutrons out the side of the moderator will be 4>/6. (We get a factor of 3 because it is only the normal velocity that counts and another factor of 2 because there are no inward coming neutrons.) If S is the surface area of the moderator, we must have finally S4> 6 Taking the moderator to be a cylinder 2 ft high and It ft in diameter, the surface area S is 105 cm 2 • The computed flux near the surface is then 90 neutrons cm 2 /sec. From the observed activity of one of the silver isotopes, we may write 4> =

~ na

(52)

Knowing A, n, and a we may thus determine 4>. In determining the value of n, we must count only those nuclei that are actually counted by the G-M tube. Remember that only those beta particles produced within a stopping dis­ tance of the inside of the silver cylinder will be counted. Also, one must correct for the solid angle subtended by the active region of the G-M tube. These estimates can be made only very approximately. You may wish to do this, however, as a way of determining neutron flux. 26

neutron activation

le

3

a or ty of

NP-5

other materials Other materials which may be conveniently used for neutron activation are indium and gold. In Table 2 we summarize the relevant information for these materials as well as for silver. Write the nuclear reactions for indium and for gold. What differences would you expect in the observed activities when using indium or gold?

)p

to

be

ne

TABLE 2

Isotope l~iAg 12~Ag 1~~Ag l~SAg ll~In 11~In ll~"In

l1r

lUln

Id s?

Abundance (%)

)n

Neutron cross section Metastable, b

Stable, b 40

2.4 min

1.77

48.18 24 sec

2.87

72 sec 50 days

1.98

54 min 14 sec

1.00 3.29

64.8 hr

0.96

4.28

95.72

lWn l~gAu

Beta energy, MeV

51.82

11~"'In l~~Au

Lifetime

100

2

82

61

2

150

50

99

x, ,ss ve te i? of 3 2 Ie

le ~O

questions 1

Plot Eq. (51) on semilog paper for some suitable choice of initial activities and lifetimes. Why is it desirable to use semilog paper for this plot? From the composite curve determine "1 and "2 separately. Also find the initial activities separately. Can this separation always be made?

2

Explain why the normal neutron flux at the surface of the can is one-sixth what the flux is a few centimeters inside the can.

3 For measurements of mean counting rate performed with a vacuum tube voltmeter, explain why the fractional bounce of the meter reading increases as the counting rate decreases. How might you estimate the actual counting rate from the amount of bounce?

4 Find other uses for neutron activation in addition to those mentioned in i2)

the experiment. What is the sensitivity of the method? How does it compare with spectroscopic analysis?

of )e.

is­

1St >e. do

27

In ~

R

7

II

~

• ~

,

I !I R

7

II

~

• ~

.,

1

iii

• -­

-

berkeley p hysic I borato r y. 2d edition

a lan m. por is, u niversity of c alifornia. berkeley hugh d . young, carneg/e .mellon u nlver.slty semiconductor diodes t unnel diodes and r elaxation osclll tors the transIstor

SE-J SE-2

E-

transistor amplifiers

SE-4

positive feedback and oscillation

E-5

negatIve feedback

mcgraw-hill b new york

st lou is

0

SE­

company

san franc isco

dUsseldo rf

rr M

M M M M M

rr:

M·

M· M· M·

M-

el· EI­

semiconductor electronics Copyright © 1971 by McGraw· Hill, Inc.

All rights reserved. Printed in the United States of America.

No part of this publication may be reproduced, stored in a

retrieval system, or transmitted, in any form or by any means,

electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher.

Library of Congress Catalog Card Number 79-125108

E'· EI·

E'· E'·. fie F·' F·:;:

F·3 F·4

07-050492-x

1234567890 BABA 79876543210

F·5

The first edition of the Berkeley Physics Laboratory

copyright 1963,1964,1965 by Education Development

Center was supported by a grant from the National Science

Foundation to EDC. This material is available to publishers

and authors on a royalty·free basis by applying to the

Education Development Center.

el.

This book was set in Times New Roman, printed on

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the designer was Elliot Epstein. The editors were Bradford

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supervised production.

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elf! EC EC EC

EC EC

semiconductor electronics

ts

work ental units nient ,tions table 'S are

INTRODUCTION

:s/mole

!/weber

•

Semiconductors are a class of materials whose electrical properties are inter­ mediate between good metallic conductors and good insulators. They are of enormous practical interest, forming the basis for a wide variety of devices used in electronic circuitry, including diodes, transistors, photocells, particle detectors, and integrated circuits. Semiconductor devices have completely revolutionized the whole field of electronics in the past 15 years. The physical basis for the properties of semiconductors is discussed in many textbooks, '" and only a brief review need be given here. The simplest semiconductors are the elements silicon and germanium. The electrical con­ ductivity of these materials is much less than that of most metals, but it increases very rapidly with temperature, unlike the metals, in which the conductivity nearly always decreases with increasing temperature. Further­ more, the presence of certain kinds of impurities in silicon or germanium, even in extremely small concentrations, can increase the conductivity enormously. The conductivity of any material depends on the existence of electrons which are more or less free to move within the materiaL In metals there are many mobile electrons, even at low temperatures. Germanium at low tem­ peratures has no free electrons, because of its crystal structure. Each atom has four valence electrons, and in the crystal lattice each atom has four nearest-neighbor atoms situated at the corners of a regular tetrahedron. Each valence electron participates in a covalent (shared-electron) bond with one of the nearest neighbors; thus all valence electrons are bound to indi­ vidual atoms and none are free to move. But only a small amount of energy is needed to break one of these bonds, 1.1 eV for silicon, and only 0.7 eV for germanium. This energy can be supplied by thermal motion; hence as the temperature increases, more and more bonds are broken and electrons become free to participate in conduction. The positively charged vacancies or holes can also move by successive replacement of adjacent electrons, so these also contribute to conductivity. Thus, we see why the conductivity increases rapidly with temperature. This type of conductivity is called intrinsic conductivity to distinguish it from impurity conductivity, to be dis­ cussed next. If some atoms of an element having five valence electrons, such as arsenic, are added, four are involved in the covalent bonds, but the fifth is very loosely bound (binding energy of the order of 0.01 eV), and even at low temperatures can break away and move freely through the lattice. Such an impurity is called a donor impurity or an n-type impurity, since the atoms donate negatively charged current carriers. Conductivity of an n-type semi­ conductor at ordinary temperatures is chiefly due to the electrons from n-type impurities. See, for example, Hugh D. Young, Fundamentals of Optics and Modern Physics, McGraw-Hill Book Company, New York, 1968, Chap. 11.

semiconductor electronics

FIGURE 1

( a)

(h)

+e

+e +e

+e

+e

+e

- e

-e

+

+

-e

-e

+

+

- e

- e

+

+

p

+e

+c

E

+

-e

Electron s and holes in the vicinity of a p-n junction will diffuse across the junction and recombine. (ti l

+ e_

_-e + + -

4----

+

+ p

( e)

- c_

+e

+

+ n

If we apply a n electric field in the for­ ward direction, we get a large recom­ bination current. (II

+

-e _

_+ e

+

+

-----~,, ------

_+e

\'

-e____..

+ p

.

+ n

Ell

If we apply the field in the reverse directio n, we get only a small current from th ose carriers thermally generated in the junction. 2

e

- - - - -..... E II

As a result, an electric field is e tab­ Iished, which prevents further reCom­ bination.

-

- e +

+

11

P

n

-e

+

+

+

+

E

+

+e

- e

+ +"

-I'

+

+

+

P

-e

+e

-c

+

n

We rep resent (above) a m atrix of im­ purity a toms, which are distributed th rough the semiconductor. (c I

+e

Th us, the 1- V graph of a p-n junction is strongly asymmetric.

ltyof is the

le for­

'ecom­

introduction

S£-1

Correspondingly, an impurity with only three valence electrons, such as gallium, can take an electron from an adjacent germanium atom to complete its four bonds. This leaves a hole at the adjacent atom, and this hole can migrate through the lattice, contributing to the conductivity. Such an impurity is called an acceptor impurity, and the resulting material is called a p-type semiconductor. Junctions It is remarkable in semiconductor technology that a material can be

produced which has different impurities in different regions, changing smoothly from n type at one_end to p type at the other. In this case we have a surplus of electrons at one end and of holes at the other. Each tends to diffuse across the boundary or junction into the opposite region, in which it becomes a minority carrier. However, as Fig. 1 shows, the electric fields resulting from this redistribution of charge limit the extent of this diffusion. Now suppose an electric field is imposed across the junction. If the direction of the field is fromp to n region, the resulting forces tend to drive the holes across the boundary into the n region, and the electrons in the opposite direction into the p region, resulting in a considerable current flow across the junction. But if the E field has the opposite direction, then both electrons and holes are pushed away from the junction, resulting in a negligible current flow. Another way to say the same thing is that an electric field from p to n region helps the diffusion process in pushing carriers across the junction, while the opposite direction of E opposes this diffusion. Thus a p-n junction is strongly directional in its electrical properties, acting as a rectifier. It is a good conductor from p to n, but poor from n to p. The properties of p-n junction diodes are studied in Experiment SE-l. By combining two junctions in a "back-to-back" arrangement (p-n-p or n-p-n), we obtain an even more interesting device called a transistor. The action of a transistor is based on the same principles as for the diode, but the third region opens new possibilities; transistors can be used as various sorts of amplifiers. The basic properties and behavior of transistors are studied in Experiment SE-3, and a few of their applications to useful electronic circuits are investigated in Experiments SE-4 through SE-6.

nction

3

experiment

S E-1

semiconductor diodes

introduction The simplest semiconductor diode is just a p-n junction, as discussed in the INTRODUCTION. It is possible to derive an analytical expression for the voltage-current relationship for the device. This derivation makes use of the fact that a given potential difference across the junction corresponds to a cer­ tain electron energy, and the number of electrons having this much energy is determined by the Maxwell-Boltzmann distribution. Thus, it should not be surprising that the expression contains the factor exp (e V/kT), where e is the electron charge, V the potential difference, k Boltzmann's constant, and T the absolute temperature. The result of this analysis, which will not be given in detail here, is (1)

where 10 is the maximum reverse current for large negative V, a constant characteristic of the particular diode involved. Thus, the characteristics of a diode are strongly temperature-dependent; this dependence can be studied. Figure 2 shows graphs ofEq. (1) at several temperatures. The terms forward bias and reverse bias are often used in describing semi­ conductor device behavior. Forward bias refers to a voltage in the direction FIGURE 2

I

----------------------------~---------------------v

5

semiconductor electronics

of easy current flow across a junction, from p to n region ; reverse bias refers to the opposite situation. We may use a reverse-biased j u nction diode as an electronically variable capacitor, termed a varactor diode . From F ig. I we observed that as we increase the reverse voltage on the diode, both electrons and holes move away fro m the junction , leaving the intervening region electrically charged. Let us consider the simple situation shown in Fig. 3. With a reverse-bias V across the diode, the electrons and holes have moved back by a distance d p

FIGURE 3 0

n

0

0

0

0 0

0 0 Ii

0 0

0 0

0

• • •• • • •• •• • • • +d

- p + p

- - ++ - - + + - - + + - - + + - ++

- d

()

+~ 2

z

A

on each side of the junction, exposing media of charge density plus and minus p. The charge distribution then resembles a capacitor, with an effective charge on each side given by

Q

(2)

pAd

=

and it remains to determine the relation between V and d. The situation shown in Fig. 3 is different from the usual parallel-plate capacitor, where the field between the plates is uniform. For the diode junction the electric field in­ creases linearly to a maximum value Q/eA at the center of the j unction (c is the dielectric constant) decreasing to zero at z = d. The two ki nds of field and potential configurations are compared in Figs. 4 and 5. T he potential across the junction is given by V

=-

f

Edz

=

Qd f.A

(3)

Eliminating d by Eq. (2), we obtain

V =

e~ (~y

(4)

In most applications of the reverse-biased diode, a small ac signal is impressed in series with the bias voltage. Under these circumstances we are interested in what is called the small-signal capacitance:

(5) Solving Eq . (4) for Q, we obtain Q = A(ep V)1/2

(6)

Differentiating Eq. (6) with respect to V and substituting into Eq. (5), we obtain p )1 /2 (e 2 V

= ~

c v

(7)

The significance of this result is that when the dc voltage V is varied, the capacitance C for small ac signals varies as V- 1 / 2 • H ence, we have a variable capacitor controlled by a voltage! 6

' rs

an

e

ns

on

V d

nus

FIGURE 4

FIG URE 5 (a)

(a )

4

~

I+:I-Q I·

0

A

In an ordinary ca pacitor, charges + Q and - Q lie on plates of a rea A sepa­ rated by a medium of dielectric con­ stant 8.

0

In the reverse-biased junction diode, the charges + Q and - Q a re dis trib­ uted uniformly.

~ge

(b)

(b)

(2)

E

wn eld

12­

111­

&

,; IS

eld tial

-d

+d

(3)

(4) I is are

In an ordinary capacitor, the electric field is unifonn in the region between the plates.

.In the junction diode. the electric field increases linearly to the center and then drops linearly.

( e)

(e)

(5)

(6)

----1------/------ r----z

---.-------F-------+---z

we -d

+t1

(7)

the .ble

In an ordinary capacitor, the potential increases linearly between th plates.

-d

For the junction diode , the vari ation in potential is parabolic. 7

semiconductor electronics

experiment 1 diode characteristics

To study the behavior of a semiconductor diode, the circuit shown in Fig. 6 is suggested. The diode is a general-purpose germanium junction diode. FIGURE 6

+H----------------Q Low-voltage

power supply

700

n

The arrow on the diode shows the direction of easy current flow, usually called the forward direction. A voltmeter is used to measure the voltage across the diode and across the resistor; the current is obtained from the latter. Plot I as a function of V. Then reverse the polarity of the diode to obtain the reverse half of the characteristic, for which you may want to use a compressed voltage scale on the graph. This part of the curve may be used to determine 10 in Eq. (I). This data may also be used to check the validity of Eq. (I). It is convenient to transform that equation as follows:

In (l

+ 10) =

eV kT

+ In 10

(8)

Thus, if we plot the quantity (l + 10 ) as a function of Von semilog paper, using the logarithmic scale for the current, the result should be a straight line. The intercept on the current axis is In 10 , and the slope is e/kT. From the graph, using the known values of e and T, determine Boltzmann's constant, and compare with the accepted value. 2 low temperature

You may wish to repeat the observations with the diode dipped in liquid nitrogen. Plot the data on the same sheet as the first set, and observe the differences. Again make a semilog plot; this time use your data to determine T. 3 oscilloscope display

A very convenient way to display diode characteristics, although it is less precise than the method used above, is to display the V-I curve on a scope. The circuit of Fig. 7 is suggested; an alternating voltage is applied to the diode, and the scope plots current as a function of voltage. Again dip the diode in liquid nitrogen or spray with Freon and observe the change in the charac­ teristic curve. 4 varactor diode

The behavior of the varactor diode is most easily observed by using the diode capacitance as part of a resonant LC circuit, and measuring the resonant 8

semiconductor diodes

SE-1

FIGURE 7

lOOkU

frequency. The circuit shown in Fig. 8 is suggested; the dc power supply in series with the sine-wave generator gives a constant biasing voltage V plus a small ac component. FIGUREB

Low-voltage power supply

25mH

Junction diode

--

CAUTION

Be sure the polarities are as shown in the figure. Since there is no current-limiting resistance in the circuit, if the diode is connected in the forward direction it will almost certainly be overloaded and burned out. The resonant frequency is determined by varying the frequency and finding a maximum signal on the scope. Determine the resonant frequency for several values of biasing voltage. For each, calculate the capacitance from the usual relation (f)

f

=

2n

2n(LC) 1/2

(9)

Remember that this capacitance is the junction capacitance and the scope input capacitance in parallel. How should you correct for this additional capacitance? 9

semiconductor electronics

Plot junction capacitance as a function of V. Alternatively, plot C as a function of V- 112 • What shape should this graph have if Eq. (7) is obeyed?

questions 1 When electrons diffuse into a p region in a diode, why do they not immediately recombine with holes in that region? 2 Is there any upper limit to the reverse negative voltage which may be applied to a p-n junction? What happens if this limit is exceeded? 3 What determines the maximum forward voltage which may be applied to a junction diode? The crucial factor is power dissipation. (Why?) How is this related to the maximum voltage? 4 What advantages do semiconductor diodes have compared to vacuum diodes? What disadvantages? 5 What happens if a varactor is connected backwards by mistake? If the voltage were limited to a nondestructive value, would it function as a varactor? Explain. 6 Semiconductor diodes always become less effective at high temperatures. Why? There are at least two reasons; one has to do with Eq. (1); the other has to do with thermal generation of holes and electrons corresponding to intrinsic conductivity. 7 What practical applications might a varactor diode have?

8 The circuit used to observe varactor action has some series resistance as well as inductance and capacitance. Why? What effect does this have on the determination of the junction capacitance?

70

experiment

S E-2

tunnel diodes and relaxation oscillators

a 11

s Iy

introduction In past experiments we have encountered various types of nonlinear devices. A nonlinear device, by definition, is one for which current is not proportional to voltage, in contrast to an ideal resistor which obeys Ohm's law for which current is a linear function of voltage. Figure 9 shows two different kinds of nonlinear behavior.

.a tis lIn

FIGURE 9

I

I

he

a

--------,r--------v

es. ter to (a) Tungsten filament

(b) Semiconductor diode

Even more remarkable are devices for which the V-I curve has regions of

rell the

negative slope, commonly called regions of negative resistance. It is customary

to distinguish two different kinds of negative resistance; typical examples are shown in Fig. 10. In this experiment we study one example ofeach type; type I is represented by the tunnel diode, type II by the glow-discharge tube. FIGURE 10

I

L---------------v (a) Type I

~--------~----v (b) Type II

Tunnel Diode A tunnel diode is similar to an ordinary p-n junction diode; the differences

are in thickness of the 'transition or junction region and in the impurity densities. The basis of tunnel diode operation is shown in Fig. II. As the figure shows, the negative-resistance region of the characteristic results from the "tunnel current" which is appreciable only at small forward voltages. Suppose we include a tunnel diode in a circuit with a voltage source and a series resistor, as in Fig. 12. The diode characteristic gives one relation 11

semiconductor electronics

FIGURE 11

(a)

(b)

+e

-e

+

I

+

+

-e

+

+

+

+e

----------~~----------v

-e

+

+

+

p

n

In an ordinary diode, the electron and hole densitie s are mode rate and the depletion region is broad. (c)

+e

+e

+ +e

+e

+e

+e

+ + +e

+e

+e

+e

+ + P

-e

-e

+

+

-e

-e

+

+

- e

-e

+

+

-e

-e

+

+

-e

-e

+

The forward current IS associated with recombination and the reverse current with the generation of carri­ ers. (d ) +e

+e

-e

+ +e

+

-e

-e

+

+

+

+

+

+

-e

-e

+e

+e +e

+e

+e

+e

--­

+

+

+

p

n

-e

+

-e

+

+

-e

-e

+

+ n

• Eo

In a tunnel diode, the electron and hole densities are very high and the depletion region is very narrow.

(f)

(e)

I

12

With a small applied potential, dec­ trons can leak through the junction a nd recombine with holes.

I

----~~------~~-------v

----~---------------------v

T his leakage (or tunneli ng) of carriers produces an addi tional current at low potentials.

The actual characteristic of the tun­ nel diode is the sum of a recombina­ tion and a tunnel current.

tunnel diodes and relaxation oscillators

FIGURE 12

-

SE-2

I R

between V and I for the diode, and another can be obtained by applying Kirchhoff's voltage (loop) law to the circuit. We obtain Vo = V

+ IR

(10)

Plotting I as a function of Von the same axes used for the diode characteristic, we obtain graphically a simultaneous solution of the two relations, as in Fig. 13, for any given values of Vo and R. Increasing Vo causes the line to

d e 1­

FIGURE 13

------~~------.--------------v

move parallel to itself, and changing R changes the slope without changing the horizontal intercept, as can be seen from the figure. Such a liJ,le is called a load line corresponding to the values Vo and R; the resistor R is called the load. Suppose we attempt to obtain the V-I characteristic of the device by gradually increasing Vo and measuring Vand Ifor each value of Vo. Figure 14 shows a series of load lines for R = 1000 and Vo ranging from 0 to 0.8 V. Note that as Vo is increased from 0 to 0.3 V, the load line intercepts the

:;­

In

FIGURE 14

Tunnel diode

n­

aV, Volts 13

semiconductor electronics

characteristic at a single point and the current gradually rises. In the vicinity of 0.4 V, the load line intercepts the characteristic at three points, so there are three solutions of the V-I relations. As Va approaches 0.5 V, the load line breaks away from the hump in the characteristic, and again we have only one solution. A plot of I as a function of Va is shown in Fig. 15. Note that there is a voltage range where two values of current are possible. The actual current depends FIGURE 15

V o, Volts

on whether Va is increasing or decreasing. If we now try to reconstruct the V-I characteristic from the data of Fig. 15, we find that we are not able to obtain the section of the characteristic between the two circles in Fig. 14. Neon-Glow Lamp A second type of negative-resistance device is the familiar neon-glow lamp. A typical V-I characteristic is shown in Fig. 16. As the voltage is increased from FIGURE 16

I

Extinction Ignition

zero, only an extremely small current flows until the voltage reaches the ignition potential, at which point it is sufficient to ionize enough gas atoms to set up a self-sustaining glow discharge. The presence ofthis ion density makes the tube a much better conductor, and the voltage very quickly drops to a smaller value which is very nearly independent ofcurrent. This independence makes such tubes very valuable, in modified form, as voltage regulators. When the voltage is decreased below this value, called the extinction potential, the ionization cannot be maintained, the gas deionizes, and current drops to 14

tunnel diodes and relaxation oscillators

inity here . line only

SE-2

its former very small level. To initiate the discharge again, the voltage must be raised back to the ignition potential.

Relaxation Oscillators

ltage ends

Both kinds of negative-resistance devices can be used to make an oscillator, or wave generator, of a particular kind called a relaxation oscillator. The simplest example of a relaxation oscillator is an ordinary door buzzer, shown in Fig. 17a. When the switch is closed, current in the circuit grows exponenR

FIGURE 17

~~!II

T

Contact

(a)

I

tthe Ie to

4.

.mp.

rom

the 1S to akes to a ence :ors. !tial, )s to

(b)

tially with a characteristic time LjR, as studied in Experiment EC-2. But when the current reaches a certain critical value, the magnetic field of the inductor pulls the contact apart, opening the circuit, and the current goes to zero and starts over, as shown in Fig. 17b. An analogous example of a relaxation oscillator is the circuit of Fig. 18, incorporating an inductor and a tunnel diode. The mode of operation is as follows. FIGURE 18

When we connect the battery, the current through the tunnel diode begins to build up. When it exceeds the peak current Ip shown in Fig. 19, the voltage across the diode jumps to V2 • The circuit components are chosen so that, with the increased drop across the diode, the current is forced to drop along the characteristic. The current continues to drop until it reaches the valley current lv, below which the voltage drops discontinuously to Vt. Now the current begins to rise again and repeats the cycle. The time dependence of the current and voltage is shown in Figs. 20 and 21. 15

semiconductor electronics

FIGURE 19

I

Iv V

FIGURE 20

V V2

Vv Vp

t

FIGURE 21

I

L-----------'r-----------'~------t

FIGURE 22

R

c

NE-2

A relaxation oscillator may also be made with a neon-glow lamp, as shown in Fig. 22. The theory of operation of this device is much simpler than that ofthe tunnel diode, because ofthe nearly ideal characteristic of the glow lamp. When the potential Vo is applied to the circuit, the capacitor C begins to charge through the resistor R to the potential Vo. But as the potential reaches the firing potential Vr, the lamp fires and discharges the capacitor almost instantaneously. Finally the voltage across the capacitor reaches the extinc­ tion potential V., the lamp deionizes, and the capacitor recharges through R to repeat the cycle as shown previously in Fig. 16. The time dependence of the voltage is shown in Fig. 23. The period Tl during which the capacitor C charges through R is usually very much longer than the period during which the capacitor discharges through the glow lamp. This is because the series resistor R is ordinarily much larger than the resist­ ance of the lamp during the discharge. 16

tunnel diodes and relaxation oscillators

FIGURE 23

We can obtain a simple analytical expression for Tb and thus for the frequency of oscillation. From Fig. 23, we have the relations

v.

=

Vo(l - e- to / RC )

Vr = Vo [1 -

(11)

e-(/o+TtlIRC]

(12)

Solving Egs. (11) and (12) for T 1 , we obtain T 1 =RCln

V - V 0 • Vo Vc

(13)

experiment 1 tunnel diode

A low-current tunnel diode is used in this experiment. The circuit shown in Fig. 24 is suggested. The diode current is determined from the voltage across the 10-Q resistor. What do the other resistors do? An oscilloscope display of FIGURE 24

109 Low-voltage power supply

own that IlIlp. is to .ches nost tinc­

2.7g

the characteristic is also possible. The circuit shown in Fig. 25 can be used for this purpose. The power supply and sine-wave generator together provide a sinusoidal voltage whose average is different from zero, and the resistors serve the same as in the previous circuit. You may wish to compare the scope trace with the plot of your direct measurements of V and I.

ghR

d Tl ,nger unp. esist­

SE-2

2

neon-glow lamp To obtain the V-I characteristic of the NE-2 neon bulb, the circuit of Fig. 26 may be used. The power supply and battery in series provide a voltage variable from about 45 to 80 V. You will not be able to obtain the full characteristic of this device. Can you explain why? How large a series resistor would be needed to obtain the full characteristic of the NE-2? What type of 17

semiconductor electronics

FIGURE 25

1O\V

Low-voltage power supply

4.5 V

FIGURE 26

+.H------I- 1I1I r--+---,

Low-voltage power supply NE-2

voltmeter would be needed? Identify the negative-resistance region of this device. 3 diode relaxation oscillator

To make a relaxation oscillator using a IN3720 tunnel diode, assemble the circuit of Fig. 18, using the following component values: Vo

= 3V

L = 241lH Observe the diode voltage on the scope; calibrate the vertical axis to measure the various voltages, and compare your values with predictions obtained from the diode characteristic. It is also interesting to look at the inductor voltage on the scope. 4 neon-lamp oscillator The neon-lamp relaxation oscillator, Fig. 22, will operate with a wide range of values of Rand C. There is however a minimum value of R. Can you find it? For each pair of values of Rand C measure the frequency by using Lissa­ jous patterns with a sine-wave generator. Using the ignition and extinction voltages determined previously, with Eq. (13) compute a value for the frequency, and compare with the measured value. Can you think of reasons for any discrepancies that occur?

questions 1 How can the full characteristic of the tunnel diode (including the negative­

resistance portion) be obtained? 18

tunnel diodes and relaxation oscillators

SE-2

2

Why can you not obtain the full V-I characteristic of the neon bulb? How large a series resistor would be needed to obtain the full characteristic? What minimum input impedance would the voltmeter have to have?

3

What limitations are there for the frequencies at which each of the two relaxa­ tion oscillators will function?

4

Explain the functions of the various resistors in Fig. 24.

5

What is the minimum value of R for the relaxation oscillator using a NE-2 bulb?

6

In a gas tube called a thyratron, a third element is introduced which permits varying the ignition voltage of the tube. What effect would this have if the tube were used in place of the neon bulb in a relaxation oscillator? What might such a device be used for?

7

Do you think relaxation oscillators have anything to do with the horizontal sweep generator in an oscilloscope?

is

Ie

'e d

;e

d

l­

n e

.S

19

experiment

SE-3

the transistor

Introduction In this experiment we undertake a study of the basic properties of a junction transistor and some of its applications in simple amplifier circuits. In later experiments in this series, we study circuit applications in more detail. The junction transistor may be thought of as a pair ofjunction diodes with a common n or p region. Ifthe n region is common, we designate the transistor as pnp. Ifthe p region is common, we designate the transistor as npn. We have already discussed in Experiment SE-I the characteristics of a p-n junction. The electrical properties of a p-n junction are reviewed in Fig. 27. In the forward direction holes cross the junction into the n-type region where they recombine with electrons. Electrons similarly cross the junction into the p-type region to recombine with holes. When the potential is reversed, majority carriers move away from the junction on their respective sides. A small current in the back direction remains from those carriers of both signs which are generated within the region of the junction. The behavior of a transistor is described in Fig. 28. With a forward bias across the p-n junction on the left, as shown in Fig. 28c, holes are injected into the central or base region. Most of these holes diffuse across the base to the collector junction. A small fraction of the holes recombine with electrons within the base, resulting in a current flow out of the base. The collector junction is biased in the reverse direction as shown in Fig. 28e, resulting in a very weak injection of electrons into the base. The usual circuit nomenclature for a transistor is shown in Fig. 29. There are three currents, conventionally defined as flowing into the device, and three voltages, defined as shown. From Kirchhoff's laws, we have the relations

IE

+

Ic

VBE

+

+ IB V CB

0 = VCE

Thus, the electrical state of the device is completely described by any two voltages and any two currents. The arrow on the emitter lead suggests the injection of carriers into the base by the emitter. For a pnp transistor the direction is as shown; for an npn transistor the direction of the arrow is reversed. Transistor behavior can be represented in several ways. One useful way is to plot base current as a function of base-to-emitter voltage, for various constant values of collector-to-emitter voltage, as in Fig. 30. Note that for VCE 0, the curve resembles that for an ordinary pn junction diode. In this particular case the minority carriers injected into the base have no incentive to continue to the collector, and most are drawn off at the base, which then acts with the emitter as a simple diode. As VCE is increased, more and more 21

semiconductor electronics

FIGURE 27 (a)

(b )

~HOle

•

'::: Electron

Drift

Drift

•

• E

------~... E

Semiconductors have two kinds of carriers: electrons, which have a nega­ tive charge . . .

and holes, which represent a deficit of electrons and act as if they have a positive charge. (d

(c)

•• • • • •• • • •\

0

0

0

• • ••• • • •• • • • • •

o

0

0

0 00

0

p

n

p

n

-

0

0

0

0

. 0

0

0

0

0 0

-

00 0

0

0

\- Junction

- :11: + A semiconductor junction separates an n-type region, in which majority carriers are electrons, and a p-type region, in which majority carriers are holes.

({)

( e)

p

II

.--­

If we apply a positive potential to the p-type region, we draw a heavy 'for­ ward" current across the junction. Electrons flow to right and holes to left.

.•• •••••

o

0

0

0 0

0 0

~

0

v

+ 111; ­ If we apply a negative potential to the p-type region, we draw only a very light "reverse" current. 22

Thus, a p-n junction is a current recti­ fier.

SE-3

the transistor

FIGURE 28 (b)

(a)

p

o o

0

0-­

••

0 0

0

p

rl

0

••• ••

0

o o

p

n

•• •• • • •

0 0 0

1--0 0

0

n

0 0

0

o

0 0

•• • • •• •

1 A transistor is fo rmed from a pair of

or the p -type region in the center.

p-n junctions with either the n-type

region in the center ... (d )

(e)

Emitter 00

Base

••

Collector

Emitter

o

0

0

0 ••

0

0

0

0

0

p

·0 0

•

• ••

0

0

0

0 0

0

0

0

0

0

0

Base •• 0 •• 0

• ••

•• 0

Collector 00

0 0

0 0

0

p

n

If we bias one junction in the forward direction, we inject minority carriers into the base region.

(e)

These carriers diffuse to the second junction, where they are collected. Ideally all the injected carriers are col­ lected and there is no base current. (f)

Emitter 0

0 0

0

0

0 0

P

Base

• ••• • •• 11

Collector

0

Emitter

0

0

0

0

0 0

0

0

0

0

0

0 0

Base

•• • •• • •

Collector 0

0 0

0 0

0 0

P

P R

If we place a resistor in the collector circuit, we obtain a voltage propor­ tional to the emitter current. The col­ lector is biased in the reverse direction to prevent injection across the collector junction.

R

In the usual transistor amplifier we introduce a signal at the base and take the amplified signal from the collector.

23

semiconductor electronics

FIGURE 29

Emitter FIGURE 30

-70

-60 -50

«

::t

.....~

-40 -30 -20 -10

-0.1

-0.2 VBE,V

minority carriers are drawn to the collector, resulting in a decreased base current and an increased collector current (not shown). Since the base current is very nearly an exponential function of the voltage VBE , a reduction in base current by a constant factor simply shifts the curve to the right. These curves are called the input characteristics of the transistor. Another set of characteristics, called the output characteristics, are shown in Fig. 31. The particular choice of variables for these graphs may seem arbitrary, but its usefulness will become clear when we study the behavior of transistor amplifiers. This set of curves is most closely analogous to the plate characteristics of a vacuum triode or pentode. It should be noted that ordinarily the base current is much smaller than the emitter current, the order of 2 percent, unless the collector-to-emitter voltage is nearly zero. Thus, in ordinary operation nearly all the minority carriers injected by the emitter reach the collector; the remaining ones recombine with majority carriers in the base and contribute to lB' The simplest example of a transistor amplifier is the circuit shown in Fig. 32. The input and output currents are very nearly equal, but the voltage and resistance on the output side may be much larger than on the input side; hence this circuit functions as a voltage and power amplifier, with a relatively low input impedance. The input and output are actually in series, as in a 24

the transistor

FIGURE 31

-3

FIGURE 32

Input

Output

grounded-grid vacuum-tube circuit, and the current is controlled by the small voltage input.

experiment 1

:em rof late that :der ., in

tter

SE-3

transistor characteristics To measure the output characteristics of a transistor, the circuit shown in Fig. 33 is suggested. The circuit is arranged so that all voltages, including those used to determine currents, can be measured with a single vacuum-tube voltmeter with its ground side connected to the emitter, which is the common terminal between "input" and "output" in this circuit. The base current is

FIGURE 33

's in

Fig. and ide; vely in a

Oto3 V

25

semiconductor electronics

adjusted by means of the 500-kQ variable resistor, and the collector-to­ emitter voltage is varied by changing the tap on the 7!-V battery. Battery polarities shown are for a pnp transistor; for an npn transistor they must be reversed. Any general-purpose germanium transistor may be used; a few representa­ tive types are pnp

npn

2N1303 2N1305

2N1302 2N1304

Using the circuit of Fig. 33, determine the input and output characteristics of a pnp transistor. Note that it is possible to take data for both of these sets of curves at the same time. You may wish to compare the results with those of a comparable npn type. 2

oscilloscope display

Transistor characteristics can be displayed conveniently on a scope. For example, to observe input characteristics, we hold VCE constant and vary VBE cyclically, and plot VBE on the horizontal axis, with a voltage proportional to IB on the vertical axis. Changing VCE' we obtain another curve in the family, and so on. One circuit for accomplishing all this is shown in Fig. 34. Similarly, the output characteristics can be observed using the circuit of Fig. 35. Note that in Figs. 34 and 35 the sine-wave generator is "floating" above ground. To accomplish this, it will be necessary to disconnect the instrument case from the ground.

FIGURE 34

IV p-p

4~ V

+

26

the transistor

0­

SE-3

FIGURE 35

ry

be lOY p-p

a-

lOon ICs

!ts ,se

or .ry tal he 14. of g" he

questions 1

In Fig. 29, do the positive directions of VBE and VCB correspond to the "easy" direction of current flow across the junctions, or otherwise?

2

In measuring the output characteristics of a transistor, could an ordinary 20,OOO-Q;V multimeter be used to measure the voltages across the resistors in Fig. 33? Explain.

3

What is the "input impedance" for the circuit of Fig. 33? That is, if VBE is increased slightly, with a corresponding increase in IB' what is the ratio ofthese increments?

4

What factors limit the maximum voltages or currents that a transistor can handle without damage?

5

In a pnp transistor, what determines the conductivity of the material in each section? Can you think of any advantages of having different conductivities in the two p sections? How might this difference be achieved?

6

Suppose that in the circuit of Fig. 33 the 3-V battery is replaced by a sine­ wave generator. Would the voltage across the I-ill "output resistor" also be sinusoidal?

7

In the "common-base" characteristics for a transistor, one plots collector current as a function of collector-to-base voltage, for various fixed values of emitter current. Sketch your prediction for the appearance of these curves.

27

experiment

S E-4

transistor atnplifiers

introduction An amplifier is any device which increases the voltage, current, or power of a signal. A familiar example is a record-playing system, in which the pickup transducer typically feeds a signal of 0.001 V into a load of 50 kn, for an input power of the order of 10 -11 W. This signal is amplified to a few volts with a comparable load, by means of a voltage amplifier, and is then sent to a power amplifier with a final output into a low-impedance load (typically 8 n) of 10 to 100 W, fora total power gain ofafactor ofthe order of 10 12 , or 120 dB. One way to describe the function of an amplifier is to specify the ratio of output to input, for current, voltage, or power. Ordinarily amplifiers are designed to be either voltage or current amplifiers, but may be both; and nearly all amplifiers increase power. Thus one refers to the voltage gain, current gain, or power gain ofan amplifier. This is often expressed in decibels, defined as in Experiment EI-2 as 10 times the log (base 10) of the output­ input ratio. Other important characteristics of an amplifier are the input and output impedance, defined the same way as for the instruments studied in the Electronic Instrumentation (EI) series of experiments, and the frequency response, which is the dependence of the gain on frequency. Nearly all amplifiers have a response which falls off at sufficiently low and high fre­ quencies; some are designed for a uniform response over a broad band of frequencies, others for a sharp maximum in gain at a particular frequency. The latter are usually called tuned amplifiers. The most common electrical amplifiers make use of thermionic vacuum tubes or transistors. Until 20 years ago the vacuum tube was used universally, but transistors are now replacing tubes in many applications because of their small size, long life, mechanical ruggedness, and greater efficiency because no power is required to heat a cathode as in the vacuum tube. Tubes still have many important applications, however. In this experiment we study several kinds of simple circuits using transistors. Of course, an amplifier cannot manufacture power out of thin air. The initial source of the output power is some sort of power supply, which is often a battery or a rectifier arrangement which converts 110-V ac line power into dc. The basic function of the amplifier is to convert this dc power into output power which is controlled by the input. Often the objective of an amplifier design is for a varying input signal to give an output signal which is proportional at each instant to the input. Such an amplifier is called a linear amplifier. We might consider a circuit similar to that of Fig. 33 (Experiment SE-3) with the 3-V "input" voltage replaced by a variable voltage, such as the sinusoidal voltage source shown in Fig. 36. The difficulty is that output is proportional to input only when the input voltage is negative; only then can the emitter inject carriers into the base. 29

semiconductor electronics

FIGURE 36

Oto7 1 V - ­ 2 ­

+

lOOkn

lkQ

When the input is positive, the emitter-base junction is reverse-biased, and no injection can take place. The way out of this dilemma is to add a constant "bias" voltage to the input, large enough so that the sum of it and the varying voltage always has the correct polarity. Thus even in the absence of an input signal there is a certain input (base) current, with a corresponding output (collector) current. These values are called the operating point or quiescent currents. The addition­ al currents associated with an ac input signal are then thought ofas temporary excursions to one side or the other of this "operating point" condition. Clearly, the maximum input signal that can be amplified without distortion is limited by the input biasing current. Often in circuit diagrams the biasing voltages are omitted, especially when only the ac components of signals are being analyzed, but they are essential for the operation of the amplifier. There are three basic ways to use a transistor as an amplifier. Figure 37 shows the common-emitter configuration; in this configuration, the "ground" sides FIGURE 37

ofinput and output are both connected to the emitter; there is no ac compon­ ent of potential difference between them. The input signal is applied between base and emitter, and the output signal is taken from the collector. This circuit gives a current gain of the order of 50; the load resistor R may also be considerably larger than the input resistance, resulting in a power gain of several thousand. Figure 38 shows a common-base circuit. Here the input and output currents 30

transistor amplifiers

SE-4

FIGURE 38

Output

are very nearly equal, but the output resistance may be several times the input resistance. This circuit has a much lower input impedance than the common­ emitter circuit, and is ordinarily useful only when working into a very high impedance load. Finally, the common-collector circuit shown in Fig. 39 has a voltage gain of unity, but the output resistance may be substantially lower than the input resistance, resulting in an appreciable power gain.

and

the

has

is a :ent. .ion­ rary lion. :tion lsing s are mer.

R

FIGURE 39

Output

R

lOWS

-

sides

lpon­ ween This so be .in of

'rents

In Figs. 37 through 39, the bias sources are represented by individual batteries; in practice, the input bias is often derived from the output voltage supply, as we shall see in the experimental procedure section. All transistor amplifiers have a high-frequency cutoff. Often the principal reason is the junction capacitance in the input side of the transistor. At low frequencies the input terminals behave as a pure resistance (the input resistance), but at higher frequencies they behave as a resistance and a capacitance in parallel. The effective input capacitance can be measured by finding the frequency at which the gain drops to 1/J2 of its low-frequency value; at this frequency the impedance l/wC of the capacitor is equal in magnitude to that of the resistor, which is measured at low frequencies. Graphical Analysis

The transistor characteristic curves obtained in Experiment SE-3 can be used to make quantitative calculations concerning the input and output im­ pedances and the gain of a transistor amplifier circuit. As an example, we consider the circuit of Fig. 40. In this figure the bias voltages are omitted, and the following analysis refers entirely to the ac components of voltages and currents. Note: These are denoted by small letters to distinguish them from dc or operating-point values, usually denoted by capital letters. The input source is represented as a voltage generator Vs in series with a resistance R. to represent the internal resistance of the source, which may, in fact, be the 31

semiconductor electronics

FIGURE 40

Output Input

output impedance of another amplifier or of a transducer of some sort. The load is represented as a resistor R L . We first note that Kirchhoff s voltage laws give us two relations among the various voltages and currents. For the input loop, we have (14)

and for the output loop (15)

With respect to the inpu(side, the input characteristics shown in Fig. 30 and repeated in Fig. 41, give an additional relation between Vbe and i b , depending FIGURE 41

on the value of Vee" Recalling that Vbe and ib represent departures from the operating-point values VBE and IB' we plot the operating point on Fig. 41, and then represent Eq. (14) graphically as a straight line (a load line) with slope R", not passing through the operating point, but displaced to the left by a distance equal to VS' Thus for a given value of vs • the actual values of Vbe and 32

transistor amplifiers

SE-4

ib are given by the intersection of the load line with the characteristic curve for the appropriate value of Vee' The problem is that we do not know the value of Vee, since it presum ably varies with input voltage. Nevertheless, the above analysis gives a relation ­ ship between ib and Vee; this relation is plotted as a solid line in F ig. 43. W e may now find another relation between these two variables by analyzing the output side of the circuit. Figure 42 shows the output characteristics, on

FIGURE 42

-1

eE'V

Ig which we have plotted the line representing Eq. (15). The intersections of this output load line with the output characteristics give a second relation between ib and vee; this relation is plotted in Fig. 43 as a broken line. Finally, the FIGURE 43

- 20 .}

-10

+10

+20

he

.1, ith by

nd

intersection of these two curves gives the values of ib and Vee corresponding to a given value of VS' This graphical analysis is somewhat cumbersome in practice, and it is useful to make an approximate analytical representation of the information 33

semiconductor electronics

obtained from Figs. 41 and 42. We proceed as follows: In Fig. 41 we note that Vbe increases with increasing I b , but decreases with increasing (i.e., less negative) Vee' Thus for small increments from the operating-point values it is reasonable to represent the ac component Vbe as a linear function of Vee and ib as follows: (16)

In this equation ri and 11 are parameters chosen to fit the equation to the empirical curves, represented in tbis approximation as uniformly spaced straight lines. The quantity r, is called the short-circuit input impedance, and 1/11 is the open-circuit reverse-voltage transfer ratio. Similarly, the small signal components of the voltages and currents in Fig. 42 are represented approximately by the relation (17) In this equation f3 is the short-circuit forward-current transfer ratio (or amplification), and Yo is called the open-circuit output admittance. Taken literally, Eqs. (16) and (17) suggest that the small-signal behavior of a transistor in the grounded-emitter circuit can be represented by an equivalent circuit as shown in Fig. 44, consisting of'resistors, a voltage source, and a

FIGURE 44

c

B

.L

Vbe .

Yo

E

current source, the latter two controlled by tenninal voltages or currents. Having established this equivalence, we may now use this equivalent circuit, or Eqs. (16) and (17), which contain the same information, to compute all the necessary information about the amplifier, such as voltage, current, or power gain, input and output impedances. Typical values of the circuit parameters used for the equivalent circuit are

rl = 2700

n ~=

3

X

10 - 4

Yo = 1.5

X

10 -

5

mho

and

f3

=

50

J1

As an example, we may compute the voltage gain Av = Vce/ Vbe for this circuit; we need to solve Eqs. (15), (16), and (17) simultaneously to eliminate ib and ic and obtain the ratio of the two voltages. This is a straightforward 34

transistor amplifiers

te 'ss

algebraic procedure, the details of which need not be repeated. The result is

it

(

ld 6)

ne ed

ld

(yoRL + l)ri _ ~)-l fJRL If.

(18)

Often in calculations of this sort, approximations can be made which are based on the relative magnitudes of the parameters involved. For example, the load resistance RL is often the order of 1 to 5 kil, in which case the product YORL is much smaller than unity and may be neglected in Eq. (18). In addition, ordinarily If. is much larger than fJ, so if rj and RL are comparable in magnitude, then fJRL may be neglected in comparison with Wj' Thus, when these approximations are valid the voltage gain becomes simply

illl ed 7)

SE-4

Av

fJ RL rj

(19)

In general, the power gain t; is the ratio of input to output power, and is given by

or (20)

of mt la

When the approximations above are valid, using Eq. (19), the power gain becomes (21)

There are cases, of course, in which these approximations are not valid. The calculations then become more complicated, but the same general methods as well as the small-signal equivalent circuits may still be used. For sufficiently large signals, Eqs. (16) and (17) also become inadequate, and one must resort to graphical analysis. In this case, since output voltage and current are no longer proportional to input, the amplifier is no longer linear, and a sinu­ soidal signal suffers harmonic distortion. For other types of circuits, such as the common-base and common­ collector circuits discussed earlier, other kinds of equivalent circuits are sometimes more convenient. The terminology used for the parameters of the equivalent circuit is not universal, and in some handbooks the symbolism must be carefully translated.

ts. it,

le

er :rs

50

lis te rd

experiment 1 common-emitter circuit

A typical common-emitter circuit is shown in Fig. 45. For convenience we use the same battery to establish the operating point of both the input and the output circuits. Adjust the 500-kil variable resistor for a collector current of about 1 rnA. Measure the voltage at the collector with respect to emitter VCE and the voltage at the base with respect to emitter VBE • Determine the base current from a measurement of the voltages in the input circuit. Note the values of VCE, VBE, Ic, and IB , which establish the operating point of the transistor. You may use the circuit shown in Fig. 46 to inject a signal current into the base. 35

semiconductor electronics

500kO

FIGURE 45

L5kO to 1.5 V

+

5OOkO

FIGURE 46

L5kO

2 input resistance

Measure the input signal vbe at the base of the transistor for a 400-Hz input signal. Compute the apparent input resistance rj of the base. Measure the signal voltage Vee at the collector, and then compute the current gain fJ = (ie/ib)'

3 current gain

Now measure the current gain fJ as a function of frequency. At what fre­ quency does the gain drop to 1/J2 of its low-frequency value? Using one of the techniques described in Experiment EI-4, measure the frequency dependence of the phase shift in fJ. Interpret these observations in terms ofan effective input capacitance as discussed in the Introduction to this experi­ ment, and determine the value of this capacitance.

4 input and output impedances

Using whatever additional 0 bservations are needed, determine the input and output impedances of this grounded-emitter circuit. How are they related to the parameters appearing in the equivalent circuit of Fig. 44 ? 36

transistor amplifiers

SE-4

questions , In the grounded-emitter circuit studied in this experiment, suppose the input voltage is a sinusoidal signal with rms value I V. What bias voltage is required if this signal is to be amplified without distortion? Suppose the bias voltage is only half this value; sketch the output signal which results. 2 What is the relative phase ofinput and output, for sinusoidal signals, for each

of the three types of amplifier circuits discussed? 3 Compute the input and output impedances for the grounded-emitter amplifier circuit. 4 Is the power gain of an amplifier equal to the product of the voltage gain and

the current gain? Never? Always? Sometimes? Under what conditions? 5 Which of the three amplifier types has the largest input impedance? Which

has the smallest? 6 Which of the three amplifier types has the largest output impedance? Which

has the smallest? 7 What determines the maximum power that can be handled by a transistor

amplifier without a damaging overload ?

.put the ~=

fre­ eof ncy fan eri­

ilnd

ito

37

experiment

S E-5

positive feedback and oscillation

Introduction The term feedback refers to any situation in which part of the output of a system, such as an amplifier, is fed back into the input. Feedback can be either a nuisance or a useful phenomenon. In a public·address system, if too much sound from the loudspeakers finds its way back to the microphone, the result is a most unpleasant howl or squeal. Feedback in an amplifier circuit, on the other hand, can change the properties of the amplifier in very useful ways. In this experiment we consider the use of feedback in an oscillator circuit which produces a sinusoidal voltage. In Experiment SE-6 we study feedback in a more general way, and examine some of its possible effects on amplifier performance, such as frequency response and gain. To introduce the basic idea of an oscillator, we return to the circuit shown in Fig. 47, which was studied in detail in Experiment EC-3. We discovered FIGURE 47

c

L

R

there that during each half·cycle of the square wave, the circuit undergoes a damped oscillation with frequency W

_1 _ ~)1/2 ( LC 4L2

(22)

and decay of amplitude characterized by the time constant 2L R

(23)

If this decay constant is very long relative to the period 2rr/w ofone cycle, then the frequency is given approximately by w = (LC)-1/2. If R = 0, the frequency is exactly (LC) - 1/2 and the decay constant is infinite, so the oscillations continue indefinitely without delay. In reality, of course, there is always resistance in the circuit, in the inductor windings particularly, and so the case R = is not physically realizable. Suppose now that some way can be found to replace the energy dissipated in the resistor R (which represents the total circuit resistance). If this is

°

39

semiconductor electronics

possible, then the oscillations can continue indefinitely. The trick is to feed in the energy synchronously with the oscillations. If we draw a small current from the circuit, amplify it, and return it to the circuit, the effect is to inject energy into the circuit in a manner which is controlled by the instantaneous current and which is therefore automatical1 y synchronized with it. A transistor amplifier may be used to do just this, as we shall see. Before we proceed with the details of the circuitry, we return briefly to the equivalent circuit for a transistor, discussed in Experiment SE-4 and re­ produced here as Fig. 48a. This circuit can often be simplified further by FIGURE 48

c

B

1... Yo

E Equivalent circuit ( a)

c

B

E Approximate eq uivalent circuit

(b)

making reasonable approximations. First, for many common transistors the resistor llyo has a value the order of 100 kn. Thus, jf the circuit to which the emitter and collector are connected has a much smaller resistance than this, which is often the case, then the current through llyo is negligible and it may be omitted. Second, since l /J-t is typically very small, the order of 10-4, it is usually a good approximation to neglect the voltage veel J-t in comparison with the ordinarily much larger voltage ib'j across the resistor rio With these 40

positive feedback and oscillation

SE-5

seemingly drastic, and yet quite reasonable, approximations, the equivalent circuit reduces to the very simple arrangement of Fig. 48b. As with the original equivalent circuit, this circuit represents the behavior with respect to the ac components of the voltages and currents; the operating point or de components are not included. In this approximation, then, the transistor is represented by a resistor ri and a current generator controlled by an input current i b , with a current amplification of p. To use this device in conjunction with the LC resonant circuit, the input and output must be connected to different parts of the circuit, otherwise input and output would be identical, and there could be no amplification. One arrangement commonly used is to split the capacitance C into two capacitors of value 2C in series; the effective capacitance is then still C. The input is taken across one of the capacitors, and the output is fed to the other, as in Fig. 49. T4e resistor R f controls the amount of current fed back into the resonant circuit. FIGURE 49

t

2C

The operation can be understood qualitatively as follows. During the part of the oscillation when the top capacitor has a positive charge on its top plate, current ib flows into the base through rio The current Pib generated by the current generator flows through Rf to the left and through the bottom capacitor, increasing the positive charge on its top plate. Since this capacitor, like the top one, is already charged with this polarity, this current adds to the charge and hence adds energy to the resonant circuit. During the opposite half of the cycle, all the polarities are reversed, and the same thing happens. Ofcourse, it is possible to analyze this circuit in detail to obtain quantitative expressions for the various currents and charges, and thus determine the new frequency and relaxation time. This analysis will not be given in detail, but a few results are of particular interest. If the damping is slight, the frequency is still given by co = (LCr 1/2. The decay constant "C, formerly given by "Co = 2L/R, is now dependent on the amplifier parameters:

R

~ = 2L - 8[r;

P-I + (P +

I)RrJC

(24)

We note that if the current amplification factor Pis only unity, or if either or Rc is excessively large, the second term on the right side becomes very small, and Eq. (24) reduces to Eq. (23). In fact, however, Pis ordinarily much larger than unity, so that Eq. (23) may be written approximately as ri

R

P

(25) 47

semiconductor electronics

It may happen that rj and Rf have the same order of magnitude, in which case is negligible compared to PRf' and we obtain the further simplified ex­ pression rj

R T

2L

(26)

8Rf C

The essential point is that the amplifier has the effect of decreasing liT and thus increasing T. This is, of course, expected, since the original reason for including the amplifier was to compensate for the resistive power losses and thus sustain the oscillations for a longer time. As Eq. (26) shows, there is a critical value of R f for which l/T = 0, and there is no relaxation at all. In this case the oscillations continue indefinitely at their original amplitude with the circuit losses just matched by the amplifier contributions. The critical value of Rr is given by L =~ 4RC 8C

(27)

What happens when Rr is further reduced? Taking Eq. (25) literally, we should expect a negative relaxation time, corresponding to growing oscilla­ tions. This, in fact, is what happens, up to a point, but when sufficiently large amplitudes are reached, the effective value of P decreases so as to limit the final amplitude of oscillations.

experiment 1 relaxation time

Assemble the circuit shown in Fig. 50, with L = 25 mR and 2C 0.05 j.1F. The feedback resistor Rr may be a 5-kn potentiometer. Note that an npn transistor is used, so that the square-wave generator performs the dual FIGURE 50

L

v 2C

2C

42

1.5kQ

positive feedback and oscillation

SE-5

function of exciting the LRC circuit oscillations and of providing a forward bias for the emitter junction. The collector bias is supplied by the battery. First assemble the circuit with Rc removed and observe the decay of oscillations. Set the amplitude of the square-wave generator so that the peak voltage at the emitter is about 2 V. One way to accomplish this is to set the amplitude so that the average voltage at the emitter, as measured with a de voltmeter, is about I V. Measure the relaxation time To using the rising edge of the square wave. (Why not the other edge?) Using Eq. (23), compute the effective circuit resistance R. Disconnect the inductor and measure its resistance; can you explain why it is not equal to the R just obtained? 2. ct'i't\ca\ ieedbac\{

Reconnect the inductance, and connect R f • Gradually decrease it, observing the decay of oscillations. Find the value of Rc for which the relaxation rate is zero. Remove Rc without changing its setting, and measure its resistance with an ohmmeter. Using reasonable values for ri and p, compare this result with that predicted by Eq. (27). 3 oscillation

To construct a continuously operating oscillator, assemble the circuit shown in Fig. 51, in which the square-wave generator has been replaced by a fixed

FIGURE 51

25mH

.05 p.F

+

+

Ii V .05p.F

1.5k!l

f*

It-V battery to provide forward bias for the emitter. Adjust the feedback resistor R f until the circuit just breaks into oscillations. The required value Rc is larger than before; can you explain why? What starts the oscillations in the first place? Decrease the value of Rc further and observe the waveform of the oscilla­ tions. At sufficiently high oscillation levels you may find that the waveform is distorted, indicating that nonlinearities in the transistor are becoming significant. 43

f

'i

1'! 1

semiconductor electronics

questions 1 How does the square-wave generator provide the proper bias for the emitter

junction of the transistor? What changes in the circuit would be needed to use a pnp instead of an npn transistor? 2 Could an oscillator circuit be designed using a split inductance instead of a split capacitance? Draw a circuit diagram for such a circuit. What advantages or disadvantages might it have compared with the split-capacitor design ? 3 Do the two capacitors in the resonant circuit used in this experiment have to be equal? Why or why not? 4 Suppose the series resistance of the resonant circuit is increased, by adding an additional resistor, until the critical damping resistance is reached. Can the circuit still be made to oscillate by using a feedback amplifier? 5 Oscillators are sometimes discussed in terms of equivalence of the amplifier

to a negative resistance in the circuit. What properties would a negative resistance have? How might it be useful in a resonant circuit? Might a tunnel diode (Experiment SE-2) be used as a negative resistance? 6 In the analysis of the oscillator circuit of this experiment, is the neglect of the

resistance l/yo in the equivalent circuit justified? 7 In the circuit of Fig. 51, how is the maximum amplitude of oscillations related

to the emitter biasing voltage?

44

experiment

S E-6

negative feedback .itter d to ofa ages ~?

Introduction

Ie to In Experiment SE-5 we studied how oscillations in a resonant circuit can be sustained by sampling the current in the circuit, amplifying the current, and feeding it back into another part of the circuit to replace the energy losses in the circuit resistance. In this experiment we approach the matter of feedback from a more general point of view and study some of its other applications. We consider first a voltage amplifier having a voltage gain A. That is, in Fig. 52a, V2/Vl = A. Suppose now we take a certain fraction! of the output

ding Can ifier ltive tmel

'the

lted

FIGURE 52

"') 0

I-

0-­

Av}

-L

-

-

1

lv,

A

(a)

v,1

1·,

A

,.-0­

fvot I

f

1

tvo

(b) voltage and feed it back to the input, as in Fig. 52b. How does this change the function of the circuit? In particular, is the voltage gain changed? In Fig. 52b, the input and output voltages of the composite system are denoted as Vi and vo' and we have the following relations:

(28) 45

semiconductor electronics

We combine these two equations to eliminate VI and solve for the ratio voIv j , which is the new gain of the composite system. We denote this quantity as A'. We obtain A A' =--1 - fA

(29)

When f is positive, the feedback increases the overall gain of the system, as might be expected, since the feedback adds to the input signal a fraction of itself. In fact, whenfis large enough so fA approaches unity, the voltage gain approaches infinity, and the amplifier produces an output signal even in the absence of an input. In some cases this corresponds to an undesirable instability, but in some cases it is useful. In Experiment SE-5, for example, the self-sustaining oscillations result from this feedback. In this case the feedback network, incorporating the resonant circuit, is such that the con­ dition fA = 1 is satisfied only at one particular frequency, the resonant frequency of the circuit, so the "instability" manifests itself as a sinusoidal signal with that frequency, which may be just what is desired. Whenfis negative, A' is less than A; this may seem to be an undesirable result if the amplifier is to be used for voltage amplification. There are compensating advantages, however. Suppose we are concerned with the frequency response of the system. The response of the original amplifier may be characterized by the quantity dA/dw, that is, the rate of change of gain with frequency. The corresponding quantity for the feedback amplifier, with gain A' given by Eq. (29), is oA '

(1 - fA)(oA/ow) - A[ - f(oA/ow)]

ow

(1 - fA)2 1

oA

(30)

(1 - fA)2 ow

Iffis negative, this is clearly less than dA/dw. This result is not significant in itself, since A' is also less than A. The proper comparison is for the fractional change in gain with frequency, that is, 1 oA '

1 oA

A' ow

1 - fA A ow

(31)

We see that the fractional change is reduced by the factor (1 - fA), and correspondingly the high-frequency cutoff is increased by this same factor. Negative feedback also makes the gain A' less sensitive to individual circuit paraqleter changes than the original gain A. Suppose the current gain f3 of a transistor changes. Using a calculation just like the one above for the frequency, we can show that the fractional change in A' with f3 is less than that of A, again by the factor (1 - fA). Thus the feedback amplifier has improved stability with respect to change of circuit parameters. In fact, rewriting Eq. (29) as follows: 1 A' =--l/A -f

(32)

we see that when A is sufficiently large the gain A' is almost independent of A and depends only on! In this case the stability of the amplifier is deter­ mined almost entirely by the stability of the feedback network and is affected very little by changing transistor characteristics, supply voltages, and the like. 46

FIGURE !:

negative feedback

JV

FOURIER SERIES

j,

I

as

29)

em, jon age [lin Lble pie, the !on­ .ant idal

SE - 6

The frequency response of an amplifier, that is~ the variation of its gain with frequency , for a sinusoidal signal, is directly related to its transient response, that is, its response to a sudden pulse or a square wave. This relation has already been studied in a particular case in Experiment EC-3 . To develop the connection, we use the fact that any periodic wave may be represented as a superposition of sinusoidal waves whose frequencies are that of the periodic wave and integer multiples of it. That is, any periodic wave J(t) with period T and corresponding angular frequency w = 2n/ T can be represented as a series of the form J(t) =

L an sin nwt + L bncos nwt

(33)

n= O

n=l

Such a series is called a Fourier series; represen tation of a complicated wave shape by a series of sinusoidal functions is a very useful technique. When a musician speaks of harmonics or overtones, he is referring to the higher­ frequency components which are present in a complicated wave shape corresponding to a tone produced by a certain musical instrument; this terminology expresses the basic idea of Fourier analysis. For a Fourier series to be useful, we need to determine the coefficients an and b n , which are the amplitudes of the various sinusoidal components of the complex wave. We illustrate the determination of these coefficients with a particular example, the square wave, as in Fig. 53, This wave has the property

tble are the nay ~am

vith '/GURE 53

1

(30)

;ant the

(31)

and :tor. iual ~ain

, the .han has 'act,

a)

b

that for any value of t ,J(t) = - J( - L). That is, it is an odd function. Thus, we expect that all the cosine terms, which are even functions, will be zero, and the series will have the form <Xl (34) J(t) = L an sin nwt n= 1

To determine the coefficients an, we multiply both sides ofEq. (34) by sin mw!, where m is a positive integer, and integrate from 0 to T (or 2n/w). We obtain from the left-hand side

(32)

It of !tercted like.

T

LHS

= Vo

ft Jo

sin mwt dt - Vo

fT iT

The right-hand side is

RHS =

fT ~ Jo

sin mwt dt

=

VoT [1 - cos mn] (35)

mn

an sin nwt sin mwt dt

(36)

n= 1

47

semiconductor electronics

Consultation with an integral table shows that every one of the terms in this sum integrates to zero except the one in which m and n are equal, in which case we obtain simply am{T/ 2) = amen/OJ). Equating this result to Eq. (35) and solving for am, we obtain

2Vo (l - cos mn) nm

am = -

(37)

T hus, the Fourier series for a square wave of amplitude Vo and angular frequency w is V(t) = Vo(sin wI + 1- sin 3m! + t sin 50Jt + ... ) (38) Now if we know the response of the system for a sinusoidal input of arbitrary frequency, and if we know that the system is linear (that is, the output is a linear function of input), then the response to a complex wave can be obtained by simply adding the responses to the individual Fourier (sinusoidal) components. Transient Response We now return to the matter of the transient response of a transistor amplifie r and the effect of feedback on this response. We consider a grounded­ emitter circuit like that studied in Experiment SEA. As we have seen, the behavior df the transistor in this situation is approximated reasonably well by the equivalent circuit shown in Fig. 54a. However, the d ropoff of gain at FIGURE 54

c

B

E

(a)

c

B

E

(b) 48

negative feedback

this lich 35)

37)

SE-6

high frequencies suggests the presence of an effective input capacitance, as might be represented by Fig. 54b. The cutoff frequency, by definition, is the frequency at which the gain drops to I/J2 of its midrange value, and this occurs when the two parallel impedances have equal magnitudes, that is, when r i = l/wci. This pair of impedances has an effective time constant 1: = ricl . Comparing these two relations, we see that this time constant is just the reciprocal of the cutoff frequency

tlar (39)

1:=

W

38) . of the ::an ,ier

The response of the circuit of Fig. 54b to a step function or square-wave input current can be analyzed in detail. We assume that for t < 0, there is no charge on the capacitor Ci' At t = 0, we introduce a constant current i. into the base. What is the time dependence of the charge qb? From conservation of charge (Kirchhoff's current law), we have .

:tor

0

dqb

lb-

dt

(40)

ed­

the veil 1 at

From equality of potential (Kirchhoff's voltage law), we have qb

(41)

Eliminating the current ib between Eqs. (40) and (41), we obtain dqb

-d t

1

+ rici qb =

. Is

(42)

This is the familiar relaxation equation; its solution is

e- tl1

qb = isr(l

(43)

where r = ric i • Finally, combining Eqs. (40) and (43) and using the relation ie = /3oi, we obtain the collector current (44)

where /30 is the low-frequency limit of /3. Thus, we see that the same RC equivalent circuit may be used to represent both the frequency response and the transient response. What is the origin of this effective capacitance observed in Experiments SE-4 and SE-6? As discussed in Experiment SE-3, the current which crosses the collector junction results from a diffusion of minority carriers through the base. Thus, strictly speaking, the collector current depends not on the emitter or base current but rather on the amount of minority charge within the base. Ifwe are speaking ofa pnp transistor, the minority carriers are holes. If we let p stand for the total number of holes within the base, we may write dp

.

.

ep

e-=l+l-- dt e C r

(45)

where 1: is the recombination time within the base. Now in order to preserve the electrical neutrality of the transistor, the sum of the three currents must be zero: (46) 49

semiconductor electronics

Substituting into Eq. 45, we obtain

= _ (e dp + ep )

i

dt

b

(47)

't

Thus, we see that the base current contains two terms. The first is equal and opposite to the rate of buildup of minority charge within the base. This is a majority carrier component which just maintains electrical neutrality. The second term is equal and opposite to the recombination current. This part of the base current replaces those majority carriers which are lost through recombination. By comparing Eqs. (42) and (47) we see that qb = ep is the compensating majority charge in the base, and 't is the minority carrier recombination time within the base. Negative Feedback Finally, we discuss negative feedback and the way in which it may be used to produce an apparent reduction in relaxation time. In Fig. 55 we show a FIGURE 55

B

J

E

transistor amplifier with some feedback from the collector back into the base. The effect of this feedback is to inject additional current into the base pro­ portional to the collector voltage. Again we inject a signal current is into the base. The feedback network introduces an additional current •

Vc

Ie = Re =

f'

(48)

Ie

where f = Rd(Re + R L ) is the fraction of collector current fed back. We now modify Eq. (42) to take the form dqb

1

-dt + -qb = 't

.

.

I -fl S

C



(49)

Substituting ic = f30qb/'t as in Eq. (44), we obtain dqb -d

t

50

+

I

+ ff30 't

qb

=

. Is

(50)

negative feedback

SE-6

We see by inspection ofthis equation that the apparent relaxation time is now

shortened by a factor (l + ffJo). Solving Eq. (50) for a step current is> we

f7)

obtain i = POqb =

nd

,a

C

he

I

r

Pois (1 _ + fPo

e-(l

+ fPo)t,,)

(51)

Notice that the low-frequency current gain is reduced by exactly the same factor as is the relaxation time. (It is a general result that the "gain-band­ width product" of an amplifier is independent of feedback.) Insert a 6.8-kO resistor for R f • How large is the feedback fraction f1 Measure the relaxation time. By what factor is it reduced 1 Compare with 1 + fPo. From the size of the step in Vee you may compare the low-frequency gain. By what factor is the low-frequency gain reduced 1 If we differentiate Eq. (51) with respect to the time, we obtain

ui

gh he ier

ed a

T

die

=

Pois e-(l + fPo)!/!

(52)

r

Note that the initial slope of ie versus time is not affected by the feedback.

experIment 1

FIGURE 56

transient response To study the transient response of a common-emitter circuit, the circuit of Fig. 56 is suggested. This is essentially the same circuit used in Experiment 5OOkO

1.5kO

se.

'0­

he

4~ to15V

= +

~8)

~9)

)()

SE-4. The battery provides the operating-point current for the output circuit and also, through the 470- and 500-kO resistors, for the input circuit. Adjust the 500-kO variable resistor for a collector current of about 1 rnA. With the square-wave input as shown, observe the waveform obtained at the collector. Does it resemble the waveforms observed in Experiment EC-3 1 51

semiconductor electronics

Measure the relaxation time r. Compare this value with the value obtained ftom Eq. (39) and the high-frequency cutoff observed in Experiment SE-4. 2 negative feedback

Now modify the circuit by adding the elements Rr and the 0.5 Ilf capacitor, as shown in Fig. 57. The feedback resistor Rr feeds back to the base a current 500 kU

FIGURE 57

Rf

O.5J.LF

41 to15V­ 2

­

+ lOOkU

proportional to the collector voltage. The 0.5)iF capacitor blocks dc components of current so that the operating point is not changed. For ac components of the frequencies used here, this capacitor acts as a short circuit and can be ignored. Start with a value Rr = 6.8 kQ. How large is the feedback fraction f? Again measure the relaxation time. By what factor is it reduced? Compare this factor with the quantity (1 + fff). The midfrequency gain may also be measured by observing the amplitude of the step in Vee and comparing it with the value when feedback is not included. By what factor is this gain reduced? 3 frequency dependence

You may wish to try other values of the feedback resistor R r• It is also interest­ ing to measure the frequency dependence of the gain with negative feedback, and compare with the results obtained in Experiment SE-4. Are your results consistent with those obtained from the transient response? 4 Fourier analysis

The representation of a square wave in terms of its Fourier components may be studied experimentally, using the circuit of Fig. 58. We drive a series reso­ nant LC circuit with the output of a square-wave generator. A sinusoidal

signal at the same frequency as the square wave is applied to the horizontal deflection plates of an oscilloscope, and the voltage across the capacitor is applied to the vertical plates. When the frequency of the square wave is the 52

negative feedback

SE-6

same as the resonant frequency of the LC circuit, the Lissajous pattern will open up. If we reduce the square-wave signal to one-third the resonant frequency of the LC circuit, we shall observe a second resonant maximum with just one-third the amplitude of the first maximum. We may think of this

ed ·4. )r, nt FIGURE 58

L

c

-

--

signal as being excited by the third harmonic of the square wave. We may reduce the square-wave frequency to one fifth, observing a signal of one-fifth amplitude, and so on. The reason for applying a sinusoidal signal to the horizontal plates is to make it possible to identify the harmonic at sight.

dc ac Jrt

(? iCe be ith d?

questions 1

Suppose an amplifier has positive feedback. How will its frequency range compare to that of the same amplifier with no feedback?

2

Does negative feedback reduce the effect of changing parameters in the feedback circuit itself? Explain.

3

A certain amplifier is nonlinear in the sense that a sinusoidal input yields an output which has a sinusoidal component with the same frequency as the input, plus a component having twice that frequency. This is an example of harmonic distortion. What effect does negative feedback have on this phenomenon?

8t:k, Its

4 Find the Fourier-series representation of a sawtooth wave such as used in

the horizontal sweep of an oscilloscope. Compare the form of this series with that for the square wave .

ay

•0­

lal tal . is :he

5

The feedback circuit used in this experiment contains a resistor and a capaci­ tor in series. At sufficiently low frequencies the capacitor no longer acts as a short circuit for ac components. What minimum frequency must be used for this capacitor to be negligible? 53

semiconductor electronics

6 Is it possible in the circuits studied in this experiment to have so much

negative feedback that the gain is less than unity? Explain. 7 Can feedback be used in a grounded-base amplifier circuit? Try to design a

possible circuit. Should the feedback be based on the output current or on output voltage? Explain. 3

2

0

1.1

I: 10

•II •... ~

II.

:::I

I::

1

9

8

z 6

5

4

3

54

0

I~§~ E~~

Q

" 7. 6

m

5 4

3D

• Hittta

M

2

m ttfttt

1

9

II

8

1

6

00 ~

R

-

5 .\

m

m

3

::1=

~

2

=At 1

-m

9

§=

m

8

7. 6 5

E

i

4

3_ .,

1

~ IE

m

m

r­

g IIIm I

I I I

I I

I I

m

I

I