PHYSICS IN YOUR KITCHEN LAB
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Physics in Your Kitchen Lab Edited by Academician I.K. Kikoin Translated from the Russian by A. Zilberman
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Mir Publishers Moscow
First published 1985 Revised from the 1980 Russian edition
EDITORIAL
BOARD
Academician I . K . Kikoin (chairman), Academician A . N . K o l m o g o r o v (deputy chairman), I . S h . Slobodetskii (scientific secretary), Cand. Sc. (Phys.-Math.), Corresponding member of the Academy of Sciences of the U S S R A . A . Abrikosov, Academician B . K . Vainstein, Honoured teacher of the Russian Soviet Federative Socialist Rep u b l i c B . V . Vozdvizhenskii, Academician V . M . Glushkov, Academician P . L . K a p i t s a , Prof. S . P . K a p i t s a , Corresponding member of the A c a d e m y of Sciences of the U S S R Y u . A . O s i p y a n , Corresponding member of the A c a d e m y of Pedagogical Sciences of the U S S R V . G . Raz u m o v s k i i , Academician R . Z . Sagdeev, M . L . S m o l y a n s k i i , Cand. Sc. (Chem.), Prof. Y a . A . S m o r o d i n s k i i , Academician S . L . Sobolev, Corresponding m e m b e r of the Academy of Sciences of the U S S R D . K . Phaddeev, Corresponding member of the A c a d e m y of Sciences of the USSR I.S. Shklovskii.
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Contents E d i t o r ' s Note A Demonstration of Weightlessness by A. Dozorov A Cartesian Diver by A. Vilenkin An A u t o m a t i c S i p h o n by V. Mayer and N. Nazarov Exercises Vortex R i n g s by R.W. Wood On Vortex R i n g s by S. Shabanov and V. Tornado Models by V. Mayer
7 9 12 13 17 17 23
Shubin
The Aerodynamics of Boomerangs by Felix Hess A H y d r o d y n a m i c Mechanism in a F a l l i n g Test T u b e by G.I. Pokrovsky A n Instructive E x p e r i m e n t w i t h a Cumulative Jet by V. Mayer Exercises Magic w i t h Physics by V. Mayer and E. Mamaeva A D r o p on a H o t Surface by M. Golubev and A. Kagalenko Surface Tension Draws a Hyperbola by I. Vorobiev Experiments w i t h a Spoonful of Broth by V. Mayer H o w to Grow a Crystal by M. Kliya Crystals Made of Spheres by G. Kosourov A Bubble Model of Crystal by Ya. Geguzin
33 37
51
53 54 56 58 61 64 71 74 85
Contents Determining the Poles of a Magnet by B. Aleinlkov A Peculiar P e n d u l u m by N. Minz Lissajous Figures by N. Minz Exercises
101 106 118
Waves in a Flat Plate (Interference) by A. Kosourov H o w to Make a R i p p l e Tank to E x a m i n e W a v e Phenomena by C.L. Stong An Artificial Representation of a Total Solar Eclipse by R.W. Wood Believe I t or Not by G. Kosourov Colour Shadows by B. Kogan W h a t Colour Is B r i l l i a n t Green? by E. Pal'chikov A n Orange Sky by G. Kosourov The Green Red L a m p by V. Mayer Measuring Light W a v e l e n g t h w i t h a Wire by N. Rostovtsev Exercises Measuring Light w i t h a Record by A. Bondar A Ball for a Lens by G. Kosourov
99
118
128
140 144 150 152 154 161
164 172
Phonograph 172 177
To Georgii Ivanovich Kosourov
Editor's Note
Physics is an experimental science since it studies the fundamental laws of nature by direct experimentation. The experimenter asks questions of nature in any experimental work, but only correctly formulated questions are answered. This means that unless a physical experiment is set up correctly, the experimenter will not get the desired results. An experimenter's skill, therefore, depends on his ability to formulate experiments correctly. The experimental physics is a fascinating science, which enables us to understand, explain and, sometimes, even discover new phenomena in nature. The first step in becoming an accomplished physicist is mastering of the techniques of physical experimentation. Modern experimental physics uses very sophisticated and expensive apparatus, housed, for the most part, in large research institutes and laboratories where many of the readers of this book may one day conduct their own original research. Until then, however, the engaging experiments described in this book can be performed right at home. Most of the experiments included here were first published separately in the journal Kvant. Just as "a picture is worth a thousand words", an experiment once performed is worth a thou-
8
Editor's Note
sand descriptions of one. I t is recommended, therefore, that readers perform the experiments described themselves. The means for this are readily available, and it should soon become obvious that experimentation is a captivating pastime. The experiments presented here need not be confining; they may be varied and expanded, providing, in this way, an opportunity for real scientific investigation. The book is dedicated to Georgii Ivanovich Kosourov, one of the founding fathers of Kvant. Kosourov, who edited the experimental section of the journal in its first year of publication, has contributed several very interesting articles to this collection. Among the other authors of this book are a number of famous physicists, as well as young researchers just beginning their careers. W e hope this book w i l l fascinate not only students already interested in physics who intend to make it their lifework but also the friends to whom they demonstrate the experiments in a laboratory made right at home.
A Demonstration of Weightlessness by A.
Dozorov
The weightless state is achieved in free flight. A satellite in orbit, a free falling stone, and a man during a jump are all in a state of weightlessness. A weight suspended from a string weighs nothing in a free fall and, therefore, does not pull on the string. I t is easy to make a device that will let you "observe" weightlessness. Figure 1 depicts such a device schematically. I n its 'normal' state, weight G pulls the string
Fig. 1
Fig. 2
taut, and elastic plate EP bends, breaking the contact between terminals K\ and K2 of the
10
A. Dozorov
circuit. Naturally, lamp L, connected to the circuit, does not light up in this case. If the entire device is tossed into the air, however, weight G becomes weightless and does not tighten the string. The elastic plate straightens out and the terminals connect, which switches on the lamp. The lamp is lit only when the device is in a weightless state. Note that this state is achieved both when the device is thrown up and as it returns to the ground. The adjustment screw S makes it possible to place the terminals so that they have a small clearance when the device is stationary. The device is fastened to the inside of a transparent box, as shown in Fig. 2. A little practical advice about construction. In order to provide for the use of a large-cell (flat) battery or a small one-cell battery, reserve space for the larger battery. Access to the battery compartment should be facilitated since battery may have to be replaced frequently. The battery can be secured to the outer surface of the device, and two holes for connecting wires should be provided in the casing. Any thin elastic metal strip can be used as an elastic plate, even one half of a safety razor blade (after fastening the blade to the stand, you will see where to co i nect the string for the weight). The design can be simplified further, if the adjustment screw and terminal K\ are combined and the plate functions as terminal K 2 (Fig. 3). Figure 4 shows a design that has no adjustment screw at all. If you think a little, you can probably come up with an even simpler design.
A Demonstration of Weightlessness
Fig. 3
Fig. 4
11
12 A Cartesian Diver by A.
Vilenkin
A toy ship made of paper w i l l float easily, but if the paper gets soaked, the ship sinks. W h e n the paper is dry, it traps air between its bell and the surface of the water. If the bell gets soaked and begins to disintegrate, the air escapes the bell and the ship sinks. B u t is it possible to make a ship whose bell alternately keeps or releases air, making the ship float or sink as we wish? I t is, indeed. The great French scholar and philosopher Rene Descartes was the first to make such a toy, now commonly called the 'Cartesian Diver* (from Cartesius, the L a t i n spelling of Descartes). Descartes' toy resembles our paper ship except that the 'Diver' compresses and expands the air instead of letting it in and out. A design of the 'Diver' is shown in Fig. 5. Take a m i l k bottle, a small medicine bottle and a rubber balloon (the balloon will have to be spoiled). Fill the m i l k bottle with water almost to its neck. Then lower the medicine bottle into the water, neck down. Tilt the medicine bottle slightly to let some of the water in. The amount of water inside the smaller bottle should be regulated so that the bottle floats on the surface and a slight push makes it sink (a straw can be used to blow air into the bottle while it is underwater). Once the medicine bottle is floating properly, seal the m i l k bottle with a piece of rubber cut from the balloon and fastened to the bottle with a thread~wound' r around the neck. ""Press down the piece of rubber, and the 'Diver* will sink. Release it, and the 'Diver' will rise.
An Automatic Siphon
13
This is because the air inside the milk bottle is compressed when the piece of rubber is pushed in. The pressure forces water into the medicine
Fig. 5
bottle, which becomes heavier and sinks. As soon as the pressure is released, the air in the medicine bottle forces the extra water out, and the 'Diver' floats up.
A n Automatic Siphon by V. Mayer and N.
Nazarov
Most o you probably studied the workings of the siphon, the simplest device for pumping liquids, while still in grade school. The famous American physicist Robert Wood is said to have begun his scientific career when still a boy with just such a siphon. This is how W . Seabrook
14
V. Mayer and N. Nazarov
described that first experiment in his book about Robert Wood. He wrote that there was an elevation over a foot high around a puddle, and everybody knew that water would not flow uphill. Rob laid a hose on the ground and told one of the boys to seal its end with his finger. Then he started filling the hose with water u n t i l it was full. Already a born demonstrator at that age, Rob, instead of leaving his end of the hose on the ground, let it dangle over a high fence which separated the road from the ditch. Water flowed through the siphon. This was apparently Wood's first public scientific victory. The conventional siphon is so simple that almost no improvement in its design seems possible. Perhaps its only disadvantage is that it is necessary to force the air from bends in the siphon prior to operation. Yet even this problem was solved, thanks to human ingenuity. Once inventors had understood the shortcoming in the design, they removed it by the simplest possible means! To make an automatic siphon*, you will need a glass tube whose length is about 60 cm and whose inner diameter is 3-4 m m . Bend the tube over a flame so that it has two sections, one of which is about 25 cm long (Fig. 6). Carefully cut a small hole (1) in the shorter arm about 33-35 m m from its end with the edge of a needle file (wet the file first). The area of the hole should not be more than 0.5-1 mm 2 . Take a ping-pong * This version of the a u t o m a t i c siphon, S . D . P l a t o n o v , was described in Zavodskaya 4, No. 6 (1935) (in Russian).
invented by Laboratoriya,
15
An Automatic Siphon
ball, and make a small hole in it which is then reamed with the file u n t i l the tube can be pushed into the ball and is held there tightly. Push the
Fig. 6
Fig. 7
tube into the hole u n t i l its end nearly touches the side of the ball opposite the hole (Fig. 6). The tube should fit tightly in the opening. If the hole is too large, fill the gap with plasticine. Make another hole (2) close to the end of the tube
16
R. W. Wood
inside the ball. Its initial diameter should be about 1 m m . Quickly lower the arm of the siphon with the ball at the end into a glass of water. The tube will fill almost immediately with a rising column of water broken by a series of air bubbles. When the water reaches the bend, it will move down the second arm of the siphon (Fig. 7), and in a few moments a continuous stream of water will begin to flow from the end of the tube! If the experiment is unsuccessful at first, simply adjust the siphon slightly. The correct operation of the automatic siphon depends on the appropriate choice of diameters of the holes in the tube and the ball. Faulty positioning of the glass tube and the ball or an inadequate seal between the tube and the ball may also spoil the siphon operation. The second hole in the ball can be gradually enlarged with a needle file to improve the performance of the siphon. As soon as the siphon is operating satisfactorily, glue the ball to the tube. How does the automatic siphon work? Look at Fig. 6 again. When the ball is lowered into the glass, water floods simultaneously into opening 2 and into the open end of the glass tube. Water rises in the tube at a faster rate than in the ball. The water rising to opening 1 in the wall of the tube seals the tube. As the ball floods with water, the air pressure inside it rises. W h e n equilibrium is reached, a small air bubble is forced through the opening 1. The bubble cuts off a small column of water and carries it upward. The water that follows reseals opening 1, and the compressed air forces another air bubble into the tube, cut-
17
On Vortex Rings
ting off another portion of water. Thus, the section of tube w i t h the ball has an air/water mixture whose density is lower than that of water. Under hydrostatic pressure this mixture rises to the bend and flows down the second arm of the tube. W h e n the ball is completely filled w i t h water, pressure creates a continuous flow of water and the siphon begins to operate. EXERCISES Show experimentally that water floods the ball at a slower rate than the tube. Explain why. 2. To be certain that the explanation of the operation of the siphon is correct, replace the opaque ball with a small glass bottle with a rubber stopper. The design of the setup with the bottle should be an exact replica of the original. The glass tube should go through the rubber stopper. Since the bottle is transparent, you will be able to see the air/water mixture forming in the tube. 3. Determine whether the rise of the column of water depends on the water depth o£ the ball. 4. Make an automatic siphon by replacing the glass tube with a rubber hose.
Vortex Rings by R. W.
Wood
I n the course of some experiments preparatory to a lecture on vortex rings I have introduced certain modifications which may be of interest to teachers and students of science. The classic vortex-box is too well known to require much description. Our apparatus, which is rather larger than those in common use, is a pine box measuring about a metre each way, * Nature, 2-01544
February
28, 1901, pp. 418-420.
18
R. W. Wood
w i t h a circular hole 25 cms in diameter in one end. Two pieces of heavy rubber t u b i n g are stretched d i a g o n a l l y across the opposite or open end, w h i c h is then covered w i t h black enamel cloth tacked on rather loosely. The object of the rubber chords is to give t h e recoil necessary after t h e expulsion of a ring to prepare the box for a second discharge. Such a box w i l l project air vortices of great power, the slap of the ring
Fig. 8
against the brick w a l l of the lecture h a l l being d i s t i n c t l y a u d i b l e resembling t h e sound of a flip w i t h a towel. A n audience can be given a v i v i d idea of the quasi-rigidity of a fluid i n r o t a t i o n by projecting these i n v i s i b l e rings i n r a p i d succession i n t o t h e a u d i t o r i u m , the i m p a c t of the r i n g on t h e face r e m i n d i n g one of a blow w i t h a compact t u f t of cotton. For rendering rings visible I h a v e found t h a t by far the best results can be o b t a i n e d b y c o n d u c t i n g a m m o n i a and hydrochloric acid gases i n t o t h e box t h r o u g h rubber tubes l e a d i n g to two flasks in w h i c h N H 4 O H and HG1 are b o i l i n g . Photographs of large rings m a d e i n t h i s w a y are reproduced i n F i g . 8, the side view being p a r t i c u l a r l y interesting, showing the comet-like t a i l formed b y the s t r i p p i n g off of the outer portions of the
Vortex Rings
IJ
ring by atmospheric friction as it moves forward. The power of t h e air-rings can be shown b y directing t h e m against a flat pasteboard b o x , stood on end at some distance from the vortex apparatus, the box being at once overturned or even driven oft 011 to the floor. A large cluster of b u r n i n g gas jets can be extinguished b y the i m p a c t of a ring. For showing the elasticity of the rings b y bouncing one off the other, I find t h a t the best p l a n is to drive t w o i n r a p i d succession from t h e box, the second being projected w i t h a s l i g h t l y greater velocity t h a n the first, all experiments t h a t I have m a d e w i t h t w i n boxes h a v i n g yielded unsatisfactory results. Though the large vortices obtained w i t h an apparatus of t h i s description are most s u i t a b l e for lecture purposes, I find t h a t m u c h more b e a u t i f u l and s y m m e t r i c a l rings can be m a d e w i t h tobacco smoke b l o w n from a paper or glass tube about 2.5 cm i n diameter. I t is necessary t o practice a l i t t l e to learn just the nature a n d strength of t h e most suitable puff. R i n g s b l o w n i n this w a y i n s t i l l air near a l a m p or i n f u l l s u n l i g h t , when viewed l a t e r a l l y , show the spiral stream lines i n a most b e a u t i f u l m a n n e r . I h a v e succeeded i n p h o t o g r a p h i n g one of these rings i n the following w a y . A n instantaneous drop shutter was fitted to t h e door of a dark room a n d an arclamp focussed 011 its aperture b y means of a large concave m i r r o r . The shutter was a s i m p l e affair, merely a n a l u m i n i u m slide operated w i t h an elastic b a n d , g i v i n g an exposure of 1/300 of a second. A p h o t o g r a p h i c plate was set on edge i n the dark room i n such a position t h a t it w o u l d 2*
20
R. W. Wood
be i l l u m i n a t e d by the divergent beam c o m i n g from t h e i m a g e of t h e arc when the shutter was opened. A r u b y l a m p was placed i n front of the sensitive film. As soon as a good r i n g , s y m m trical in form and not m o v i n g too fast, was seen to be i n front of t h e plate, a string leading t o the shutter was p u l l e d and the p l a t e i l l u m i n a t e d w i t h a d a z z l i n g flash. The ring casts a perfectly sharp shadow o w i n g to the s m a l l size and distance of
the source of l i g h t ; the resulting picture is reproduced i n F i g . 9. The ring is seen to consist of a layer of smoke a n d a layer of transparent air, w o u n d u p i n a spiral of a dozen or more complete turns. The angular velocity of r o t a t i o n appears to increase as t h e core of the r i n g is approached, t h e i n n e r portions being screened from friction, if we m a y use the t e r m , by t h e r o t a t i n g layers s u r r o u n d i n g t h e m . This can be very nicely shown b y differentiating the core, f o r m i n g an air ring w i t h a smoke core. I f we m a k e a s m a l l vortex box w i t h a hole, say 2 cm i n d i a m e t e r , fill i t w i t h smoke a n d push very gently against the diap h r a g m , a fat r i n g emerges w h i c h rotates i n a very lazy fashion, to a l l appearances. I f , however, we clear t h e air of smoke, pour i n a few drops of a m m o n i a and brush a l i t t l e a strong HC1 around the lower part of the aperture, the smoke forms
21
On Vortex Rings
i n a t h i n layer around t h e under side of t h e hole. G i v i n g the same gentle push on the d i a p h r a g m , we find t h a t the smoke goes to the core, the rest of the ring being i n v i s i b l e , the visible part of the vortex s p i n n i n g w i t h a surprisingly h i g h v e l o c i t y . Considerable k n a c k is required to form these t h i n cressent-like vortices, the best results being
Fig. 10
Fig. 11
u s u a l l y attained after q u i t e a n u m b e r of a t t e m p t s h a v e been m a d e . A d r a w i n g of one of these smoke-cores is shown i n F i g . 10. The a c t u a l size of the vortex being i n d i c a t e d by dotted lines, it is instructive as showing t h a t the air w h i c h grazes the edge of the aperture goes to t h e core of the ring. The experiment does not work very well on a large scale, t h o u g h I have had some success by v o l a t i l i s i n g sal a m m o n i a c around t h e upper edge of the aperture b y means of a zig-zag iron wire heated by a current. B y t a k i n g proper precautions we can locate the smoke elsewhere, f o r m i n g a perfect half-ring, as is shown i n F i g . 11, i l l u s t r a t i n g i n a s t r i k i n g
22
S. Shabanov and V. Shubin
m a n n e r t h a t the existence of the ring- depends in no w a y on the presence of the smoke. The best w a y to form these half-rings is to breathe smoke very g e n t l y i n t o a paper t u b e a l l o w i n g it to flow along t h e b o t t o m , u n t i l the end is reached, when a r i n g is expelled by a gentle puff. A large test t u b e w i t h a hole b l o w n i n the b o t t o m is perhaps preferable, since the c o n d i t i o n of things inside can be w a t c h e d . I t is easy enough to get a ring, one h a l f of w h i c h is w h o l l y i n v i s i b l e , the smoke ending a b r u p t l y at a sharply defined edge, as shown i n F i g . 11, requires a good deal of practice. I h a v e tried f u l l y half-a-dozen different schemes for getting these half-rings on a large scale, b u t no one of t h e m gave results w o r t h m e n t i o n i n g . The hot wire w i t h the sal a m m o n i a c seemed to be the most p r o m i s i n g m e t h o d , b u t I was u n a b l e to get t h e sharp cut edge w h i c h is t h e most s t r i k i n g feature of t h e s m a l l rings b l o w n from a tube. I n accounting for the f o r m a t i o n of vortex rings, t h e rotary m o t i o n is often ascribed to friction between the issuing air-jet and the edge of t h e aperture. I t is, however, f r i c t i o n w i t h the exterior air t h a t is for the most part responsible for t h e vortices. To illustrate t h i s p o i n t I have devised a vortex box in w h i c h friction w i t h the edge of the aperture is e l i m i n a t e d , or rather compensated, by m a k i n g it equal over the entire cross-section of the issuing jet. The b o t t o m of a cylindrical t i n box is drilled w i t h some 200 s m a l l holes, each about 1.7 m m in diameter. I f the box be filled w i t h smoke and a sharp puff of air delivered at the open end, a b e a u t i f u l vortex ring w i l l be thrown off from the cullender surface (Fig. 12). W e may even
23
On Vortex Rings
cover the end of a paper t u b e w i t h a piece of l i n e n c l o t h , t i g h t l y stretched, and blow smoke rings with it. I n experimenting w i t h a box provided w i t h two circular apertures I h a v e observed the fusion of two rings m o v i n g side b y side i n t o a single large ring. I f t h e rings h a v e a h i g h v e l o c i t y of r o t a t i o n they w i l l bounce apart, b u t if t h e y are
Fig. 12
sluggish they w i l l u n i t e . A t the m o m e n t of u n i o n the form of the vortex is very u n s t a b l e , b e i n g an extreme case of the v i b r a t i n g e l l i p t i c a l r i n g . I t at once springs from a h o r i z o n t a l d u m b - b e l l i n t o a vertical dumb-bell, so r a p i d l y t h a t t h e eye can scarcely f o l l o w the change, and then s l o w l y oscillates into the circular form. T h i s s a m e p h e n o m enon can be shown w i t h two paper tubes h e l d i n opposite corners of the m o u t h and nearly p a r a l l e l to each other. The air i n the room m u s t be as s t i l l as possible in either case.
On Vortex Rings by S. Shnbanov
and V.
Shubin
Formation of the Vortex R i n g s To study vortex rings in the air under laboratory conditions, we used the apparatus designed
24
S. Shabanov and V. Shubin
by Professor Tait (Fig. 13). One end of this cylinder, the membrane, is covered with a flexible material such as leather. The other end, the diaphragm, has a circular opening. Two flasks, one containing hydrochloric acid (HC1), the other ammonium hydroxide ( N H 4 O H ) , are placed in the box, where they produce a thick fog (smoke) of ammonium chloride particles 'NH4CI). By
Membrane
HC1
Fig. 13
Diaphragm
NH4OH
Fig. 14
tapping the membrane, we impart a certain velocity to the smoke layer close to it. As this layer moves forward, it compresses^the next layer, which, in turn, compresses the layer followinglit, in a chain reaction that reaches the diaphragm where smoke escapes through the opening and sets formerly still air in motion. Viscous friction against the edge of the opening twists the smoky air into a vortex ring. The edge of the opening is not the m a i n factor in the formation of the vortex ring, however. W e can prove this by fitting a sieve over the opening in the Tait's apparatus. If the edge were
On Vortex Rings
25
i m p o r t a n t , m a n y s m a l l vortex rings w o u l d f o r m . Y e t they do not. Even w i t h a sieve, we s t i l l observe a single, large vortex r i n g (Fig. 14). I f the m e m b r a n e is substituted b y a plunger t h a t is set to m o t i o n , a continuous smoke jet w i l l appear on the edge of the opening instead of vortex rings. I t is essential t o provide for t h e
Fig. 15
Fig. 16
i n t e r m i t t e n t outflows of smoke t h r o u g h the opening. Vortex rings can be produced in water using an ordinary p i p e t t e and i n k . Let a few drops of i n k fall from a h e i g h t of 2-3 centimetres i n t o an a q u a r i u m w i t h very s t i l l water, w h i c h has no convection flows. The f o r m a t i o n of the i n k rings w i l l be very obvious in the clear water (Fig. 15). The set u p can be changed s l i g h t l y , t h e stream ofj'ink can be released from a pipette submerged i n water (Fig. 16). The vortex rings o b t a i n e d i n this case are larger. Vortex rings i n water form s i m i l a r l y t o those in the air, and t h e b e h a v i o u r of the i n k i n water is s i m i l a r to t h a t of smoke i n the air. I n b o t h cases viscous friction plays a v i t a l role. Experiments show t h a t t h e analogy is complete o n l y j n the first m o m e n t s after the f o r m a t i o n of tlje
26
S. Shabanov and V. Shubin
vortex, however. As it develops further, the vortex behaves differently in water and the air. Movement of the Environment Around the Vortex Rings W h a t happens to the environment when a vortex forms? W e can answer this question with the right experiments. Place a lighted candle 2-3 metres away from the Tait apparatus. Now direct a smoke ring so
Fig. 17
that it passes the candle but misses the flame narrowly. The flame will either go out or flicker violently, proving that the movement of the vortex involves not only the visible part of the ring, but also adjacent layers of the air. How do these layers move? Take two pieces of cloth, and soak one in hydrochloric acid, and the other in ammonia solution. Hang them up about 10-15 centimetres apart. The space between them w i l l immediately be' filled with smoke (ammonium chloride vapour). Now shoot a smoke ring from the apparatus into the vapour cloud. As the ring passes through the cloud, the ring expands while the cloud starts moving circularly. From this we can conclude that the air close to t,he vortex ring is circulating (Fig. 17).
On Vortex Rings
27
A similar experiment can be set up in water. Put a drop of ink in a glass full of water that has been stirred slowly and continues to circulate. Let the water get still. Y o u w i l l see ink fibres in the water. Now put ink ring into the glass. W h e n this ring passes close to the fibres, they twist. Vortex Rings in Water W e decided to study the behaviour of vortices in water further. W e know that a drop of ink
Fig. 18
placed into an aquarium from a height of 2-3 centimetres will form an ink vortex ring. This ring soon develops into several new rings, which, in turn, break into other smaller rings, and so on to form a beautiful "temple" in the aquarium (Fig. 18).
28
S. Shabanov and V. Shubin
W e found that the division of the initial ring into secondary rings was preceded by expansions in the large ring itself. How this can be explained? Since the environment through which the ink ring moves is nonuniform, some of its parts move faster than others, some lag behind. The ink (which is heavier than water) tends to collect in the faster sections, where it forms swelling due to surface tension. These swellings give birth to new droplets. Each droplet on the initial vortex behaves independently, eventually producing a new vortex ring in a cycle that repeats several times. Interestingly, we could not determine any regularity in this cycle: the number of rings in the "fourth generation" was different in each of ten experiments. W e also found that vortex rings require "living" space. W e tested this by placing pipes of different diameters in the path of rings in water. W h e n the diameter of the pipe was slightly larger than that of the ring, the ring disintegrated after entering the pipe, to produce a new ring with a smaller diameter. W h e n the diameter of the pipe was four times larger than the ring diameter, the ring passed through the pipe without obstruction. I n this case the vortex is not affected by external factors. Smoke R i n g Scattering W e conducted several experiments to study interaction between the smoke ring and the opening of different diameter. W e also studied the relationship between the ring and a surface at various angles. (We called these experiments scattering tests.)
On Vortex Rings
29
Consider a r i n g h i t t i n g a d i a p h r a g m whose aperture is smaller t h a n the ring. Let us e x a m i n e two cases. First, the r i n g m a y collide w i t h t h e d i a p h r a g m when t h e forward m o t i o n v e l o c i t y of the ring is perpendicular to the d i a p h r a g m p l a n e and the centre of t h e r i n g passes t h r o u g h the centre of the d i a p h r a g m . Collision, on the other h a n d , m a y be off-centre if the centre of t h e r i n g does not a l i g n w i t h t h e centre of the d i a p h r a g m . I n the first case, t h e r i n g scatters when it h i t s
Diaphragm
Fig. 19
the d i a p h r a g m , a n d a new ring w i t h a smaller diameter forms on t h e other side of the d i a p h r a g m . This smaller r i n g forms just as it w o u l d i n t h e Tait apparatus: t h e air t h a t moves around the o r i g i n a l r i n g passes t h r o u g h the aperture and entrains the smoke of t h e scattered vortex w i t h i t . A s i m i l a r s i t u a t i o n can be observed when a r i n g collides centrally w i t h an aperture of e q u a l or somewhat larger d i a m e t e r . The effect of an offcentre collision is even more interesting: the newly formed vortex emerges at an angle t o the original direction of the m o t i o n (Fig. 19). Try to explain why. Now let us consider an interaction of the r i n g w i t h a surface. E x p e r i m e n t s show t h a t if the
30
S. Shabanov and V. Shubin
surface is perpendicular to the velocity of the ring, the ring spreads without losing its shape. This can be explained as follows. W h e n the air stream inside the ring hits the surface, it produces a zone of elevated pressure, which forces the ring to expand uniformly. If the surface is at a slant relative to the original direction, the vortex recoils when colliding with it (Fig. 20). This phenomenon can be explained as the effect
Fig. 20
of elevated pressure in the space between the ring and the surface. Interaction of Rings The experiments with interacting rings were undoubtedly the most interesting. W e conducted these experiments with rings in water and in the air. If we place a drop of ink into water from a height of 1-2 cm and., a second later, let another drop fall from 2-3 cm, two vortices moving at different velocities will form. The second drop will move faster than the first (y2 greater than i\). When the rings reach the same depth, they begin to interact with each other in one of three possible ways. The second ring may overtake the first
On Vortex Rings
31
one w i t h o u t t o u c h i n g i t (Fig. 21a). I n t h i s situat i o n the water currents generated by the rings repel one another. Some of the i n k from t h e first r i n g flows over to t h e second r i n g because t h e more intensive currents i n the second r i n g p u l l the i n k w i t h t h e m . O c c a s i o n a l l y , some of this
(a)
Ib)
(c)
Fig. 21
i n k passes t h r o u g h the second r i n g a n d forms a new, smaller r i n g . The rings t h e n b e g i n to break d o w n , and the process continues as we observed earlier. The second r i n g m a y , on the other h a n d , touch the first w h i l e o v e r t a k i n g it (Fig. 216). As a result, the more intensive flows of the second r i n g destroy the first one. N o r m a l l y , new smaller vortices emerge from the r e m a i n i n g i n k cluster of the first ring. F i n a l l y , the rings m a y collide centrally (Fig. 21c), in w h i c h case t h e second r i n g passes through the first and shrinks, whereas the first
32
V. Mayer
ring expands. As before, this is a result of the interaction of the water currents of both rings. The rings begin to break d o w n at a later stage. The interaction of smoke rings i n the air was investigated using a Tait apparatus w i t h two apertures. The results of the experiments greatly depended on the force and d u r a t i o n of the i m p a c t on the m e m b r a n e . I n our setup the m e m b r a n e was struck w i t h a heavy p e n d u l u m . If t h e distance I between the apertures is less t h a n the diameter d
Fig. 22
of each aperture (I is less t h a n d) the two air currents m i x and produce a single vortex ring. I f d < I < 1.5d, no ring appears at a l l . I n all other cases two rings appear. I f I is more t h a n 4d, the rings do not interact. I f 1.5d < I < 4d, the rings converge at first, a l t h o u g h they occasionally separate again before d i s i n t e g r a t i n g . The formation of an " i m a g i n a r y " ring i n the space between the rings explains this convergence (Fig. 22). The i m a g i n a r y ring moves i n the opposite direction from the planes of the real rings, which begin to t u r n toward one another and g r a d u a l l y move closer. W e were not able to determine why the rings eventually disintegrate c o m p l e t e l y .
Tornado Models
33
Tornado Models by V. Mayer The tornado is one of the most awesome a n d mysterious p h e n o m e n a in nature. I t s power is so great t h a t almost n o t h i n g can w i t h s t a n d its force. H o w are tornadoes able to carry heavy objects over such considerable distances? H o w do they form? Modern science has yet to answer these and m a n y other questions completely. Is it possible t o s i m u l a t e a tornado i n t h e laboratory? W i t h the f o l l o w i n g two experimental setups, y o u can m a k e a water model of a t o r n a d o even at home. 1. Solder a brass-or tin-plate disc about 40 m m in diameter and 0.5-1 m m thick to the shaft of a micromotor l i k e those c o m m o n l y used i n toy machines. The disc m u s t be exactly perpendicular to the shaft to ensure t h a t the disc w i l l r u n true. Use grease or m i n e r a l oil to seal the bearings of the shaft. The contact studs and t h e soldered wires leading to t h e m should be protected w i t h a plasticine layer. A t t a c h a plasticine cake about 5 m m t h i c k t o the bottom of a glass (or jar) about 9 cm i n diameter and 18 cm h i g h . A t t a c h the m i c r o m o t o r to i t , a l l o w i n g a clearance between the lower end of its shaft and the plasticine cake. The wires from the m i c r o m o t o r should be fastened t o t h e inside w a l l of the glass w i t h adhesive or plasticine. Fig. 23 shows the setup ready for operation. F i l l the glass w i t h water. Then pour in a layer of sunflower oil 1-2 cm t h i c k . W h e n the wires from the m i c r o m o t o r are connected to a flashlight 3-01544
34
V. Mayer
b a t t e r y , the disc begins r o t a t i n g and causes the l i q u i d i n the glass t o circulate. This circulation disturbs the surface between the water and o i l , and a cone filled w i t h t h e oil soon forms. The cone grows u n t i l i t touches t h e disc, which then breaks t h e oil i n t o drops, t u r n i n g the l i q u i d t u r b i d . After the m i c r o m o t o r is shut off, the oil
Fig. 23
drops return to the surface where they reform a c o n t i n u o u s layer. The experiment can then be repeated. Figures 24a and 24b show photographs of the f o r m a t i o n of the air cone. W e modified the exp e r i m e n t s l i g h t l y here b y filling t h e glass w i t h water o n l y . 2. A n even more c o n v i n c i n g m o d e l of a tornado can be constructed by soldering a piece of copper wire (or a k n i t t i n g needle) about 25 cm i n length a n d 2 m m in diameter to the shaft of a m i c r o m o t o r . Solder a rectangular brass or t i n plate a b o u t 0.5 X 10 X 25 m m in size at right
Tornado Models
35
angles t o the wire (Fig. 25a). S w i t c h on the motor to check the operation of this stirrer. I f necessary, straighten its extended shaft (the wire) to minimize w o b b l i n g . Lower the stirrer vertically i n t o a glass of water 15-20 cm i n diameter and 25-30 cm h i g h . S w i t c h on the m o t o r . A cotie w i l l grow g r a d u a l l y
o n j t h e water surface to form a t o r n a d o t h a t extends to the r o t a t i n g p l a t e (Fig. 256, c, d). As the tornado touches the plate, m a n y air bubbles appear, s i g n i f y i n g a vortex around the p l a t e . If you hold the m o t o r in your h a n d , the t o r n a d o w i l l behave very m u c h like a l i v i n g creature. Y o u can spend hours w a t c h i n g its "predatory" surges. C o n t i n u e the experiment by p l a c i n g a wooden block on the water surface. The block w i l l be sucked in by the tornado. Try to adjust the rotation speed of the stirrer so that the block remains underwater at the same depth for a long t i m e . 3*
36
Felix Hess
The tornado w i l l suck, in bodies lying ]on the b o t t o m of the glass before the stirrer is switched on if their density is greater t h a n t h a t of water (which is not true of the wooden block). A l i g n the shaft of the m o t o r w i t h the axis of the glass. Y o u w i l l see a cone moving down the shaft and air bubbles w h i c h m a r k its continuation under the plate (Fig. 25e). I f you place some well washed river sand on the bottom of the
The Aerodynamics of Boomerangs
37
glass, you w i l l be able t o observe t h e structure of the tornado u n d e r the plate. These experiments show t h a t tornadoes are always caused b y a vortex in a l i q u i d or gas.
The Aerodynamics of Boomerangs * by Felix
Hess
I m a g i n e t h r o w i n g a piece of wood i n t o t h e air, m a k i n g it fly a r o u n d i n a large circle a n d h a v i n g it come to rest g e n t l y at your feet. Preposterous! Y e t of course t h i s is exactly w h a t a b o o m e r a n g does, provided t h a t i t has the right shape and is t h r o w n properly. 'As is well k n o w n , boomerangs originated a m o n g the a b o r i g i n a l i n h a b i t a n t s of A u s t r a l i a . A l t h o u g h boomerang-like objects h a v e been found i n other parts of the w o r l d as well (in E g y p t a n d I n d i a , for instance), these objects are not able to r e t u r n , as far as I k n o w . The reader m a y be a l i t t l e disappointed t o learn t h a t most Australian boomerangs also do not return. A u s t r a l i a n boomerangs can be r o u g h l y d i v i d e d i n t o t w o types: war boomerangs a n d return boomerangs. Those of the first t y p e are, as their name i m p l i e s , m a d e as weapons for fighting and h u n t i n g . A good war boomerang can fly m u c h farther t h a n an o r d i n a r y thrown stick, b u t i t does not r e t u r n . R e t u r n boomerangs, w h i c h exist i n m u c h smaller numbers, are used almost exclusively for p l a y . A c t u a l l y t h i n g s are not q u i t e as s i m p l e as this. There are m a n y k i n d s of a b o r i g i n a l weapons * An abridged version of an article that first appeared in the November issue of Scientific American for 1968.
38
Felix Hess
in A u s t r a l i a , a number of which look like boomerangs, so t h a t the d i s t i n c t i o n between boomerangs and t h r o w i n g or s t r i k i n g clubs is not a sharp one. Neither is t h e d i s t i n c t i o n between war boomerangs a n d return boomerangs. The shape of boomerangs can differ from t r i b e to tribe (Fig. 26). W h e t h e r a given boomerang belongs to the return type or not cannot always be inferred easily from its appearance. R e t u r n boomerangs, however, are usually less massive and have a less obtuse angle between their t w o arms. A t y p i c a l return boomerang m a y be between 25 and 75 centimeters long, 3 to 5 centimeters wide and 0.5 to 1.3 cm t h i c k . The angle between the arms m a y vary from 80 to 140 degrees. The weight may be as m u c h as 300 grams. The characteristic banana-like shape of most boomerangs has h a r d l y a n y t h i n g t o do w i t h their a b i l i t y t o return. Boomerangs shaped l i k e the letters X, V, S, T, R, H, Y (and probably other letters of the a l p h a b e t ) can be m a d e to return q u i t e well. The essential t h i n g is the cross section of the arms, w h i c h should be more convex on one side t h a n on the other, l i k e the w i n g profile of an airplane (see F i g . 27). I t is only for reasons of s t a b i l i t y t h a t the overall shape of a boomerang must lie more or less i n a plane. Thus if y o u m a k e a boomerang out of one piece of n a t u r a l wood, a smoothly curved shape following the grain of the wood is perhaps the most obvious choice. I f you use other m a t e r i a l s , such as p l y w o o d , plastics or metals, there are considerably more possibilities. H o w does one throw a return boomerang? As a rule it is taken w i t h the right h a n d by one of
Felix Hess
40
i t s extremities and held v e r t i c a l l y u p w a r d , the more convex, or upper, side to t h e left. There are t w o possibilities: either t h e free extremity points f o r w a r d — a s is the practice a m o n g the Austral i a n s — o r i t p o i n t s b a c k w a r d . T h e choice depends entirely on one's personal preference. Next, the
Fig. 27
r i g h t arm is brought b e h i n d t h e shoulder and t h e b o o m e r a n g is t h r o w n forward i n a h o r i z o n t a l or s l i g h t l y upward direction. For successful t h r o w i n g , t w o things are i m p o r t a n t . First, the p l a n e of t h e boomerang at t h e m o m e n t of its release s h o u l d be nearly v e r t i c a l or somewhat i n c l i n e d to the r i g h t , b u t certainly not h o r i z o n t a l . Second, t h e boomerang s h o u l d be given a r a p i d r o t a t i o n . This is accomplished b y s t o p p i n g the t h r o w i n g m o t i o n of t h e r i g h t arm a b r u p t l y just before t h e release. Because of its inertia the boomerang w i l l rotate m o m e n t a r i l y around a p o i n t situated i n the thrower's r i g h t h a n d . Hence i t w i l l acquire a forward v e l o c i t y and a rotationa l v e l o c i t y at the same t i m e . A t first t h e boomerang just seems to fly a w a y , b u t soon its p a t h curves to the left and often u p w a r d . Then i t m a y describe a w i d e , more or less circular loop and come d o w n somewhere near
The Aerodynamics of Boomerangs
41
the thrower's feet, or describe a second l o o p before d r o p p i n g t o the g r o u n d . Sometimes t h e second loop curves to the r i g h t , so t h a t t h e p a t h as a whole has the shape of a figure eight (Fig. 28). I t j i s a splendidf'sight if "the* b o o m e r a n g , " q u i t e near again after Mescribing' a^loop, loses speed,
Fig. 28
hovers some 3 meters above your head for a w h i l e and then slowly descends l i k e a h e l i c o p t e r . Every boomerang has its own characteristics w i t h respect t o ease of t h r o w i n g , shape of p a t h and hovering a b i l i t y . Moreover, one b o o m e r a n g can often describe very different orbits d e p e n d i n g on t h e w a y i t is t h r o w n . The precision of r e t u r n depends t o a large extent on the s k i l l of t h e thrower, who m u s t t a k e i n t o account such factors as the influence of w i n d . The greatest distance d u r i n g a flight m a y be 40 meters, b u t it can also be m u c h less or perhaps twice as m u c h ; t h e
42
Felix Hess
highest p o i n t can be as h i g h as 15 meters above the g r o u n d or as low as 1.5 meters. I have heard t h a t w i t h modern boomerangs of A u s t r a l i a n m a k e distances of more t h a n 100 meters can be a t t a i n e d , s t i l l followed by a perfect return, b u t I regret t o say t h a t so far I h a v e not been able to m a k e a boomerang go beyond a b o u t 50 meters. I n the foregoing general description it was t a c i t l y assumed t h a t t h e thrower was righthanded and used a "right-handed" boomerang. I f one were to look at an ordinary right-handed boomerang from its convex side w h i l e it was in flight, its direction of r o t a t i o n w o u l d be counterclockwise. Hence one can speak of the leading edge a n d the t r a i l i n g edge of each boomerang a r m . B o t h the leading and t h e t r a i l i n g edges of an a b o r i g i n a l boomerang are more or less sharp. The l e a d i n g edge of a modern boomerang arm is b l u n t , l i k e the leading edge of an airplane w i n g . S o m e t i m e s t h e arms h a v e a slight t w i s t , so t h a t their l e a d i n g edges are raised at the ends. The entire phenomenon m u s t of course be explained i n terms of the i n t e r a c t i o n of the boomerang w i t h the air; i n a v a c u u m even a boom e r a n g w o u l d describe n o t h i n g b u t a parabola. This interaction, however, is difficult to calculate exactly because of the complicated nature of the p r o b l e m . Let us nonetheless look at the matter in a s i m p l e way. I f one throws a boomerang in a horizontal direction, w i t h its plane of r o t a t i o n vertical, each boomerang arm w i l l " w i n g " the air. Because of the special profile of the arms the air w i l l exert a force on them directed from the flatter, or lower, side to the more convex, or upper, side
The Aerodynamics of Boomerangs
43
(Fig. 29). This force is the same as the l i f t i n g force exerted 011 the wings of an a i r p l a n e . I n a right-handed throw the force w i l l be directed from the right to the left as viewed by the thrower.
Lift (L)
Velocity (r)
Fig. 29
Fig. 30
This force alone, however, is not sufficient to m a k e a boomerang curve to the left. F o l l o w i n g one boomerang arm d u r i n g its mot i o n , one can see t h a t its velocity w i t h respect to the air is not constant. W h e n t h e arm p o i n t s u p w a r d , the forward velocity of the boomerang adds to the velocity due to the r o t a t i o n ; when it points d o w n w a r d , the t w o velocities are i n opposite directions, so t h a t the resultant speed w i l l be smaller or even vanish at some p o i n t s (see F i g . 30). Thus on the average the boomerang experiences not o n l y a force from the right to the left b u t also a torque acting around a h o r i z o n t a l axis, which tends to cant the boomerang w i t h
44
Felix Hess
its upper part to the left. A c t u a l l y this t u r n i n g over w i l l not be observed because the boomerang is s p i n n i n g r a p i d l y and hence behaves l i k e a gyroscope. N o w , a gyroscope (which really is n o t h i n g more t h a n a r a p i d l y s p i n n i n g flywheel) has the property t h a t , w h e n a torque is exerted on i t , it does not give w a y t o t h a t torque b u t changes its orientation around an axis t h a t is perpendicular to both the axis of rotation and the axis of the exerted torque; i n the case of a boomerang the orientation turns to the left. This m o t i o n is called precession. Thus the boomerang changes its o r i e n t a t i o n to the left, so t h a t its plane w o u l d m a k e a g r a d u a l l y increasing angle to its p a t h were it not for the r a p i d l y increasing forces t h a t t r y t o direct the p a t h p a r a l l e l t o the boomerang p l a n e a g a i n . The result is that, the p a t h curves t o t h e left, the angle between boomerang p l a n e and p a t h being kept very s m a l l . I n a c t u a l practice one often sees t h a t , a l t h o u g h the p l a n e of the boomerang is nearly vertical at the start of the flight, i t is a p p r o x i m a t e l y horiz o n t a l at the end. I n other words, the plane of the boomerang slowly turns over w i t h its upper part to the r i g h t ; the boomerang i n effect "lies d o w n " . Let us now consider the question i n more d e t a i l . Because m u c h is k n o w n a b o u t the aerodynamic forces on airfoils (airplane wings), it is convenient to regard each boomerang arm as an airfoil. L o o k i n g at one such w i n g , we see t h a t it moves forward and at the same t i m e rotates around the boomerang's center of mass. W e e x p l i c i t l y assume t h a t there is no m o t i o n perpendicular t o t h e p l a n e of the boomerang. W i t h a cross
The Aerodynamics of Boomerangs
45
boomerang (as] w i t h the rotor of a helicopter) the center of mass lies at the intersection of t h e wings, b u t t h i s is not the case w i t h an o r d i n a r y boomerang. Here one arm precedes t h e center of mass, whereas t h e other arm follows i t . W e call both arms "eccentric", the eccentricity being the distance from the arm to the center of mass. A preceding arm has a positive eccentricity, a following arm a negative eccentricity. A fixed p o i n t of a w i n g feels an airstream t h a t changes continuously i n m a g n i t u d e and direction w i t h respect to t h a t part of the w i n g . Sometimes the airstream m a y even blow against the t r a i l i n g edge of the w i n g profile, w h i c h can easily be imagined if one t h i n k s of a slowly r o t a t i n g boomerang w i t h h i g h forward velocity and looks at the arm p o i n t i n g d o w n w a r d . W h a t are the forces on an airfoil m o v i n g in this special manner? Let us first look at a simpler case: an airfoil m o v i n g in a straight line w i t h a constant velocity v w i t h respect to the air (Fig. 31). I t is customary to resolve the aerodynamic force i n t o two components: the l i f t L (perpendicular to v) and the drag D (opposite to v). These are b o t h proportional to v2. I f the spanwise direction of the w i n g is not perpendicular to the v e l o c i t y , v has a component p a r a l l e l to the w i n g t h a t has no influence; therefore we replace v by its component perpendicular to the w i n g , or the "effective velocity" V e f i . I n this case the forces are proportional to {V e ff) 2 . L o o k i n g at the boomerang arm a g a i n , i t is clear t h a t each p o i n t 011 the arm takes part i n the boomerang's forward velocity v. The velocity
Felix Hess
46
w i t h respect to the air due to the r o t a t i o n , however, is different for each p o i n t . For a rotational velocity 10 and a p o i n t at a distance r from the axis of r o t a t i o n (which passes t h r o u g h the boomerang's center of mass), t h i s velocity is cor. For each p o i n t on the arm one can reduce the velocities v and cor to one resultant velocity. Its component perpendicular to the arm is V e f j
Center mass
lo
Fig. 31
Fig. 32
(Fig. 32). Of course, the v a l u e of V e f f for a particu l a r p o i n t on the arm w i l l change continuously d u r i n g one period of r e v o l u t i o n . One assumes t h a t the c o n t r i b u t i o n s to the l i f t and drag of each part of a boomerang arm at each moment are again proportional to ( V e f f Y . C a l c u l a t i o n s were m a d e of the f o l l o w i n g forces and torques, averaged over one period of revolution: the average lift force L\ the average torque T, with its components 71, around an axis parallel to v (which makes the boomerang turn to the
The Aerodynamics of Boomerangs
47
left) and T2 a r o u n d an axis perpendicular t o v (which makes the boomerang "lie down") (Fig. 33); the average d r a g Z ) , w h i c h slows down the forward velocity v, a n d t h e average drag torque T D, which slows d o w n t h e r o t a t i o n a l v e l o c i t y co. I t turns out t h a t none of these q u a n t i t i e s except T 2 depends on t h e eccentricity for boomerang
Fig. 33
Fig. 34
arms t h a t are otherwise i d e n t i c a l ; T2 is exactly proportional to t h e eccentricity. The forces a n d torques acting on a boomerang as a whole are o b t a i n e d by a d d i n g their values for each of its arms. The c o n t r i b u t i o n s to T2 by arms w i t h opposite eccentricity m a y p a r t l y or completely cancel each other. N o w we come to t h e i m p o r t a n t question: H o w does a boomerang m o v e under the influence of these aerodynamic forces and torques (and of the force of g r a v i t y , of course)? As m e n t i o n e d earlier v
48
Felix Hess
the average torque T causes the gyroscopic precession of a boomerang. Let us take a closer look at the gyroscope. I f a gyroscope spins around its axis w i t h a r o t a t i o n a l v e l o c i t y co a n d one exerts a torque T on i t , acting around an axis perpendicu l a r to the spin axis, the gyroscope precesses around an axis perpendicular to b o t h the spin axis and the torque axis (Fig. 34). The angular velocity of the precession is called Q . A very s i m p l e connection exists between Q , to, T and the gyroscope's m o m e n t of inertia / , n a m e l y Q = T i l - ® . W e h a v e seen t h a t for a boomerang T is p r o p o r t i o n a l to coy, so t h a t the velocity of precession Q m u s t be p r o p o r t i o n a l to coiV/w, or vll. Hence the velocity of precession does not depend on co, the r o t a t i o n a l velocity of the boomerang. A n even more s t r i k i n g conclusion can be d r a w n . The velocity of precession is p r o p o r t i o n a l to v / I , the factor of p r o p o r t i o n a l i t y depending on the exact shape of the boomerang. Therefore one can write Q = cv, w i t h c a characteristic parameter for a certain boomerang. N o w let the boomerang have a velocity twice as fast; it t h e n changes the o r i e n t a t i o n of its p l a n e twice as fast. That imp l i e s , h o w e v e r , t h a t the boomerang flies through the same curve! T h u s , roughly speaking, the diameter of a boomerang's orbit depends neither on the rotat i o n a l velocity of t h e boomerang nor on its forward velocity. This means that a boomerang has its p a t h diameter more or less b u i l t i n . The dimensions of a boomerang's flight p a t h are proportional to the m o m e n t of inertia of t h e boomerang, and they are smaller if the profile of
The A e r o d y n a m i c s of
Boomerangs
49
1.110 arms gives more lift. Therefore if one wants a boomerang to describe a small orbit (for instance in a room), it should be made out of light material. For very large orbits a heavy boomerang is needed w i t h a profile giving not much lift (and of course as l i t t l e drag as possible). Now one has everything needed to form the equations of m o t i o n for a theoretical boomerang. These equations can be solved numerically on a computer, giving velocity, orientation and position of the boomerang at each instant. How do these calculated paths compare w i t h real boomerang flights? For an objective comparison it would be necessary to record the position of a boomerang during its flight. This could be done by means of two cameras. I n order to control the i n i t i a l conditions, a boomerangthrowing machine would be necessary. As yet I have had no opportunity to do such experiments, but I did manage to record one projection of experimental boomerang paths w i t h a single camera. I n the wing t i p of a boomerang a t i n y electric l a m p was mounted, fed by two small 1.5-volt cells connected in series, placed in a hollow in the central part of the boomerang (see Fig. 35). I n this way the boomerang was made to carry during its flight a light source strong enough to be photographed at n i g h t . Some of the paths recorded in this manner are shown in Fig. 36; calculated orbits are added for comparison in Fig. 37. Because the camera was not very far from the thrower, those parts of the trajectories where the boomerang was close to the camera appear exaggerated in the photographs. This effect of perspec4-01544
A Hydroflynamic M e c h a n i s m in a Palling Test, Tube
51
live was taken i n t o account in the a c c o m p a n y i n g calculated paths. The reader m a y decide for himself whether or not he finds the agreement between theory a n d experiment satisfactory. A t any rate, the general appearance and peculiarities of real boomerang paths are reproduced reasonably well by this theory.
A Hydrodynamic Median ism in a Falling Test Tube by G. I.
Pokrovsky
Fill a standard test, tube w i t h water, a n d , h o l d i n g it a few centimeters above the table t o p ,
let it drop v e r t i c a l l y (see Fig. 38). The surface of the table should be sufficiently hard to produce 4*
V. Mayor an elastic i m p a c t . D u r i n g i m p a c t , the meniscus of the water i n the test t u b e , w h i c h is n o r m a l l y concave because of c a p i l l a r y force, w i l l r a p i d l y level o u t , and a t h i n stream of water w i l l suddenly burst upward from the centre. Figure 38 si lows the water surface before i m p a c t (broken line) and after i m p a c t (solid line). The stream u p w a r d separates i n t o drops, and the uppermost drop reaches a height s u b s t a n t i a l l y higher than t h a t from w h i c h the t u b e is dropped. This indicates t h a t the energy i n t h e water is redistributed d u r i n g i m p a c t so t h a t a s m a l l fraction of water close t o t h e centre of the meniscus shoots out of the t u b e at h i g h v e l o c i t y . A device t h a t redistributes energy is called a m e c h a n i s m . Usually, t h i s word is applied t o solid parts (levers, toothed wheels, etc.), alt h o u g h there are l i q u i d a n d even gaseous mechanisms. The water i n the t u b e is just one example of such a m e c h a n i s m . H y d r o d y n a m i c mechanisms are especially imp o r t a n t w h e n very great forces t h a t cannot be w i t h s t o o d b y c o n v e n t i o n a l solid parts are involved. The force of explosive m a t e r i a l i n a cartridge, for example, can be p a r t i a l l y concentrated b y m a k i n g a concave c a v i t y i n the cartridge, w h i c h is lined w i t h a m e t a l sheet. The force of the explosion compresses the m e t a l and produces a t h i n m e t a l l i c jet whose velocity (if the shape of the l i n i n g is correct) m a y reach t h e escape velocity of a rocket. Thus, this modest experiment on a very s i m p l e phenomenon in a test t u b e relates to one of the most interesting problems of the h y d r o d y n a m i c s of u l t r a high speeds.
Aii Instructive E x p e r i m e n t with a C u m u l a t i v e .let
53
An Instructive Experiment with a Cumulative Jet by V. Mayer I n Professor G . I . Pokrovsky's s i m p l e a n d elegant experiment on the h y d r o d y n a m i c s i n a test tube, a t u b e p a r t i a l l y filled w i t h water is dropped from a few centimetres above a hard surface, t h u s p r o d u c i n g a jet of water from the tube u p o n i m p a c t . Since the water at the edges clings s l i g h t l y to the glass of the t u b e , t h e meniscus is concave. U p o n i m p a c t t h e t u b e and the water in it stop sharply, w h i c h causes t h e water to accelerate r a p i d l y . The water behaves as if it were very heavy and its surface levels out. The water around the edges resides, a n d a t h i n jet of water gushes out from the center of the tube for a short t i m e . Y o u can set up a s i m i l a r , perhaps even more striking experiment. Carefully cut off the b o t t o m of a test t u b e to m a k e a glass p i p e 15 m m i n diameter and about 100 m m long. Seal the fluted end of the p i p e w i t h a piece of rubber cut from a toy b a l l o o n . F i l l the pipe w i t h water a n d , covering the open end w i t h your finger, lower that end i n t o a glass of water. Remove your finger and adjust the p i p e so t h a t about a centimetre of water is left inside the pipe. The water inside the pipe should be level with the water surface in the glass. Fix the pipe vertically to a s u p p o r t . Now s l i g h t l y t a p the piece of rubber stretched over the pipe. A c u m u l a t i v e jet of water w i l l i m m e d i a t e l y rise inside the pipe and reach the piece of rubber itself.
54 V. Mayer
Figures .'!!) arid 40 show drawings of photographs taken at different moments during this experiment. They depict different stages of the formation and disintegration of various cumulative jets. The top two pictures show the jet proper; the bottom two depict the break-up of the jet into i n d i v i d u a l drops. Try to explain the results of this experiment b y comparing it with the one described by Pokrovsky. This setup is especially interesting because it allows us to observe the actual format i o n of a cumulative jet, which is more difficult i n the experiment w i t h a f a l l i n g test tube because the h u m a n eye is not fast enough to register the phenomena that take place during impact. Nevertheless, we advise you to return to the experiment, with the f a l l i n g tube once more to examine the details of the formation of the jet. W i t h this in mind we suggest you solve the following problems. EXERCISES 1. Determine whether the shape of the test tube bott o m affects stream formation. Does the stream develop because the bottom directs the shock wave i n the water? To answer this question, solder t i n bottom of any shape (plane or concave, for example) to a thin-wall copper pipe. Use these modified test tubes in the experiments described above to prove that the shape of the b o t t o m does not influence the formation of the stream. Thus, the results of this experiment cannot be explained as the direction of the wave by the bottom. 2. Determine whether it is necessary for the l i q u i d to wet the walls of the test tube. Place a small piece of paraffin inside a glass test tubp, and melt the paraffin over the flame of a dry fuel. Rotate the tuhe over the flame to coat the inside with a t h i n paraffin film. Now,
(a)
Fig. 39
Fig. 40
(A)
V. Mayer and E. Mamaeva
56
repeat the Pokrovsky experiment with this coated test tube. The cumulative jet should not form, which means that the walls of the tube must be wet for the experiment to work properly. 3. W h a t other experiments can be set up to obtain a cumulative jet in a tube that is stationary relative to the observer?
Magic with Physics by V. Mayer
and E.
Mamaeva
Take a glass pipe, one end of which is tapered l i k e t h a t of a pipette, and show the pipe to your audience. H o l d a glass of water (heated to 8090 °C) b y its r i m in your other h a n d , and show it to your audience, too. N o w , lower the tapered end of t h e p i p e i n t o the glass, and let the pipe fill w i t h water. Close the upper end of the pipe w i t h your finger and remove the p i p e from the glass (Fig. 41). Y o u r audience w i l l be able to see air bubbles appear at the lower end of the pipe. They grow, leave the walls of the p i p e , and rise t o the t o p of the p i p e . B u t the water stays in the pipe! N o w , e m p t y the pipe back i n t o the glass b y r e m o v i n g your finger from the upper end, and wave t h e pipe in the air several times before t a k i n g some more water. Close the upper end w i t h your finger again, and q u i c k l y p u l l the pipe out of t h e glass and t u r n it upside down (Fig. 42). A strong stream of water over a metre high will burst out of the pipe. A l t h o u g h the secret of this trick is very simple indeed, your audience is u n l i k e l y to guess it,
Magic with Physics
^
The glass contains water heated to 80-01) °C whereas the pipe is room temperature (about 20 °C). You should be able to explain why no water leaves the pipe at first without any hints.
M. Golubc.v and A. Kagalenko
58
The e x p l a n a t i o n for the powerful stream of water is more c o m p l i c a t e d . W h e n hot wafer enters the pipe from the glass, the air in the upper part of t h e pipe remains at room temperature because the p i p e conducts heat poorly. After you h a v e closed the upper o p e n i n g and turned the p i p e upside d o w n , the hot water streams downw a r d , h e a t i n g the air q u i c k l y . The pressure rises, a n d t h e e x p a n d i n g air shoots t h e r e m a i n i n g water out t h r o u g h the tapered end of the pipe. Use a glass p i p e 8-12 m m i n diameter a n d 30-40 cm long for this experiment. The smaller o p e n i n g should be a b o u t 1 m m i n diameter. Between tricks the p i p e should be well cooled (you can even blow t h r o u g h i t ) because the h e i g h t of the f o u n t a i n w i l l depend on the temperature difference between the air and the water i n the p i p e . The o p t i m a l a m o u n t of wafer in the p i p e fluctuates from 1/4 to 1/3 of its v o l u m e and can easily be determined e m p i r i c a l l y .
A Drop on a Hot Surface by M. Golubev
and A.
Kagalenko
T u r n i n g an iron upside down and levelling it h o r i z o n t a l l y , let a l i t t l e water drop on its hot surface. I f the temperature of the iron is s l i g h t l y over 100 °C, the drop w i l l diffuse as expected and evaporate w i t h i n a few seconds. I f , however, (lie iron is much hot lei' (300-350 °C,). something u n u s u a l w i l l h a p p e n : the drop w i l l bounce between ] and 5 m i l l i m e t r e s off the icon (as a ball bounces off the floor) and w i l l then move over
A Drop oil a H o t Surlaco
50
the hot surface without touching i t . The stability of such a state depends, first of all, on the temperature of the surface: the hotter the iron, the calmer the drop. Moreover, the "longevity" of the drop, the t i m e before it evaporates completely, increases m a n y times over. The rate of evaporation depends on the size of the drop. Larger drops shrink quickly to 3-5 m m , whereas smaller drops last longer, w i t h o u t noticeable changes. I n one of our experiments a drop 3 m m in diameter remained for about 5 minutes (300 seconds) before evaporating completely. W h a t is the explanation for this strange phenomenon? W h e n the drop first touches the heated surface, its temperature is about 20 °C. W i t h i n fractions of a second, its lower layer are heated to 100 °C, and their evaporation begins at so fast a rate that the pressure of the vapour becomes greater than the weight of the drop. The drop recoils and drops to the surface again. A few bounces are enough to heat the drop through to boiling temperature. I f the iron is well heated, the drop calms down and moves over the iron at a distance slightly above the iron. Obviously, the vapour pressure balances the weight of the drop in this condition. Once such a steady state is reached, the drop is fairly stable and can "live" a long time. W h e n the drop is small, its shape is roughly that of a sphere. Larger drops are vertically compressed. On a hot surface the drop seems to be supported by a vapour cushion. The reaction force that develops as a result causes the deformation of the drop. The larger the drop, the more noticeable the deformation.
60
I. Vorobiev
Oscillations, for e x a m p l e , compression, tension, or even more complex oscillation, m a y develop, especially in large drops (Figs. 43 and
Pig.
43
Fig. 44
44). The photograph in Fig. 43 shows a dark spot in the centre of the drop. This is v a p o u r b u b b l e . I n large drops several such bubbles m a y
Warfare Tension Draws a Hyperbola
CI
appear. O c c a s i o n a l l y , a drop assumes 1 lie shape of a ring w i t h a single, h i g v a p o u r b u h h l e i n the m i d d l e . W h e n t h i s occurs the e v a p o r a t i o n proceeds so intensively t h a t the drop shrinks v i s i b l y . Figure 44 shows one of the most interesting k i n d s of oscillations: a " t r i g o n a l " drop. Keep the f o l l o w i n g advice in m i n d when cond u c t i n g experiments like those described above. 1. The iron should be as smooth as possible, without scratches or irregularities. W h e n a drop runs i n t o such an irregularity, its life is considera b l y reduced. W h y ? 2. The i r o n s h o u l d be fastened t o a support h o r i z o n t a l l y . I n our experiments we used a t r i p o d for geodetical instruments. 3. Safety precautions should not be neglected. The conductors of the iron should be r e l i a b l y i n s u l a t e d , t a k e care not to scald your h a n d s w i t h boiling water.
Surface Tension Draws a Hyperbola by I.
Vorobiev
The coefficient of the surface tension of a l i q u i d can be determined b y measuring the rise of wett i n g l i q u i d i n a c a p i l l a r y tube. C a p i l l a r y tubes and a microscope for measuring their inner diameters are not always readily a v a i l a b l e , however. F o r t u n a t e l y , the tubes can be easily replaced w i t h two glass plates. Lower the plates i n t o a glass of water, and draw them together gradually. The water between them w i l l rise: it is sucked in by the force of surface tension (Fig. 45).
1. Vorolnov
TJio coefficient of surface tension o can easily be calculated from t h e rise of the water y a n d the clearance between the plates d. The force of the surface tension is F = 2 o L , where L is the length of the p l a t e ( m u l t i p l i e d by 2 because the water contacts both plates). This force re-
where
p
is
the
water
density.
Consequently,
2 oL = p Ldyg flence, t h e coefficient of the surface tension is o = l / 2 p gdy
(1)
A more interesting effect can be obtained by pressing the plates together on one side and leavi n g a s m a l l clearance on the other (Fig. 46). I n t h i s case t h e water w i l l rise, and the surface between t h e plates w i l l be very regular and smooth (provided t h e glass is clean and dry). I t is easy t o infer t h a t the vertical cross section of the surface is a h y p e r b o l a . A n d we can prove this by replacing d w i t h a new expression for the clearance in f o r m u l a
(1). Then d -- D j- follows
from the s i m i l a r i t y of the triangles (see Fig. 46).
Warfare Tension Draws a Hyperbola
CI
Here D is the clearance al the edge of the plates; L is the length of the plate, and x is the distance from the line of contact to the point where the clearance and the rise of the water are measured. Tims, o --= l / 2 p g y D
,
or 2 aL
1
,9*.
Equation (2) is really the equation of the hyperbola. The plates for this experiment should be about 10 cm by 20 cm size. The clearance on the open side should be roughly the thickness of a match
Fig. 47
Fig. 48
stick. Use a deep t u b like those photographers use in developing pictures to hold the water. For ease in reading the results, attach a piece of graph paper to one of the plates. Once we have a graph drawn by the water, we can check whether the curve is actually a hyperbola. A l l the rectangulars under the curve should have the same area (Fig. 47).
64 V. Mayer I f you have a t h e r m o m e t e r , you can s t u d y the dependence of surface tension of water temperature. Y o u can also s t u d y the influence of additives on water tension. The force of surface tension F is directed at right angles to the line of contact between the water and the glass (Fig. 48). The vertical component of the force is balanced b y the weight of the water c o l u m n . Try to e x p l a i n what balances its h o r i z o n t a l c o m p o n e n t .
Experiments with a Spoonful of Broth by V. Mayer The next time you are served b o u i l l o n for dinner, scoop u p a b i g spoonful, d o n ' t swallow it imm e d i a t e l y . Look carefully at the broth instead: 1'
Fig. 49
you w i l l see large drops of fat i n i t . Note the size of these droplets. N o w pour some of the broth back i n t o your soup b o w l , a n d look again at the broth i n the spoon. The drop of fat should h a v e diffused and gotten t h i n n e r b u t bigger in diameter. W h a t is the reason b e h i n d this phenomenon? First let us see under w h a t conditions a drop of fat can lie on the surface of the broth w i t h o u t diffusing. Look at F i g . 49. A drop of l i q u i d 2
Experiments with a Spoonful of Broth
65
(the fat) lies on t h e surface of l i q u i d 1 (the b r o t h ) . The drop is shaped roughly like a lens. The env i r o n m e n t 3 above t h e b o w l is a m i x t u r e of t h e vapours of l i q u i d s 1 and 2. M e d i a 1, 2, a n d 3 meet at the circumference of the drop. Isolate an increment of this circumference (close to p o i n t 0 i n F i g . 49) of A i length. Three forces of surface tension act u p o n this i n c r e m e n t . A t t h e interface of l i q u i d s 1 and 2, force F12 acts, tangential to the interface and equal i n m o d u l e t o I F
1 2
|
=O
1 2
AZ,
where o 1 2 is the surface tension at the interface of media 1 a n d 2. S i m i l a r forces F13 a n d F 2 3 act at the interfaces of 1 a n d 3: I Fi3 I =°- 13 AZ and 2 and 3: I
F23\=g23M.
Here o 1 3 a n d a 2 3 are the appropriate surface tensions. O b v i o u s l y , the drop reaches e q u i l i b r i u m if the total of all these forces equals zero F12 +
Fl3
+
F23
0
=
or their projections on coordinates X and Y (after the s u b s t i t u t i o n of appropriate absolute values and the cancellation of A I) are a
13 =
C0S
o l 2 sin 0J =
+
0
23
cos
(1)
o 2 3 sin 0 2
Here, 6X and 8 2 are the angles between the tangents to the surface of m e d i u m 2 a n d the surface 5-01544
66
V. Mayer
of m e d i u m 1. These angles are called angles of contact. I t follows from E q . (1) t h a t e q u i l i b r i u m of the drop is possible if the surface tensions are related as ^13 <
cri2 + o 2 3 .
Since surface phenomena i n a l i q u i d are practic a l l y independent of the gaseous e n v i r o n m e n t over i t , we assume t h a t =
Oi
and
a23 =
cr2.
W e call (i1 and cr2 surface tensions of l i q u i d s 1 a n d 2, respectively. I n this case these values refer to the surface tensions of b o t h the broth a n d the f a t . So, a drop of fat w i l l float on the surface of the broth w i t h o u t diffusing i f the surface tension of the broth is less t h a n the t o t a l of the surface tensions of the fat a n d the interface between the broth and the fat: <7l <
<*2 + ° 1 2
(2)
I f the drop is very t h i n (almost flat), 0X and 0 2 w i l l be s m a l l (0i = 0 2 = O) a n d the e q u i l i b r i u m c o n d i t i o n for the drop w i l l be Ol =CT2+ °12 W h e n Ox > o 2 + a 1 2 there are no angles 0 j a n d 0 2 for w h i c h E q . (1) w o u l d h o l d true. Therefore, l i q u i d 2 does not m a k e a drop on the surface of l i q u i d 1 in this case, b u t diffuses on its surface i n a t h i n layer. N o w let us try to e x p l a i n the results of our experiment w i t h a spoonful of b r o t h . Drops of f a t
Experiments with a Spoonful of Broth
07
float on t h e b r o t h , w h i c h means t h a t E q . (2) holds. W h y do they diffuse on the surface of t h e broth if t h e a m o u n t of broth is reduced? W h a t has changed? Since the surface tensions of the fat o 2 a n d t h e broth-fat interface a 1 2 r e m a i n unchanged, we h a v e to assume t h a t by p o u r i n g out some of the b r o t h , we change the surface tension of the broth o , . Hut the broth is water (to a first a p p r o x i m a t i o n ) . Can the surface tension of water be changed by s i m p l y decreasing the a m o u n t ? O b v i o u s l y n o t . The broth, however, is not p l a i n water b u t rather water covered w i t h a t h i n layer of f a t . B y p o u r i n g out some of the b r o t h , we reduce the a m o u n t of f a t , and its layer on the broth surface becomes t h i n n e r . This a p p a r e n t l y reduces the surface tension of the b r o t h and as a result, the fat drops diffuse. To test t h i s hypothesis, try the f o l l o w i n g experiment. P o u r some t a p water i n t o a clean saucer w h i c h has no traces of f a t . P u t a t i n y drop of sunflower o i l on its surface (a p i p e t t e or a clean refill from a ball-point pen can be used for the purpose). The first drop should diffuse completely over the surface, w h i l e the next drops do not diffuse b u t form a lens. Carefully pour out some of the water, a n d t h e drops w i l l diffuse a g a i n . N o w return to the experiment w i t h t h e spoonf u l of b r o t h . I f you watch the b e h a v i o u r of the fat closely, you w i l l notice t h a t the drops r u p t u r e and reunite. Figures 50-55 show p h o t o g r a p h of one such experiment. W e filled a clean glass dish w i t h t a p water coloured w i t h I n d i a n i n k t h a t contains no alcohol. W e placed eight very s m a l l drops of sunflower oil on the surface w i t h a t h i n glass t u b e (see F i g . 50). W h e n we sucked a s m a l l 5*
68
V. Mayer
Fig. 50
a m o u n t of water out w i t h a rubber b u l b , t h e drops enlarged. W h e n more water was removed, t h e drops became even larger and changed shape (because of the water currents). The b e g i n n i n g s
Experiments with a Spoonful of Broth
69
Fig. 52
Fig. 53
of future ruptures also appeared (see F i g . 51). Further modifications occued rrspontaneously. The ruptures grew and united into a single* large rupture. The drop became a~ring, w h i c h finally
M. Kliya
70
1 Fig. 54
broke and rearranged i n t o a new drop (Figs. 52-55). * Perhaps now you w i l l agree t h a t it is worthw h i l e w a t c h i n g your soup before p u t t i n g it i n t o your m o u t h !
How to Grow a Crystal
71
How to Grow a Crystal by M.
Kliya
Modern i n d u s t r y cannot do w i t h o u t a w i d e variety of crystals. Crystals are used i n watches, transistorized radioes, computors, lasers, and m a n y other machines. Even nature's enormous laboratory is no longer able to meet t h e d e m a n d s of developing technology, and special factories have appeared where various crystals, r a n g i n g i n size from very s m a l l crystals to large crystals weighing several k i l o s , are grown. The methods for growing crystals v a r y and often require h i g h temperatures a n d tremendous pressure (for example, when g r o w i n g artificial d i a m o n d s ) . B u t some crystals can be grown even i n your h o m e laboratory. The simplest crystals to grow at h o m e are potash a l u m crystals, KA1(S04)2-12H20. This absolutely harmless substance is w i d e l y available (alums are occasionally used to p u r i f y tap water). Before g r o w i n g our own crystals, however, let us t a k e a closer look at the process itself. W h e n a substance is dissolved i n water at a constant temperature, dissolution stops after a certain t i m e , and such a solution is said to be saturated. S o l u b i l i t y refers to t h e m a x i m a l quant i t y of the substance t h a t dissolves at a given temperature in 100 grams of water. N o r m a l l y , s o l u b i l i t y rises w i t h a rise in temperature. A sol u t i o n t h a t is]j saturated at one temperature becomes u n s a t u r a t e d at a higher temperature.
M. Kliya
72
I f a saturated s o l u t i o n is cooled, the excess of the substance w i l l precipitate. Figure 56 shows the dependence of potash a l u m s o l u b i l i t y on temperature. According t o the g r a p h , if 100 grams of a s o l u t i o n saturated at 30 °C are cooled to 10 °C, over 10 grams of t h e substance should precipi-
10° 20° 30° S o l u t i o n temperature
pig. 56
t a t e . Consequently, crystals can be grown by c o o l i n g a saturated s o l u t i o n . Crystals can also be grown b y e v a p o r a t i o n . W h e n a saturated s o l u t i o n evaporates, its volu m e decreases, w h i l e t h e a m o u n t of dissolved substance remains u n c h a n g e d . The excess of substance t h u s produced falls as a precipitate. To see h o w t h i s occurs, heat a saturated s o l u t i o n , a n d t h e n cover t h e jar c o n t a i n i n g t h e unsaturated sol u t i o n w i t h a glass p l a t e a n d a l l o w it to cool t o a temperature below the s a t u r a t i o n temperature. The substance m a y not p r e c i p i t a t e w i t h this method, i n w h i c h case we w i l l be left w i t h a supersaturated s o l u t i o n . This is because we need a seed, a t i n y crystal or even a speck of t h e same sub-
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stance, to form a crystal. U s u a l l y , m u l t i p l e s m a l l crystals can be generated s i m p l y b y s h a k i n g the jar or r e m o v i n g the cover. To grow large crystals, the n u m b e r of seed crystals should be l i m i t e d . As an artificial seed, a crystal grown earlier works best of a l l . The seed crystal can be grown as follows. Take t w o clean glass jars. P o u r w a r m water i n t o one of t h e m , a n d t h e n add a l u m . S t i r t h e mixture, a n d w a t c h the dissolution process closely. W h e n d i s s o l u t i o n stops, carefully d r a i n t h e sol u t i o n i n t o another jar, t a k i n g care not to pour any of t h e undissolved substance i n t o t h e second jar. Cover t h e jar w i t h a glass p l a t e . W h e n the s o l u t i o n has become cool, remove t h e p l a t e . After a short t i m e , y o u should see m a n y crystals i n the j a r . Let t h e m t a k e their t i m e t o grow, before selecting t h e ^largest as a seed crystal. N o w we are ready t o grow our o w n crystal. First of a l l , we need proper glassware. To remove undesirable nuclei from the w a l l s of the vessels, sterilize t h e m over the spout of a b o i l i n g teapot. Then m a k e another w a r m saturated sol u t i o n i n one jar, and d r a i n it i n t o another. H e a t this w a r m saturated solution of a l u m a l i t t l e more. Then cover the jar w i t h a p l a t e , a n d set it aside t o cool. As the temperature of t h e solut i o n approaches the saturation temperature, lower t h e seed crystal you m a d e earlier i n t o the jar. Since the s o l u t i o n is s t i l l u n s a t u r a t e d , the seed crystal w i l l begin to dissolve. B u t as soon as the temperature drops to the s a t u r a t i o n p o i n t , t h e seed crystal stops dissolving and starts growi n g . (If your seed crystal dissolves completel y , introduce another crystal.) The crystal w i l l
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c o n t i n u e to grow once t h e solution has cooled, i f you l i f t the cover a n d let t h e water evaporate. D o not let dust enter t h e jar. G r o w t h w i l l cont i n u e for t w o or three days. W h e n growing crystals t r y not t o m o v e or touch the jar. After a crystal has developed, re-
Fig. 57
m o v e it from the s o l u t i o n , and dry i t carefully w i t h a paper n a p k i n so i t w i l l retain its shine. The crystals develop differently, depending on whether the seed crystal is placed on the b o t t o m of t h e jar or suspended from a thread (Fig. 57). Y o u can even grow 'a necklace' by r u n n i n g a thread several times over the seed crystal before suspending i t i n the s o l u t i o n . G r o w i n g crystals is an art, and you m a y not be completely successful r i g h t away. D o not get d i s a p p o i n t e d . W i t h a l i t t l e persistence and care, you can produce b e a u t i f u l ' crystals.
Crystals Made of Spheres by G.
Kosourov
Before we can predict, e x p l a i n , or understand t h e properties of a crystal, we m u s t determine its
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structure. I f we k n o w the arrangement and the s y m m e t r y of a t o m s i n the crystal l a t t i c e , we can tell whether t h e crystal is piezoelectric, i.e., whether a certain voltage is generated at its edges under m e c h a n i c a l compression, whether the crystal is capable of ferroelectric t r a n s i t i o n , w h i c h develops at a specific temperature and is characterized b y t h e f o r m a t i o n of an i n t r i n s i c electric field, whether t h e crystal generates a l i g h t wave of double frequency when t r a n s m i t t i n g a laser beam, and so on. The structure of the crystal carries a b u n d a n t i n f o r m a t i o n about the crystal itself. Different a t o m i c arrangements i n crystals can be studied w i t h a few s i m p l e props. W e can b u i l d models of crystals, f o l l o w i n g the principles used i n nature w i t h o r d i n a r y b a l l bearings. E v e n crystallographers use these three-dimensional models, w h i c h clearly show the specifics of a t o m i c arrangements i n complex structures. Before starting our experiments, we should m a k e a few theoretical observations. Crystal lattices result from the i n t e r a c t i o n of a t o m i c forces. W h e n the atoms are close together, repulsive forces p r e v a i l and increase s h a r p l y w i t h attempts to b r i n g the atoms together. A t t r a c t i v e forces p r e v a i l at greater distances and decrease rather g r a d u a l l y w i t h distance. W h e n a t o m s are drawn together b y attractive force, the p o t e n t i a l energy of t h e i r interaction decreases, just as the potential energy of a f a l l i n g stone decreases. This p o t e n t i a l energy is m i n i m a l at the p o i n t at w h i c h attractive and repulsive forces are e q u a l , and it increases s h a r p l y as the atoms draw closer. The dependence of p o t e n t i a l energy on distance
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is shown i n F i g . 58. I n e q u i l i b r i u m , atoms assume places of m i n i m a l p o t e n t i a l energy. I f there are m a n y atoms, t h i s tendency leads t o the repeated f o r m a t i o n of the most energy-efficient configuration of a s m a l l group of atoms. This configuration is called t h e u n i t cell. Some substances h a v e very complex structures. The u n i t cell of some silicates, for example,
Fig. 58
Fig. 59
contains over 200 atoms. Other substances, m a n y metals, for example, form their crystal l a t t i c e b y a very s i m p l e a l g o r i t h m . N a t u r a l l y , we shall start from the simplest coordinations. I n our experiments a t o m s are represented b y m e t a l spheres. E l a s t i c forces, developing as a result of the c o n j u g a t i o n of the spheres, serve as the repulsive force, and the a t t r a c t i v e force is provided b y gravity. Stretch a t h i n piece of rubber (this m a y be a piece of a surgical glove) over the opening of a jar, box, or section of p i p e , and fasten it w i t h a rubber b a n d . Place t w o spheres on the piece of
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rubber. The rubber w i l l sag s l i g h t l y , a n d t h e spheres w i l l be attracted to one another. Their pot e n t i a l energy, w h i c h depends on distance, is shown a p p r o x i m a t e l y i n F i g . 59 and is very s i m i l a r t o the dependence i n F i g . 58. I f we p u t a b o u t 30 spheres on t h e piece of rubber and shake them s l i g h t l y , they w i l l arrange themselves in regular rows (Fig. 60). The centres of the spheres w i l l lie
Fig. 60
at the apexes of equilateral triangles, every side of w h i c h equals the diameter of t h e sphere. The spheres w i l l fill the w h o l e plane f o r m i n g a lattice called a hexagonal (Gr. hex six a n d gonia a corner, angle). Each sphere is surrounded b y six spheres t o u c h i n g each other. Their centres form regular hexagons. I f you t u r n t h e lattice one-sixth of a r e v o l u t i o n around a n axis t h r o u g h the centre of a n y one of the spheres, some of t h e spheres w i l l change places, b u t t h e overal arrangement of t h e system i n space w i l l r e m a i n t h e same. The l a t t i c e of spheres
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w i l l translate itself i n t o the original p o s i t i o n . After six such rotations each sphere resumes its o r i g i n a l p o s i t i o n . I n such cases crystallographers say t h a t the axis of s y m m e t r y of the sixth order, w h i c h is oriented perpendicular t o the plane of the l a t t i c e , passes through the centre of each sphere. This axis makes the lattice "hexagonal". I n addit i o n to the s y m m e t r y axes of the sixth order, third-order s y m m e t r y axes pass through the centres of the holes formed b y n e i g h b o u r i n g spheres. (The third-order s y m m e t r y axis is a straight l i n e ; each t i m e t h e l a t t i c e is rotated 120 degrees around t h i s axis, t h e o r i g i n a l configuration repeats. I r r e g u l a r l y shaped bodies have first-order s y m m e t r y axes since they return t o the o r i g i n a l p o s i t i o n after a single complete r e v o l u t i o n . Conversely, the s y m m e t r y axis of infinite order passes t h r o u g h the p l a n e of a circle at r i g h t angles since the circle translates itself i n t o the o r i g i n a l position at an i n f i n i t e l y s m a l l angle of rotation.) The f o l l o w i n g discussions w i l l be clear o n l y if you h a v e spheres for b u i l d i n g models of different crystals. Consider the holes on either side of a row of spheres (Fig. 61). Since the n u m b e r of holes i n either row equals t h e n u m b e r of spheres i n a row, an infinite l a t t i c e of spheres has twice as m a n y holes as spheres. The holes form t w o hexagonal lattices, s i m i l a r t o those formed b y the centres of the spheres. These three lattices are shifted relative t o one another i n such a w a y t h a t t h e sixth-order s y m m e t r y axes of each l a t t i c e coincide w i t h the third-order s y m m e t r y axes of the other two lattices. The second layer of spheres fills one of t h e lat-
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tices of holes t o form a hexagonal l a t t i c e of contiguous spheres, s i m i l a r t o the first such l a t t i c e . The e l a s t i c i t y of t h e piece of r u b b e r , however, m a y not p r o v i d e enough a t t r a c t i v e force t o h o l d the spheres of the second and t h i r d layers. Therefore, since we k n o w h o w t h e spheres in t h e bott o m layer lie, let us m a k e an e q u i l a t e r a l t r i a n g l e from p l y w o o d ( F i g . 62), the sides of w h i c h equal a n integer n u m b e r of spheres (seven i n our m o d e l ) , a n d fill i t w i t h spheres of t h e b o t t o m layer.
Fig. 61
Fig. 62
The second layer of spheres can fill a n y l a t t i c e of holes, b u t w h e n we reach t h e t h i r d layer, we find the holes are not e q u i v a l e n t . The centres of one of the lattices are arranged over t h e centres of t h e spheres i n t h e b o t t o m layer, whereas t h e second lattice lies over t h e v a c a n t holes of t h e b o t t o m layer. W e s h a l l begin b y p u t t i n g spheres i n t o the holes above t h e spheres of t h e b o t t o m layer. I n this case t h e t h i r d layer w i l l be a n exact replica of t h e b o t t o m arrangement of spheres; the f o u r t h layer w i l l repeat t h e second layer, a n d so on. E a c h layer w i l l repeat itself every second
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layer. O u r p y r a m i d w i l l be rather fragile (Fig. 63) because the " a t t r a c t i v e " force i n our m o d e l acts downwards o n l y , a n d the spheres i n t h e holes along the edges are easily pressed out b y the spheres of the layers above. This k i n d of p a c k i n g , called hexagonal closest p a c k i n g , is t y p i c a l of b e r y l l i u m , m a g n e s i u m , cadm i u m , and h e l i u m crystals at low temperatures and pressures over t w e n t y five atmospheres. This p a c k i n g has only one system of closely p a c k e d , parallel layers. The third-order s y m m e t r y axis, w h i c h passes through t h e centre of each sphere, is perpendicular to t h i s system of layers. The symm e t r y order is thus reduced: w h e n t h e axis passes t h r o u g h centres of the even-layered spheres, the lattice has the sixth-order s y m m e t r y ; t h e same axis passes t h r o u g h t h e centres of t h e holes i n the odd layers, a n d t h e s y m m e t r y of t h e odd layers, relative to t h i s axis, is, therefore, o n l y of the t h i r d order. Nevertheless, this p a c k i n g is called hexagonal, because i t can be viewed as t w o hexagonal lattices of even a n d odd layers. N o t e also t h a t the e m p t y holes i n a l l layers are arranged one over the other a n d channels pass through t h e whole hexagonal structure, i n t o w h i c h rods, whose diameter is 0.155 t h a t of the diameter of t h e sphere can be inserted. The centre lines of these channels are the third-order s y m m e t r y axes. Figure 63 shows the m o d e l of a hexagonal structure w i t h rods placed i n the channels. N o w let us p u t t h e spheres from the t h i r d layer i n t o the holes above the v a c a n t holes of the bott o m layer. W e can construct t w o different pyram i d s (Figs. 64 and 65), depending on the system
6-01544
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of holes we select for t h e second layer of spheres. The faces of the first p y r a m i d are e q u i l a t e r a l triangles w i t h hexagonal p a c k i n g , which do not differ from the p a c k i n g of the b o t t o m layer of t h e p y r a m i d . I n other words, our p y r a m i d is a tetrahedron, one of the five possible regular polyhedrons. This p a c k i n g has four families of closely packed layers, whose normals coincide w i t h the third-order s y m m e t r y axes of the tetrahedron w h i c h pass t h r o u g h its apexes. I n such a p a c k i n g the layers repeat every t h i r d layer. The lateral faces of the second p y r a m i d are isosceles triangles, and t h e p y r a m i d itself is part of a cube, intercepted b y the p l a n e formed b y the diagonals of the faces w i t h a c o m m o n apex (Fig. 66). The p a c k i n g i n t h e lateral faces of such a p y r a m i d forms a square lattice w i t h its rows parallel to the diagonals of the cube face. O b v i o u s l y , we h a v e o n l y one type of p a c k i n g w i t h t w o different orientations. I f we remove the spheres from the edges of the tetrahedron, the faces of the cube w i l l emerge. Conversely, by rem o v i n g the spheres t h a t m a k e u p the edges of the cube, we t u r n the cube i n t o a tetrahedron. This p a c k i n g , called cubic closest p a c k i n g , is characteristic of neon, argon, copper, g o l d , plat i n u m and lead crystals. C u b i c closest p a c k i n g possesses a l l the elements of cubic s y m m e t r y . I n p a r t i c u l a r , the third-order s y m m e t r y axes of the tetrahedron coincide w i t h the space cube diagonals, w h i c h are also third-order s y m m e t r y axes for the cube. This p a c k i n g is based on a cube of fourteen spheres. E i g h t of the spheres form t h e cube, and six form the centres of its faces. I f you look closely at the second p y r a m i d
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(Fig. 66), y o u w i l l find" this elementary cube at the apex. C u b i c closest packing can be viewed as a c o m b i n a t i o n of four s i m p l e cubic lattices. The e q u a l i t y of a l l the spheres of t h e p a c k i n g is especially noticeable from t h i s angle. Since the hexagonal a n d cubic closest packings can be produced b y superimposing hexagonal layers onto one another, the t w o obviously h a v e the same density, or space factor, despite t h e difference in symmetry.
Fig. 66
Fig. 67
If we b u i l d a square form a n d place spheres in a square lattice, we s h a l l get a n o t h e r close packi n g . A l t h o u g h the spheres i n each layer are not packed i n t h e closest possible w a y , t h e holes between t h e m are deeper, a n d , therefore, the layers lie closer t h a n i n a hexagonal structure. I f we complete t h e p a c k i n g , we w i l l get a tetrahedral p y r a m i d (Fig. 67) whose side faces are equilateral triangles i n which the spheres are packed hexagonally. I f we add another such p y r a m i d w i t h its apex d o w n w a r d , we s h a l l get a t h i r d polyhedron (after the tetrahedron a n d the cube): an octahedron w i t h eight faces. O b v i o u s l y , t h i s 6«
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is also the cubic closest p a c k i n g w i t h the faces of t h e cube parallel t o t h e p l a n e of the base. Rem o v e the spheres f o r m i n g the edges, and you w i l l find five spheres i n t h e upper intercept plane) w h i c h form the face of an elementary cube. These models can be used i n a n u m b e r of physical experiments. B y s h a k i n g the piece of rubber,
Fig. 68
Fig. 69
for example, you can s i m u l a t e the heat-induced m o t i o n of atoms. ( Y o u w i l l see how 'a rise i n temperature' destroys the p a c k i n g of the spheres.) Since eac' hexagonal layer occupies relatively shallow holes of t h e next layer, the layers are loosely b o u n d , and slippage develops easily. I f you slide one hexagonal layer against another, you w i l l see t h a t easy slippage, i n w h i c h the layers m o v e as a whole, occurs i n three directions. A s i m i l a r s i t u a t i o n can be observed i n real crystals, w h i c h explains the specifics of plastic def o r m a t i o n i n crystals. Models can be b u i l t from any k i n d of sphere. I f you do not h a v e b a l l bearings, use large necklace beads or even s m a l l apples. Figures 68 and 69 show u n i t cells of cubic and hexagonal
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packings m a d e of ping-pong b a l l s glued together. Ping-pong b a l l s , w h i c h are r e a d i l y a v a i l a b l e a n d m a k e w o r k a b l e models, are h i g h l y recommended, especially for junior-high and high-school physics classes. V a r i o u s packings of contiguous b a l l s are import a n t i n crystallography, a n d we s h a l l discuss t h e m a g a i n . M e a n w h i l e , get some b a l l s a n d b u i l d models!
A Bubble Model of Crystal by Ya,
Geguzin
On Simulation f I n t h e difficult process of i n t e r p r e t i n g experim e n t a l facts or theoretical propositions, almost everyone needs a n image, a v i s i b l e presentation, a simplified m o d e l of the subject. Perhaps one of the m o s t i m p o r t a n t skills a scholar or teacher needs is t h e a b i l i t y to construct images, analogies, a n d models, w h i c h can i l l u s t r a t e certain physical p h e n o m e n o n a n d , t h u s , enlarge our u n d e r s t a n d i n g of t h e m . W h a t s h o u l d such a model be? W h a t m u s t it be able t o show? W h a t can one expect from the m o d e l a n d w h a t are its constraints? First, we expect our m o d e l t o be a learning a i d . I t must c o n t a i n no false d a t a b u t m u s t i n c l u d e at least a fraction of the t r u t h p e r t i n e n t t o the subject. I n everyday life, of course, we scorn h a l f t r u t h . B u t ' h a l f t r u t h ' is a term of h i g h a p p r o b a t i o n i n relation to models. F i n a l l y , t h e m o d e l s h o u l d be clear a n d easily comprehensible w i t h o u t d u l l c o m m e n t a r y .
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The very best model needs no explanation at all since its clarity gives it the force of a proof. There are many convincing and elegant models in physics, particularly i n solid state physics. I n this article we shall discuss a 'live' model, which illustrates and reflects the structure flaws, and complicated interactions of real crystal very well. This is not a new model. I t was conceived by the outstanding British physicist L . Bragg in the early 1940s and realized by Bragg and his colleagues V . Lomer and D . Naem. Therefore, we shall call it the B L N model after Bragg, Lomer, and Naem. W h a t Do W e W a n t to Simulate? The answer is clear: real crystal. Real crystal is a vast set of identical atoms or molecules arranged in strict order to form a crystal lattice. Occasionally, this order is disturbed, signifying the presence of defects in the crystal. Another very important characteristic of crystals is the interaction of the atoms forming the crystal. W e w i l l discuss this interaction a bit later. Now we w i l l simply state that they do interact! W i t h o u t interaction, the atoms would form a heap of disorderly arranged atoms rather than a crystal. The maintenance of order in crystals is a direct consequence of this interaction. Another widely used model is the so-called dead model of crystal, i n which wooden or clay balls are bound by straight wires. The balls represent atoms, and the wires are the symbols of their bonds in 'frozen' state. The model is 'dead' since it 'freezes' the interaction of the atoms. I n
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this reasonable and very useful m o d e l , different k i n d s of a t o m s are represented b y b a l l s of different colour and size, and wires of different l e n g t h represent t h e distance between t h e a t o m s . Alt h o u g h the m o d e l does not reflect a l l t h e t r u t h about t h e crystal, it' does convey t h e t r u t h w i t h out false assertions. The m o d e l , of course, cannot depict the m o t i o n of the a t o m s , b u t i t reflects the order of t h e i r p o s i t i o n very clearly. The dead m o d e l is a n o u t s t a n d i n g aid i n d e p i c t i n g t h e space arrangement of atoms or i n i d e n t i f y i n g t h e most l i k e l y directions of d e f o r m a t i o n or electric current i n a crystal. The m o d e l is i n d i s p e n s a b l e i n representing t h e possible arrangement of atoms i n unidentified or l i t t l e s t u d i e d crystals on t h e basis of experimental d a t a a n d so-called general considerations. This t e c h n i q u e of modeli n g w i t h balls a n d wires aided i n one of t h e m o s t i m p o r t a n t discoveries of t h e t w e n t i e t h c e n t u r y , the i d e n t i f i c a t i o n of the structure of the DNA molecule, w h i c h certainly speaks well for t h e usefulness of the dead m o d e l ! ^ O u r objective, however, is t o s i m u l a t e a ' l i v e ' rather t h a n a ' d e a d ' crystal. O b v i o u s l y , t o do so, we need t o s i m u l a t e the i n t e r a c t i o n of atoms i n crystal, t o r e v i t a l i z e the i n t e r a c t i o n t h a t is frozen i n t h e wires and balls.
The Interaction of A t o m s i n Crystal Perhaps t h e most i m p o r t a n t characteristic of such i n t e r a c t i o n stems directly from t h e s i m p l e fact t h a t t h e distance between t w o n e i g h b o u r i n g atoms i n real crystal has a definite v a l u e at a constant temperature. ( W e are speaking, of course,
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about the distance between the positions around which atoms fluctuate i n heat induced motion. The amplitude of these fluctuations is considerably smaller than the distance between atoms.) The distance has a definite value since if we try to stretch i t , the atoms resist the effort and attract one another, and if we try to reduce the distance, the atoms repel each other. The fact that this distance is determinate allows us to conclude that atomic interaction is characterized by attraction and repulsion simultaneously. A t a certain distance between atoms (we call it determinate), the forces of attraction and repulsion become equivalent i n absolute values. The atoms i n the lattice are located at exactly this distance. I t would be useful to be able to simulate the competition of attractive and repulsive forces. Such a technique would revitalize atomic interaction in crystal. The authors of the B L N model created just such a method. Instead of wooden and clay balls, they used t i n y soap bubbles. The Interaction of Soap Bubbles on Water Surface Two soap bubbles on the surface of a body of water are not indifferent to each other: they are first attracted to one another but, after touching, are repelled. This phenomenon can be observed in a very simple experiment, for which we w i l l need a shallow bowl, a needle from a syringe, the inner balloon from a volley ball, and an adjustable elamp to control the compression of the nozzle of the balloon. F i l l the bowl w i t h
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soapy water almost t o the t o p , a n d a d d a few drops of glycerol t o s t a b i l i z e t h e b u b b l e s t h a t we w i l l b l o w onto t h e surface of t h e water. Inflate t h e b a l l o o n . Then c l a m p the nozzle, a n d insert t h e needle i n t o i t (the b u t t end first, of course). Immerse t h e other end of t h e needle i n t o t h e water (not deeply) a n d release the c l a m p i n g pressure s l i g h t l y (Fig. 70). The air t h a t escapes t h e needle at regular intervals w i l l develop i n t o i d e n t i c a l
Fig. 70
soap b u b b l e s . W e s h a l l need m a n y such bubbles i n f u t u r e experiments, b u t for t h i s first experim e n t , t r y t o m a k e o n l y two b u b b l e s , some distance from one another. I f you are not successful i m m e d i a t e l y , t r y a g a i n . Y o u s h o u l d succeed b y the f o u r t h or fifth t i m e , at least. B u b b l e s 1-2 m m i n diameter work the best. Once t h e b u b b l e s are m a d e , y o u can w a t c h their m o v e m e n t s . The bubbles w i l l m o v e ( w i t h o u t our interference) towards one a n o t h e r , slowly at first a n d t h e n more r a p i d l y . W h e n t h e y c o l l i d e , they do n o t t o u c h at a single p o i n t b u t m a k e dents on the surface. The interaction of a p a i r of ident i c a l b u b b l e s w i l l v a r y from t h a t of a p a i r of bubbles of different size. W a t c h ! N o w let us consider t h e o r i g i n of t h e force t h a t drives t h e b u b b l e s together s p o n t a n e o u s l y . T h i n k
90
Va. Geguzin
of t w o matchsticks l y i n g on the surface of the water. Since t h e b u b b l e s a n d the matchsticks are b o t h soaked b y t h e water, t h e nature of t h e i r interaction is generally the same. Two bubbles floati n g close together form a very complex surface w i t h the water, however, whereas t h a t of t h e t w o matchsticks is m u c h simpler a n d , t h u s , easier to study (Fig. 71). The force t h a t brings the
Tl7 r
^ — VL
— 2r Fig. 71
t w o floating m a t c h s t i c k s together develops as follows. W a t e r soaks the matchsticks, a n d t h e surface of the water near t h e sticks is, therefore, curved. This curvature generates force t h a t acts u p o n the l i q u i d . The force is determined b y surface tension a n d directed, i n t h i s case, u p w a r d (we s h a l l assume t h e matchsticks are c o m p l e t e l y soaked). Under the effect of this force, t h e l i q u i d rises along t h e sides of t h e m a t c h s t i c k s , t h e rise being more pronounced in the region between t h e sticks (Fig. 71). The l i q u i d appears t o stretch, a n d t h e pressure in the l i q u i d drops r e l a t i v e t o the atmospheric pressure, b y an a d d i t i o n a l pressure A p = ^r- = — , where o is the coefficient of ^ d r surface tension, d is t h e distance between t h e m a t c h s t i c k s , and r = d!2 is the curvature r a d i u s of the surface of the l i q u i d . Consequently, t h e absolute v a l u e of t h e pressure of the l i q u i d on t h e matchsticks i n t h e area between them is less t h a n
A Bubble Model of Crystal
91
the atmospheric pressure t h a t acts u p o n t h e sticks from the outside. Thus, the absolute v a l u e of t h e force t h a t draws t h e sticks together is in.
a
c
2o
|F| = A p S = —
4a 2 l
,,
M =
t
t
j
y
1
f
~ . y
From t h i s we can predict an interesting phenomenon. Since 1 Id2, i n a viscous e n v i r o n m e n t the m a t c h s t i c k s should draw together w i t h a ve-
locity t h a t increases as the distance between t h e m decreases. The bubbles also accelerate as they draw closer (Fig. 72). W e filmed the m o v e m e n t of the bubbles in our laboratory b y p l a c i n g a m o v i e camera over t h e b o w l c o n t a i n i n g t h e soap solut i o n . As soon as the bubbles started m o v i n g , we switched t h e camera on (Fig. 73). W e were able to w a t c h t h e bubbles d r a w i n g together r i g h t u p to their collision. Once they h a v e c o l l i d e d , a repulsive force starts acting. The force is caused b y an increase of gas pressure i n t h e m u t u a l l y compressed bubbles (Fig. 74), w h i c h pushes the bubbles a p a r t . Soap bubbles are apparently s u i t a b l e crystal models, if we create a number of i d e n t i c a l bubbles on t h e surface of the soap s o l u t i o n rather t h a n
92
Va. Geguzin
3
A Bubble Model of Crystal
Simply one or two. If the radius of a bubble is R = 5 X 10-2 cm, then N~ (Rp/R)2~ 4 X 4 X 10 bubbles can fit on the surface of a soap solution on an ordinary dinner plate whose radius is R p « 10 cm. Such a raft of bubbles, contained by attractive and repulsive forces, is ajtwodimensional model of crystal. The authors of this very beautiful model have shown, for example, that bubbles whose radius is 1 0 c m
Fig. 74
interact very crystals.
similarly
to
atoms
in
copper
The Model in Action The film of the B L N model in action is interesting since it shows an ideal crystal, a crystal with moving and interacting defects, and many other simple and complex processes that develop in a real crystal. I n an article, however, it is only possible to show a few photographs to illustrate the possibilities of the model.
94
Va. Geguzin
The B L N m o d e l can be used to verify certain corollaries t o the theory of crystal t h a t is absolutely free of defects, i.e. the so-called ideal cryst a l . I t is almost impossible to o b t a i n such a cryst a l in nature, b u t i t proved rather s i m p l e a n d easy to construct one m a d e of bubbles (Fig. 75). One of the most c o m m o n defects i n crystal is a vacant position at a p o i n t i n the lattice, w h i c h is not filled b y an a t o m . Physicists call this phenomenon a vacancy. I n the B L N m o d e l a vacancy is represented b y an exploded b u b b l e (Fig. 76). As b o t h c o m m o n sense a n d experiments w i t h real crystals lead us t o expect, the B L N m o d e l shows t h a t the v o l u m e of a vacancy is a l i t t l e less t h a n t h a t of an occupied position. W h e n a b u b b l e explodes, neighbouring bubbles move s l i g h t l y into the hole left b y the explosion and reduce its size. This is almost impossible to detect w i t h the naked eye, b u t if we project a photograph of the bubbles onto a screen and carefully measure the distances between bubbles, we can see t h a t the vacancy is somewhat compressed i n comparison w i t h an occupied p o s i t i o n . For physicists t h i s is evidence of both a q u a l i t a t i v e and a quantitat i v e change. Very often, crystal contains an i m p u r i t y , introduced i n the early stages of its history, t h a t deforms its structure. To solve m a n y problems of crystal physics, i t is very i m p o r t a n t t o k n o w h o w the atoms s u r r o u n d i n g the i m p u r i t y have changed position. I n c i d e n t a l l y , the presence of a n impur i t y is left not o n l y b y the nearest neighbours b u t also b y the atoms a considerable distance from i t . The B L N m o d e l reflects this clearly (Fig. 76). Most crystalline bodies are represented b y po-
95
A Bubble Model of Crystal
' W
W
^
W
1
Fig. 75
Fig. 76
Ya. Geguzin lycrystals m a d e of m a n y s m a l l , r a n d o m l y oriented crystals separated b y boundaries. W e expect m a n y properties of the polycrystals (such as mechanical strength or electrical resistance) t o depend on the structure of t h e boundaries, a n d , i n f a c t , the B L N m o d e l bears t h i s o u t . I t showed crystal physicists t h a t t h e structure of such boundaries varies according t o the m u t u a l o r i e n t a t i o n of b o u n d a r y crystals, t h e presence of i m p u r i t i e s at the b o u n d a r y , and m a n y other factors. Some parts of the polycrystals (grains), for example, m a y enlarge at the expense of others. As a result, average grain size increases. This process, called recrystallization, develops for a very explicit reason: the greater the size of the g r a i n , the less its t o t a l b o u n d a r y surface area, w h i c h means t h a t it has lower excess energy l i n k e d to the boundaries. The energy of a polycrystal is reduced i n rec r y s t a l l i z a t i o n , a n d , therefore, the process m a y occur spontaneously (since it moves the system t o a more stable e q u i l i b r i u m , at w h i c h energy storage is m i n i m a l ) . The series of photographs i n Fig. 77 illustrates a large grain ' d e v o u r i n g ' a smaller g r a i n i n s i d e i t i n successive stages. The m o v i n g b o u n d a r y between the grains appears t o ' s w a l l o w ' t h e vacancies i t comes across (this was predicted b y theorists a n d carefully studied b y experimenters i n real crystals). The b o u n d a r y does not change its structure i n t h i s process, as the B L N m o d e l clearly illustrates (Fig. 78). Restrictions of the B L N Model W i t h o u t depreciating the usefulness of the B L N m o d e l , we s h o u l d p o i n t out t h a t it does
Fig. 77 7-01544
* 'm • * * * * ZT-I
Determining the Poles of a Magnet
99
have drawbacks. The model can s i m u l a t e o n l y one s t r u c t u r e — a two-dimensional, h e x a g o n a l , and closely packed crystal. B u t real crystals h a v e a variety of structures. I n t h i s respect, t h e dead model has i n f i n i t e l y more p o t e n t i a l because atoms can be arranged in i t i n m a n y different ways, a n d , consequently, a n y structure can be simulated. The B L N m o d e l i n its contemporary modification is severely restricted b y its twodii l e n s i o n a l i t y . I t s authors tried t o m a k e a threedi lensional ( m u l t i l a y e r ) b u b b l e m o d e l , b u t its op j r a t i o n was difficult, and the m o d e l was finally re ected. I n our laboratory we b u i l t b o t h two- and three-dimensional models and f o u n d t h e latter to be i m p r a c t i c a l . Despite these a n d other weaknesses, however, the B L N m o d e l is an indispensable aid i n the s t u d y of crystals.
Determining the Poles of a Magnet by B.
Aleinikov
A t first glance it m a y seem s i m p l e to determine the poles of a m a g n e t . B u t because we cannot be sure t h a t the poles of a g i v e n m a g n e t h a v e not s i m p l y been p a i n t e d to m a k e t h e m look different, i.e. w i t h o u t reference t o their true m a g n e t i s m , the question is more complicated t h a n it appears. Occasionally, magnets are not marked at a l l , i n w h i c h case we need a m e t h o d to differentiate the positive from the negative pole. For t h i s experiment, we need a permanent horseshoe m a g n e t (the poles need not be marked; i n fact, it is even more c h a l l e n g i n g if they are not) a n d ... a television set. I t is best t o conduct 7*
100
N. Minz
the experiment i n the d a y t i m e , when a test p a t t e r n for t u n i n g the television is broadcast. T u r n the television set on, and p u t your magnet against t h e screen, as shown i n F i g . 79. The image w i l l i m m e d i a t e l y become distorted. The s m a l l circle i n the center of the test p a t t e r n w i l l F
B
Fig. 79
s h i f t noticeably u p w a r d or d o w n w a r d , d e p e n d i n g on the position of the poles. A n image on t h e television screen is produced b y an electron beam directed from inside the picture t u b e towards t h e viewer. O u r m a g n e t deviates the electrons e m i t t e d , and the i m a g e is distorted. The direction i n w h i c h the m a g n e t i c field deviates the m o v i n g charge is determined b y the left-hand rule. I f t h e p a l m is positioned so t h a t the lines of force enter i t , the fingers w h e n extended i n d i c a t e t h e direction of the current. I n this position, t h e t h u m b when held at a r i g h t angle to the fingers, w i l l show the direction i n
A Peculiar Pendulum
101
w h i c h t h e m o v i n g charge is d e v i a t e d . The lines of force go from the northern t o the southern pole of t h e m a g n e t . The direction of current according t o the left-hand rule is t h e so-called "techn i c a l " direction from plus t o m i n u s i n w h i c h pos i t i v e l y charged particles w o u l d m o v e . I n t h e cathode t u b e , however, the electrons m o v e a n d are directed towards us. This is t h e e q u i v a l e n t of positive charges h e a d i n g away from us. Therefore, the extended fingers of the left h a n d should be directed towards the screen. The rest is clear. B y the displacement of t h e central circle, we can determine whether the northern or the southern pole of t h e magnet been placed against the screen. I t is also possible to i d e n t i f y poles of an unm a r k e d battery w i t h the help of a television. For this experiment, i n a d d i t i o n t o a television, we need a b a t t e r y , an electric m a g n e t w i t h a n arched core, a resistor, a n d a conductor. Connect the battery i n series to the electromagnet, and the resistor, rated to l i m i t the current to admissible level. H o l d the electromagnet near the screen, a n d ident i f y its poles using the left-hand rule. Then use the corkscrew rule to determine t h e direction of the current a n d , consequently, the poles of the battery.
A Peculiar Pendulum by N.
Minz
The f a m i l i a r s i m p l e p e n d u l u m does not change the p l a n e i n w h i c h it swings. This property of the p e n d u l u m was used i n a well-known demon-
102
N. Minz
stration of t h e E a r t h ' s r o t a t i o n , i.e. t h e F o u c a u l t p e n d u l u m . The p e n d u l u m suspended on a l o n g wire oscillates. A circle under it is m a r k e d as a clock face. Since t h e p l a n e of oscillations relat i v e to the motionless stars does not s h i f t , w h i l e the E a r t h rotates on its axis, the p e n d u l u m passes t h r o u g h m a r k i n g s on the clock i n succession. A t either of the E a r t h ' s pole, the circle under t h e
Fig. 80
p e n d u l u m makes one complete r o t a t i o n i n twenty-four hours. This experiment was carried out b y the French physicist L . F o u c a u l t i n 1851, when a p e n d u l u m 67 metres long was suspended from the cupola of t h e P a n t h e o n i n Paris. D o a l l p e n d u l u m s keep the same p l a n e of oscill a t i o n ? The suspension, after a l l , allows oscillations in any vertical p l a n e . To m a k e the pendul u m shown i n F i g . 80a, fold a string i n h a l f , a n d attach another string in the m i d d l e . Tie t h e loose end of the second string t o a spoon, a p a i r of scissors, or a n y other object, and your p e n d u l u m is ready. (The vertical suspension should be longer or at least equal in length to t h a t of the first string.)
A Peculiar Pendulum
103
Tack t h e ends of the h o r i z o n t a l s t r i n g between the j a m b s of a doorway. N o w , p u l l t h e p e n d u l u m back (at rest, i t is i n t h e p o s i t i o n of equilibriu m ) , a n d release i t . The p e n d u l u m w i l l describe an ellipse t h a t c o n s t a n t l y changes shape, prolati n g t o one side or the other. W h y does it behave this way? A single suspension p e n d u l u m (Fig. 806) has an u n d e t e r m i n e d p l a n e of o s c i l l a t i o n . T h i s means t h a t regardless of the i n i t i a l d e v i a t i o n of the p e n d u l u m , a l l t h e forces influencing it lie i n one plane. B e careful, however, not t o propel it sideways w h e n setting it free. N o w let us d r a w a p l a n e t h r o u g h t h e i n i t i a l and deviated positions of t h e p e n d u l u m . Obviously, b o t h t h e g r a v i t y force m g , a n d t h e tension force of t h e string T lie i n t h i s p l a n e . Conseq u e n t l y , t h e resultant of t h e t w o forces, w h i c h makes t h e p e n d u l u m oscillate, acts i n t h e same p l a n e . T h u s , since there is no force t o propel t h e p e n d u l u m out of t h e p l a n e , i t keeps its p l a n e of oscillation. O u r p e n d u l u m is q u i t e another t h i n g . I n t h i s case, t h e i n i t i a l p l a n e of o s c i l l a t i o n is determ i n e d b y t h e a t t a c h m e n t of the h o r i z o n t a l string and b y t h e p l u m b l i n e of t h e v e r t i c a l string. Therefore, t h e p e n d u l u m is deviated from t h e very b e g i n n i n g so t h a t i t lies outside of t h e p l a n e . * The tension force (Fig. 80c) has a component perpendicular t o t h e i n i t i a l p l a n e , a n d the ac* Of course, if the deviation of the pendulum in the plane is strictly perpendicular to the plane of suspension, the pendulum will oscillate in this plane only. I n practice, however, a departure from this plane and velocity directed away from the plane always exist.
N. Minz
104
t i o n of this component forces the p e n d n l u m out of the plane. Since the tension force varies, its perp e n d i c u l a r component also varies. As it swings to the opposite side, the p e n d u l u m p u l l s the other h a l f of the h o r i z o n t a l string t a u t . This develops a force t h a t acts i n the opposite direction. A t the
Fig. 81
Fig. 82
same t i m e , as the experiment shows, the pendul u m oscillates i n two perpendicular planes. The curves described by our p e n d u l u m are called Lissajous figures, after the French physicist w h o was the first to describe them in 1863. A Lissajous figure results from the c o m b i n a t i o n of two perpendicular oscillations. The figure m a y be rather c o m p l i c a t e d , especially if the frequencies of l o n g i t u d i n a l and l a t i t u d i n a l oscillations are close. If the frequencies are the same, the res u l t a n t trajectory w i l l be an ellipse. Figure 81 shows the figure drawn b y a p e n d u l u m whose m o t i o n can he described as x = sin 3t, y —sin 51. Figure 82 shows the oscillations described as x = sin 31, y = sin 41,
A Peculiar Pendulum
105
The r a t i o of the frequencies can he varied b y v a r y i n g the r a t i o of the length of the vertical and h o r i z o n t a l strings. A l t h o u g h it is f a i r l y difficult to calculate the frequencies of p e n d u l u m osc i l l a t i o n s , the figures drawn b y the p e n d u l u m can be demonstrated rather easily. To m a k e the Lissajous figures visible, tie a s m a l l bucket w i t h
a perforated bottom to the p e n d u l u m . F i l l the bucket w i t h sand, and p u t a piece of dark cardboard under it on the floor. The p e n d u l u m w i l l draw a clear trajectory of its m o t i o n . Photographs of the m o t i o n of t h e p e n d u l u m can also be m a d e . P a i n t a weight or a s m a l l , heavy ball w h i l e , a n d make the suspension of dark string. P u t a sheet of dark paper 011 the floor, the paper should be m a t , since glossy paper reflects l i g h t and would spoil the pictures. Set the camera above the p e n d u l u m , w i t h the lens placed h o r i z o n t a l l y . If the exposure is long enough, the pictures w i l l show clear trajectories. Figures 83 and 84 show trajectories photographed in this way,
N. Minz
106
Changes i n the direction of the oscillations are obvious. The change is especially sudden i n Fig. 83. The exposures of the t w o p h o t o g r a p h s were different, w h i c h is obvious from the different lengths of the trajectories. The curves seem t o be inscribed w i t h i n a parallelogram, a l t h o u g h i n r e a l i t y , they s h o u l d be inscribed w i t h i n a rectangle. W e d i d not get a rectangle s i m p l y because the p l a n e of our camera was n o t s t r i c t l y horizontal. A reasonably correct trajectory can be o b t a i n e d i n experiments w i t h a p e n d u l u m if d a m p i n g is insignificant. The oscillations of a p e n d u l u m w i t h low mass a n d large v o l u m e w i l l d a m p quickl y . Such a p e n d u l u m w i l l swing several t i m e s with quickly diminishing amplitude. Naturally changes i n the oscillations of a p e n d u l u m w i t h such strong a t t e n u a t i o n can h a r d l y be photographed. Lissajous figures are c o m m o n w i t h perpendicular oscillations. They are u n a v o i d a b l e , for instance, i n t u n i n g oscillographs.
Lissajous Figures by N.
Minz
The simplest oscillations of a body are those i n w h i c h the d e v i a t i o n of the body from its equil i b r i u m position x is described as x = a sin (cot -+- cp) where a is the a m p l i t u d e , w is the frequency, a n d 9 is t h e i n i t i a l phase of oscillation. S u c h oscilla-
Lissajous Figures
107
tions are called h a r m o n i c . A s i m p l e p e n d u l u m , a w e i g h t on a spring, or v o l t a g e i n an electric circuit can oscillate h a r m o n i c a l l y . I n t h i s article we shall discuss a b o d y w i t h t w o s i m u l t a n e o u s h a r m o n i c oscillations. I f both oscill a t i o n s occur along the same straight line, the re-
s u l t a n t e q u a t i o n of the m o t i o n of the b o d y w i l l be a s u m of the equations of each m o t i o n : x = At sin (ant + cpj) + A2 sin (oy2t + cp2) I t is easy to m a k e a graph of the b o d y displacem e n t from e q u i l i b r i u m over t i m e . For this, the ordinates of the curves related to t h e first and second m o t i o n s should be added. Figure 85 illustrates how t w o h a r m o n i c oscillations can be
N. Minz
108
added (solid sinusoids). The broken l i n e represents the r e s u l t i n g oscillation, w h i c h is no longer h a r m o n i c . More complicated trajectories appear if t w o m u t u a l l y perpendicular oscillations are a d d e d . The body i n F i g . 86 moves along such a trajectory. I t s form depends on the ratios of frequencies, a m p l i t u d e s , a n d phases of the t w o m u t u a l l y perpendicular oscillations. As we k n o w , such trajectories are called Lissajous figures. The setup used by Lissajous i n his experiments is shown i n F i g . 87. The t u n i n g fork T' oscillates i n a h o r i z o n t a l plane, whereas T is vertical. A l i g h t beam passing through a lens is reflected b y a m i r r o r attached to T' towards a second m i r r o r fixed on T. The reflection of t h e second mirror is seen on a screen. I f o n l y one t u n i n g fork oscillates, t h e l i g h t spot on t h e screen w i l l m o v e along a s t r a i g h t line. I f both t u n i n g forks oscillate, the spot w i l l draw i n t r i c a t e trajectories. The trajectory of a body w i t h t w o simultaneous, m u t u a l l y perpendicular oscillations is described by a system of equations
(1)
where x and y are the projections of the body displacement on X and Y axes. For s i m p l i c i t y , assume (fi =
at,
y — A2 sin cot.
(2)
Lissajous Figures
Fig. 87
109
110
N. Minz
A* Thus, y = - j - x.
Consequently,
Eq.
(2) de-
scribes a straight line segment. Slope a respect to X .
tan a =
A«
axis is
,
Now let ? ! = q>; + x = At cos
with
( ( O j f -f- t p j ) ,
Then 1
y = A 2 sin (co2* +
^
A point with x and y coordinates determined by the above equations makes a circle of A radius. And, in fact, x2 + y2 = A2 cos2 at + A2 sin 2 at = = A 2 , which means that the trajectory of motion is a circle. Now let At =^=A2. Let us plot a trajectory for A1 = 1 and A2 = 2. At the moment of m a x i m a l displacement, x — Ai = 1, that is, cos cot — = 1, at = 0. Consequently, y = 2 sin cot = 0. Similarly, when x = 0, y equals two, and when x =
y equals ]/A2, and so on.
The graph plotted with these coordinates w i l l be an ellipse whose major semiaxis is A2 and
Lissajous Figures
111
whose m i n o r semiaxis is A t h a t is, the ellipse elongated along Y axis (Fig. 88a) * I t is easy to show that when Ax = 2 and A 2 = = 1, we get an ellipse elongated along X (Fig. 886). Clearly, by changing the a m p l i t u d e ratio, we can get different ellipses. Now let a>x = 2(0, (o2 — a,
)
(\l
r
\\ J'
e
= 0. The system of equations (3) w i l l then become x = Ai cos 2(01, y — A 2 sin tot. Transform the equation w i t h respect to x i n the following way x = Ai (cos2 cat — sin* cot) = = A t ( 1 - 2 s i n * ( o t ) = At
( l — 2-fj-).
• The fact that the system of equations x = Ay cos coi y = A 2 sin cot describes an ellipse can be shown analytically: -~T + 4 r = cos* at + sin* cot = 1, i.e., a point with coordinates x and y lies on the ellipse.
N. Minz
112
This curve is part of a parabola w i t h its axis along X and the apex at x = Aj (Fig. 89). Thus, we h a v e an open curve.
N o w let us check the effects of the frequency on the shape of the trajectory. W e w i l l assign equal a m p l i t u d e s to the lateral and l o n g i t u d i n a l oscillations described by system (3). Let us p l o t curves, for example, described b y the f o l l o w i n g equations: x = A cos cot, y = A sin 2oit, x = A cos at, y = A sin 4cot. The easiest w a y to do this is to draw a circle of A radius (Fig. 90) and
mark
the points corre-
sponding to angles cut, w h i c h equal 0, it 5n 3n 7n „ rp , . ,, . , —, -5-, -r-, -5-, j t , . . . , 2 n . t o determine the points z o 4 o w i t h coordinates x = A cos cot and y — A sin 2cot, remember t h a t for the circle whose radius is equal to u n i t y (r = 1) the cos cot is n u m e r i c a l l y equal to a projection of the vector radius /• (ait)
Lissajous Figures
413
onto X , whereas the sin co I is equal to the projection onto Y. Since we have drawn a circle of A radius, the coordinates x and y of each p o i n t of the circle are the projections of the vector r a d i i
Fig. 91
of the p o i n t s onto X and Y . Once we have determined all the points b y their coordinates, we can connect them w i t h a solid line (Fig. 90).
91fl, h). The figure in Fig. 92 is open. I t is described by 8-01544
114
tt.
Minz
t h e f o l l o w i n g system of equations x = cos 2o)t, y = sin 3cot. W h e n do open figures occur? Are there any comm o n regularities i n their origin? Consider the following equations x — cos p
„ .
.
— 2/1, S1U
pw<2 +
•
jr-*1
put2 — pat,
— Sill —
—
t% — tx W h e n t2 ~ tx — t and tf is s m a l l ) , sin
pbdtn—ptotj
«
(the difference between
pC0f3— pdttx
As a result, vx =
— Axpa
sin pcot.
S i m i l a r l y , for vv vu = A2 qto cos q(ot.
.
t2
Lissajous Figures
115
W h e n t h e velocities vx and vx ~ 0
if
pat
vy = 0
if
gwt = - ^ - - f m n .
=
are equal to zero,
kn,
From these conditions it becomes obvious t h a t Lissajous figure are open when P _ 2k q ~ 2m + 1 " The curve i n Fig. 92, for example, meets this condition. Lissajous figure can be observed on the screen of an oscilloscope. A vertical sweep indicates one h a r m o n i c oscillation, whereas another oscill a t i o n appears on a horizontal sweep. Their total m a y assume different forms if the frequency of the a l t e r n a t i n g voltage at the p l a t e of the oscilloscope is varied. A n y o n e can m a k e a s i m p l e device for observing and p h o t o g r a p h i n g Lissajous figures. Twist a s i m p l e m e t a l ruler so t h a t the p l a n e of one half of t h e ruler is perpendicular to the plane of the other h a l f . Fix one of the ends of the ruler in a bench vice. W h e n the free end is depressed and then released, it w i l l draw i n t r i c a t e Lissajous figures in the air. The m o t i o n of the free end of the ruler is the sum of t h e independent oscillations of its two parts. The first section is measured from the vice to the bend i n the ruler and the second from the bend to the free end. The oscillation of each part is perpendicular to the plane of the v i b r a t i n g section. Since the bend angle of the ruler is-77, the oscillations are m u t u a l l y perpendicular. The 8*
116
N. Minz
shape of the trajectory of the end depends on the length and the w i d t h of the rnler, as well as on t h e place of bend. The same ruler can be used to o b t a i n different figures, to vary vertical and h o r i z o n t a l oscillation ratios, s i m p l y c l a m p the ruler at different places. Since the frequency of osciIlations]depends on the length of the ruler, you can vary the fre-
Fig. 93
quency ratio of m u t u a l l y perpendicular oscillations of the end of the ruler by changing the rat i o of the length of its parts. This w i l l result in different trajectories of the, end. To photograph the figures, attach the light b u l b from a flashlight to the free end of the ruler. Connect the b u l b to a battery by wires placed along the ruler (see Fig. 93). Place this complex p e n d u l u m in a dark room, and experiment a few times to find the right exposure l i m e for photographs. A f a i r l y long exposure w i l l p r o b a b l y
A. Kosourov
118
work best. Figure 94 shows photograph obtained exactly in this w a y . Now try s i m i l a r experiments for yourself. EXERCISES 1. Prove that all curves described by the following system of equations are open: x ~ A j cos putt,
y = A2 cos qat
2. Derive an equation for a curve with the following parameters X = A J COS lot,
y = A 2 cos 2u>t
Waves in a Flat Plate (Interference) by A.
Kosourov
W a v e p r o p a g a t i o n is perhaps the most universal phenomenon i n nature. W a t e r , waves, s o u n d , l i g h t and r a d i o , even deformation transfer from one part of a solid to another are examples of this phenomenon. According to q u a n t u m mechanics, the m o t i o n of microscopic particles is also controlled by the laws of wave p r o p a g a t i o n . The physical nature, velocity of p r o p a g a t i o n , frequency and wavelength of all these waves are different, b u t despite these differences, the motion of all waves is s i m i l a r i n m a n y respects. The laws of one k i n d of wave m o t i o n can be applied almost w i t h o u t modification t o waves, of another nature. The most convenient way to stud y these laws is to study waves on the surface of a body of water.
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119
W h a t is a wave? Throw a stone i n t o a p o n d . The calm h o r i z o n t a l surface of the pond w i l l develop circles t h a t ripple outwards. P o i n t s on t h e surface of the water reached b y the wave w i l l begin to oscillate relative to t h e position of equil i b r i u m , w h i c h corresponds t o the h o r i z o n t a l surface of the water. The farther a p o i n t is from the centre of the circle, the longer the p o i n t w i l l take to 'learn' about the stone t h a t has been thrown. The disturbance travels at a determinate speed. P o i n t s t h a t are reached s i m u l t a n e o u s l y by the disturbance are said t o be i n the same stage or phase of oscillation. A l l waves disturb some physical object w i t h their action by causing the object to deviate from the state of e q u i l i b r i u m . S o u n d waves, for example, cause the periodic rise and drop of pressure. R a d i o waves and l i g h t cause r a p i d changes of tension in electric and m a g n e t i c fields. The properties of all m e d i a w i t h o u t exception are such t h a t a disturbance, w h i c h originates in a specific area, propagates by passing from one p o i n t t o another w i t h a final speed. This speed depends on the nature of t h e disturbance and the medium. The disturbance t h a t generates a wave m u s t have a source, t h a t is an external cause t h a t breaks the e q u i l i b r i u m in a certain area of the m e d i u m . A s m a l l disturbance, a stone thrown i n t o water, for example, radiates spherical waves (in this case, circles on the water surface) t h a t travel r a d i a l l y i n a u n i f o r m m e d i u m (a m e d i u m in w h i c h wave velocity does not depend on the direction of its propagation). Such sources are called point sources.
120
A. Kosourov
One of the m a i n principles of elementary wave theory is the p r i n c i p l e of wave independence, also called the p r i n c i p l e of superposition. The principle states t h a t a disturbance caused b y a wave at a p o i n t of observation is not influenced by other waves passing through the same p o i n t . The principle of superposition is, in fact, a s i m p l e rule for d e t e r m i n i n g the s u m m a r y effect of waves from different sources. A s u m m a r y oscillation is s i m p l y a sum of the oscillations caused b y each source i n d e p e n d e n t l y . Interference is a characteristic feature of wave processes. Interference is the c o m b i n a t i o n of phenomena t h a t develop i n a m e d i u m i n which waves p r o p a g a t i n g from two or more sources oscillate synchronously. The oscillations of some points of the m e d i u m m a y be stronger or weaker under the action of the two s i m u l t a n e o u s sources than they w o u l d be under the effect of either source i n i s o l a t i o n . Synchronized waves m a y even suppress each other completely. Let us t r y to produce interference t h a t we can see w i t h our own eyes. A n experienced observer can easily see the interference caused b y the waves from two stones thrown i n t o a p o n d . This method is u n s u i t a b l e for study of interference, however. W e need, instead, a stable picture of interfering waves i n the laboratory. The first t h i n g we w i l l need for t h i s experim e n t is a vessel for water. The vessel should have gently sloping walls to avoid m a s k i n g waves from the source w i t h reflections from the walls. A shallow saucer w i l l work well, if the water nearly reaches the r i m , in w h i c h case the waves roll onto the walls and are damped q u i c k l y , almost
Waves in a Flat Plate (Interference)
121
w i t h o u t reflections. A n electric hell w i t h o u t its cap is a good wave generator. W i r e the h a m m e r of the hell, and attach a cork b a l l to the wire. This cork w i l l be our wave source. B e sure that the electric wires are well i n s u l a t e d . The bell should be m o u n t e d on a swing above the saucer so t h a t it can be lowered i n t o the water at the r i m of the saucer. Power should be furnished b y an autotransformer, w h i c h w i l l enable us to vary the a m p l i t u d e of the oscillations. The autotransformer from either a toy electric train or an electric burning-out m a c h i n e w i l l serve this purpose. W h e n we switch on the setup, wo w i l l see circular waves on the water surface. The average distance between neighbouring crests, that is, the wavelength, w i l l be about 1 cm (Fig. 95). The waves can best be observed by w a t c h i n g the shadows on the b o t t o m of the saucer under direct s u n l i g h t or strong l a m p l i g h t . Every wave acts like a c y l i n d r i c a l lens and casts a bright band on the b o t t o m t h a t repeats the configuration of the wave front. Since the waves m o v e at about 10 centimetres per second, however, they m a y seem to merge if you keep your eyes fixed on the plate. They are visible only close to the source where their a m p l i t u d e is h i g h , and you w i l l need to t u r n your head q u i c k l y to trace i n d i v i d u a l waves on the surface, iust as you w o u l d need to move your head r a p i d l y to trace the m o t i o n of i n d i v i d u a l spokes in a r o t a t i n g wheel. The waves are very clear on the mat plate of a camera, especially one with a large format. By h o l d i n g such a camera by h a n d and rocking it gently, you can easily find a position from w h i c h the waves,
A. Kosourov
122
which appear to move very slowly can be seen over the entire surface. A mirror can also be used to watch the water surface. The most expedient way to observe the waves is w i t h a stroboscope. If we i l l u m i n a t e the setup w i t h short flashes of
Fig.
95
light w i t h the same frequency as the wave generator, the wave w i l l move over one wavelength from one flash to the next, and as a result, the wave picture w i l l appear stationary. To o b t a i n this effect, s i m p l y wire a small l a m p i n t o the c i r c u i t , parallel to the electric bell magnet w i n d i n g . A t a distance of 0.5-1 m 5 the l a m p w i l l i l l u m i n a t e
Waves in a Flat Plate (Interference)
123
the saucer u n i f o r m l y , and the stationary wave picture w i l l appear clearly. I t is better to use direct s u n l i g h t for photographs. N o w replace the single cork b a l l on the hammer of t h e bell w i t h a wire fork t o w h i c h two pieces of cork have been attached at the ends. The distance between the ends should be 2-3 cm. I f the corks t o u c h the water surface s i m u l t a n e o u s l y , you w i l l get t w o sources of waves t h a t oscillate not o n l y synchronously, i.e. i n t i m e , b u t in phase, w h i c h means t h a t the waves from the two sources w i l l appear i n the same instance of t i m e . The picture w i l l look a p p r o x i m a t e l y like i n F i g . 96 (here 2d/K = 4). The fan-shaped d i s t r i b u t i o n of h i g h - a m p l i t u d e zones includes i n t e r m i t t e n t 'silence' zones. The central zone of h i g h amplitudes is perpendicular to the l i n e connecting the sources, a n d b o t h types of zones are located between the sources. According t o the interference picture, the distance between n e i g h b o u r i n g peaks on the line connecting the sources is one-half of t h e distance between t w o crests, t h a t is, one-half of a wavelength. I f we change the distance between the sources, the number of h i g h - a m p l i t u d e zone w i l l change too. I n F i g . 97 the characteristic r a t i o is 2dl% = 2. The larger the distance between sources, the more 'feathers' we h a v e i n our fan. B u t the distance between crests on the l i n e connecting the sources is always one-half of a wavelength. Thus, the t o t a l number of h i g h - a m p l i t u d e zones w i l l always be twice the n u m b e r of wavelengths in the distance between sources. Hence, we can conclude t h a t if this distance is less t h a n onehalf of a w a v e l e n g t h , the waves w i l l not inter-
124
A. Kosourov
Waves in a Flat Plate (Interference)
125
fere at a l l . Such sources acl as one, producing a single system of circular waves. This can be demonstrated by g r a d u a l l y reducing the distance between sources. Note also t h a t if a wavefront continues from a h i g h - a m p l i t u d e zone to a neighbouring zone, it w i l l pass from a crest t o a trough. In other words, as a wave passes t h r o u g h the zero phase, the phase of the wave changes by onehalf of a complete cycle. N o w i m a g i n e t h a t instead of t w o cork balls creating waves i n the water we h a v e t w o l i g h t sources e m i t t i n g l i g h t waves. I f we place a screen perpendicular to the water surface in the p a t h of the lightwaves, we w i l l see i l l u m i n a t e d places, which i n d i c a t e h i g h - a m p l i t u d e zones, and shadows. N o w let us try to explain these dark and light interference bands. D r a w the two wave systems on paper, as if the waves were frozen in their tracks (Fig. 98). Indicate the crests w i t h l i g h t , solid lines and the troughs w i t h broken lines. Assign every wave a n u m b e r , g i v i n g i d e n t i c a l numbers t o those t h a t originate from the sources s i m u l t a n e o u s l y . As is clear from the drawing, waves w i t h the same number covered the distance e q u i v a l e n t for both sources s i m u l t a n e o u s l y . O b v i o u s l y , this occurs because a l l points at this distance are reached by waves t h a t travel the same distance. B y a p p l y i n g the law of superposition, we can conclude that the heights of the crests and depths of the troughs w i l l double at this distance. The resulting crests should be marked i n the d r a w i n g w i t h heavy, solid lines; mark the troughs w i t h heavy, broken lines. To the right and left of line 00 are points at w h i c h the crests of one system of waves
A. Kosourov
126
coincide w i t h the troughs of another. The waves from one source cause upward d e v i a t i o n at these p o i n t s , w h i l e the waves from the second source cause downward d e v i a t i o n , a n d , as a result, to0
Fig. 98
t a l d e v i a t i o n is close to zero. Connect all "such points w i t h a solid line. I f we analyse t h e numbers of the crests and troughs, we w i l l see t h a t all the points of the right line are reached by waves from the left, w h i c h travel half a wavelength farther t h a n waves from the r i g h t .
Waves in a Flat Plate (Interference)
127
To the right and left of the zero lines lie t h e points of intersection of the first crest w i t h the second, the second w i t h the t h i r d , and so on. I t is easy to see t h a t these are m a x i m a p o i n t s . I f we connect these points, we get a l i n e t h a t is reached by one wave system w i t h the delay of one wavelength. By analysing the d r a w i n g further we can find all the zero and m a x i m a lines. Such lines are hyperbolas. I t is now clear w h y the distance between neighbouring m a x i m a on the line connecting the sources equals one-half a w a v e l e n g t h . Indeed, the m i d p o i n t of this line is reached b y waves of the I wo systems, which m o v e i n the same phase and enhance one another. I f we m o v e off the p o i n t by one-half a wavelength, the distance travelled by one wave w i l l increase by one-half a wavelength, whereas the distance covered by another w i l l decrease by the same value. The difference between the distances travelled b y both waves w i l l equal one wavelength, and the waves w i l l enhance one another a g a i n . This w i l l reoccur every half wavelength. A m a x i m u m observed when the difference i n the distance travelled is zero is called the zero m a x i m u m or the zero order of interference. Max i m a observed when the distance is one wavelength are called first-order interference, and so on. A m a x i m a l order of interference is determined by the integer closest t o 2 d / K , where d is the distance between sources and A, is the wavelength. Now try t o predict from t h e d r a w i n g or from experiments what changes w i l l occur if one of the sources radiates waves w i t h half a period (or a
128
C. I;. Stong
fraction of a period) delay. W h a t w i l l h a p p e n if the phase shift is random? To s t u d y this phenomenon e x p e r i m e n t a l l y , s i m p l y m a k e the ends of the wire fork different lengths.
How to Make a Ripple Tank to Examine W a v e Phenomena* by C. L.
Stong
W a v e s of one k i n d or another are f o u n d at work everywhere i n the universe, r a n g i n g from g a m m a rays of m i n u t e wavelength emitted by nuclear particles to the immense u n d u l a t i o n s in clouds of dust scattered t h i n l y between the stars. Because waves of all k i n d s have i n c o m m o n the function of carrying energy, it is not surprising t h a t all waves behave m u c h alike. They m o v e i n straight lines and at constant velocities t h r o u g h uniform m e d i u m s a n d to some extent change direction and velocity at junctions where the physical properties of the m e d i u m s change. The part of a sound wave i n air t h a t strikes a hard object such as a brick w a l l , for example, bounces back to the source as an echo. By learning h o w waves of one k i n d behave the experimenter learns w h a t behaviour to expect of others, and problems solved by the s t u d y of waves in one m e d i u m can be'applied, w i t h appropriate m o d i f i c a t i o n , to those i n other m e d i u m s . The p a n of a s i m p l e ripple t a n k t h a t can be made in the home consists of a picture frame 6 * An abridged version of an article that first appeared in the November issue of Scientific American for 1962.
How to Examine Wave Phenomena
129
about five centimeters t h i c k a n d 0.6 m 2 square, closed at t h e b o t t o m b y a sheet of glass calked t o bold water, as shown i n F i g . 99. The t a n k is supported a b o u t 60 cm above t h e floor b y four sheetm e t a l legs. A source of l i g h t to cast shadows of
in
Fig. 99. Ripple tank for demonstrating wave behavior: ( i ) d o w e l s , (2) a l u m i n i z e d c a r d b o a r d , (a) h o l e f o r l i g h t , (4) l V j - v o l t m o t o r v i b r a t o r , (5) 1 0 0 - w a t t b a r e straight filament lamp w i t h f i l a m e n t v e r t i c a l , (6) picture frame with glass b o t t o m s e t in m a s t i c , (7) p a r a f f i n r e f l e c t o r , («) a l l i g a t o r clip and steel spring for a d j u s t i n g m o t o r s p e e d , (9) w h i t e p a p e r d i s p l a y screen, (10) slots for leveling
ripples t h r o u g h the glass onto a screen 9 below is provided by a 100-watt clear l a m p 5 w i t h a straight filament. Because the l a m p is suspended above the t a n k w i t h the filament axis vertical, t h e end of the filament a p p r o x i m a t e s a p o i n t source a n d casts sharp shadows. The l a m p , p a r t l y enclosed by a fireproof cardboard h o u s i n g 2, is suspended about 60 cm above t h e t a n k on a framework of dowels 1. The wave generator hangs on rubber bands from a second framework m a d e of 9-01544
130
C. I;. Stong
a pair of m e t a l brackets notched at the upper end t o receive a wooden crossbar. The distance between the wave generator and the water can be adjusted either b y c h a n g i n g the angle of the m e t a l brackets or b y l i f t i n g the crossbar from the Supp o r t i n g notches a n d w i n d i n g the rubber bands u p or d o w n as required. The agitator of the wave gen-
Fig. 100. Details of ripple generator: (J) s h a f t p a s s e s t h r o u g h h o l e in s c r e w , (2) e c c e n t r i c w e i g h t f o r a d j u s t i n g a m p l i t u d e of o s c i l l a t i o n , (3) l ' / i - v o l t m o t o r c l a m p e d i n c l o t h e s - p i n , (4) b e a d o n w i r e c a n b e t u r n e d d o w n t o g i v e p o i n t s o u r c e of w a v e s , (5) r u b b e r - b a n d s u p p o r t s
erator is a rectangular wooden rod. A wooden clothespin 3 at i t s center grasps a 1.5 v o l t t o y motor d r i v e n b y a dry-cell b a t t e r y . Several glass or plastic beads 4 are attached to the agitator b y stiff wires, bent at right angles, t h a t fit s n u g l y into any of a series of holes spaced about five centimeters a p a r t . D e t a i l s of the wave generator are shown i n Fig. 100. Attached to the shaft of the motor is an eccentric w e i g h t , a 10-24 m a c h i n e screw about 2.5cm long. The shaft 1 runs t h r o u g h
How to Examine Wave Phenomena
131
a transverse hole drilled near the head of the screw, which is locked to the motor by a nut run tight against the shaft; another nut 2 is run partly up the screw. The speed of the motor is adjusted by a simple rheostat: a helical spring of t h i n steel wire (approximately No. 26 gauge) and a small alligator clip. One end of the spring is attached to a battery terminal, and the alligator clip is made fast to one lead of the motor. The desired motor speed is selected by clipping the motor lead to the spring at various points determined experimentally. (A 15-ohm rheostat of the kind used in radio sets can be substituted for the spring-and-clip arrangement.) The inner edges of the tank are lined with four lengths of a l u m i n i u m fly screening 1 7.6 cm wide bent into a right angle along their length and covered with a single layer of cotton gauze bandage 2, either spiraled around the screening as shown in Fig. 101 or draped as a strip over the top. The combination of gauze and screening absorbs the energy of ripples launched by the generator and so prevents reflection at the edges of the tank that would otherwise interfere with wave patterns of interest. The assembled apparatus is placed in operation by leveling the tank and filling it with water to a depth of about 20 m m , turning on the lamp, clipping the motor lead to the steel spring and adjusting the height of the wave generator u n t i l the tip of one glass bead makes contact with the water. The rotation of the eccentric weight makes the rectangular bar oscillate and the bead bob up and down in the water. The height, or amplitude, of the resulting ripples can be adjusted by 9*
132
G. L. Stong
altering the position of the free n u t on the machine screw. The wavelength, which is the distance between the crests of adjacent waves, can be altered by changing the speed of the motor. The amount of contrast between light and shadow i n the wave patterns projected on the screen can be
Fig. 101. Details of tank brackets and wave absorbers: (?) w a v e a b s o r b e r , fly s c r e e n , (2) gauze bandage, (,?) s t r a p i r o n , (4) s u p p o r t f o r i i g h t s o u r c e , (S) s u p p o r t f o r w a v e g e n e r a t o r , (6) w a t e r , (7) g l a s s , (s) g l a s s s e t in mastic calking compound, (9) 1C-G a l u m i n i u m legs
altered by rotating the l a m p . The wave generator should be equipped with at least one pair of beads so that ripples can be launched from two point sources. W a v e s w i t h straight fronts (analogues of plane waves that travel,in m e d i u m s of three dimensions) are launched by turning the bead supports up and lowering the rectangular bar i n t o the water. As an introductory experiment, set up the generator to launch plane waves spaced about five centimeters from crest to crest. If the apparatus functions properly, the train of ripples will
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flow s m o o t h l y across the t a n k from t h e generator a n d disappear i n t o the absorbing screen at the front edge. A d j u s t the l a m p for m a x i m u m contrast. Then place a series of paraffin blocks (of the k i n d sold i n grocery stores for sealing jelly), b u t t e d end t o end, d i a g o n a l l y across t h e t a n k at an angle of about 45 degrees. Observe h o w the paraffin barrier reflects waves t o one side, as i n F i g . 102(top). I n p a r t i c u l a r , note t h a t the angle m a d e between the p a t h of the i n c i d e n t waves and a l i n e perpendicular t o the barrier (0;) equals the angle m a d e b y the p a t h of the reflected rays and the same perpendicular (0 r ). Set t h e barrier at other angles larger and smaller t h a n 45 degrees w i t h respect t o the wave generator and also vary t h e w a v e l e n g t h and a m p l i t u d e of the waves. I t w i l l be found t h a t the angle of incidence equals the angle of reflection whatever the position of t h e barrier, a law of reflection t h a t describes waves of a l l kinds. Next replace the paraffin barrier w i t h a slab of plate glass about 15 cm w i d e a n d 30 cm long a n d supported so t h a t its top surface is about 12 m m above t h e t a n k floor. A d j u s t the water level u n t i l i t is between 1.5 a n d 3.2 m m above the glass and l a u n c h a series of p l a n e waves. Observe h o w t h e waves from the generator slow down when they cross the edge of the glass and encounter s h a l l o w water, as shown i n F i g . 102 (bottom). As a result of the change i n speed the waves travel i n a new direction above t h e glass, just as a r a n k of soldiers m i g h t do if they marched off a dry p a v e m e n t o b l i q u e l y i n t o a m u d d y field. I n this experiment waves h a v e been diverted from their i n i t i a l direction b y refraction, an effect
F i g . 102. W a v e reflection a n d r e f r a c t i o n : t o p : (J) w a v e g e n e r a t o r , (2) s t r a i g h t b a r r i e r , ( j ) i n c i d e n t waves, (4) reflected waves; b o t t o m : (I) w a v e g e n e r a t o r (2) deep w a t e r , (3) s h a l l o w w a t e r , (4) r e t r a c t e d w a v e f r o n t s , (J) reflected w a v e f r o n t s , (6) Incident w a v e l r o n t s
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observed i n waves of all k i n d s w h e n they cross o b l i q u e l y from one m e d i u m to another i n w h i c h they travel at a different v e l o c i t y . W a t e r waves are u n i q u e i n t h a t they travel at different speeds when the thickness, or d e p t h , of the m e d i u m changes. To a very good a p p r o x i m a t i o n ' t h e ratio of wave velocity i n shallow a n d deep water is proportional to the ratio of the depths of the
Fig. 103
water. This r a t i o is i n effect t h e " i n d e x of refraction" of t h e t w o " m e d i u m s " . I n t h e case of electrom a g n e t i c waves (such as l i g h t ) or m e c h a n i c a l waves (such as sound) t h e v e l o c i t y of wave prop a g a t i o n varies w i t h the density of the m e d i u m s . The net reflection at t h e d i s j u n c t i o n between the deep a n d shallow water can be m i n i m i z e d b y beveling t h e edge of t h e glass (or a n y other s m o o t h , solid m a t e r i a l s u b s t i t u t e d for glass) as shown i n F i g . 103. W a v e energy can also be focused, dispersed a n d otherwise d i s t r i b u t e d as desired b y barriers of appropriate shape, as exemplified b y t h e parabolic reflectors used i n telescopes, searchlights, radars a n d even orchestra shells. The effect can be demonstrated i n t w o dimensions b y t h e r i p p l e t a n k . M a k e a barrier of paraffin blocks or rubber hose i n t h e shape of a p a r a b o l a a n d
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direct p l a n e waves toward i t . A t every p o i n t along t h e barrier t h e angle m a d e b y the incident waves and the perpendicular to the parabola is such t h a t t h e reflected wave travels t o a c o m m o n p o i n t : the focus of the parabola. Conversely, a circular wave t h a t originates at the focus reflects as a p l a n e w a v e from the p a r a b o l i c barrier, as shown i n Fig. 104 (top). I n this experiment the wave was generated b y a drop of water. Interference effects can be demonstrated i n the r i p p l e t a n k by a d j u s t i n g a p a i r of beads so t h a t they m a k e contact w i t h the water about five centimeters apart. A t y p i c a l interference pattern m a d e b y t w o beads v i b r a t i n g i n step w i t h each other is shown i n Fig. 104 ( b o t t o m ) . Observe t h a t m a x i m u m a m p l i t u d e occurs along paths where the wave crests coincide a n d t h a t nodes appear along paths where crests coincide w i t h troughs. The angles at which m a x i m a and nodes occur can be calculated easily. The trigonometric sine of the angles for m a x i m a , for example, is equal to nX/d, where n is the order of the maxim u m (the central m a x i m u m , extending as a perpendicular t o the line j o i n i n g the source, is the "zeroth" order, and the curving m a x i m a extending r a d i a l l y on each side are numbered "first", "second", " t h i r d " and so on consecutively), X is the wavelength and d is the distance between sources. S i m i l a r l y , m i n i m a lie along 'angular paths given by the e q u a t i o n sin 0 = (m — 1/2) X/d, where tn is the order of the m i n i m a and the other terms are as previously defined. Barriers need not be solid to reflect waves. A two-dimensional lattice of u n i f o r m l y spaced pegs arranged as i n F i g . 105 w i l l reflect waves i n
H o w t o E x a m i n e W a v e P h e n o m e n a 137
F i g . 104. R e f l e c t i o n f r o m a p a r a b o l i c b a r r i e r : t o p : ( / ) p a r a b o l i c r e f l e c t o r , (2) r e f l e c t e d w a v e f r o n t s , ( j ) n o r m a l t o p a r a b o l i c r e f l e c t o r , 1,4) i n c i d e n t w a v e f r o n t s ; b o t t o m : ( / ) c e n t r a l m a x i m u m , (2) m a x i m u m , (3) n o d e
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the ripple tank that bear a required geometrical relation to the lattice. W h e n a train of plane waves impinges obliquely against the lattice,
111 i
r y
—
| 7
-
—
circular waves are scattered by each peg and interfere to produce a coherent train of plane waves. The m a x i m u m amplitude of this train makes an angle w i t h respect to the rows m a k i n g
Fig. 106
up the lattice such that sin 0 max = nk!2d, where sin 0 max designates the direction of maximum wave amplitude, n the order, X the wavelength and d the spacing between adjacent rows of pegs (the lattice spacing) (Fig. 106).
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This e q u a t i o n , known as B r a g g ' ? l a w in honor of its B r i t i s h discoverers, t h e father-and-son t e a m of Sir W i l l i a m B r a g g and Sir Lawrence Braersr, has been w i d e l y applied in c o m p u t i n g the lattice structure of crystal solids from photographs of wave m a x i m a m a d e b y the reflection of X - r a y waves from crystals. A n o t h e r of the m a n y aspects of wave behavior t h a t can be investigated w i t h the r i p p l e t a n k is the D o p p l e r effect, first studied intensively b y the A u s t r i a n physicist Christian J a h a n n D o p p l e r . H e recognized t h e s i m i l a r i t y in wave behavior t h a t explains the a p p a r e n t increase in p i t c h of an onrushing t r a i n w h i s t l e and t h e slight shift toward t h e b l u e end of t h e spectrum i n the color of a star speeding toward t h e earth. B o t h effects are observed because i t is possible for m o v i n g wave sources t o overtake a n d in some cases t o o u t r u n their own wave disturbances. To demonstrate the D o p p l e r effect i n the r i p p l e t a n k , substitute for the agitator bar a s m a l l t u b e t h a t directs evenly t i m e d puffs of air from a solenoid-actuated bellows against the surface of t h e water w h i l e s i m u l t a n e o u s l y m o v i n g across the t a n k at a controlled and u n i f o r m speed. (A few lengths of track from a t o y train can be m o u n t e d along the edge of t h e t a n k and a puffer can be improvised on a t o y car.) W h e n t h e puffer moves across t h e t a n k at a speed slower t h a n t h a t of the waves, crests in front of t h e puffer crowd closely together, whereas those b e h i n d spread a p a r t , as shown i n F i g . 106 (top). The D o p p l e r effect is observed i n waves of a l l k i n d s , i n c l u d i n g r a d i o signals. B y m e a n s of
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relatively s i m p l e apparatus the effect can b e applied to determine the direction a n d v e l o c i t y of an artificial satellite from its r a d i o signals. These experiments merely suggest the m a n y wave phenomena t h a t can be demonstrated b y the r i p p l e tank. A n y o n e w h o b u i l d s and operates a r i p p l e t a n k w i l l find it appropriate for enough fascinating experiments to occupy m a n y r a i n y afternoons.
An Artificial Representation of a Total Solar Eclipse * by R. W.
Wood
I n preparing for polarisation experiments on the solar corona, it is extremely desirable t o have an artificial corona as nearly as possible resembling the r e a l i t y . The a p p a r a t u s described below is a i m e d at t h i s end. The artificial corona in this case resembles the real so closely, as t o startle one who has a c t u a l l y witnessed a t o t a l solar eclipse. The polarisation is r a d i a l , and is produced in the same w a y as i n the sun's surroundings, and the m i s t y gradations of b r i l l i a n c e are present as w e l l . So perfect was t h e representation t h a t I added several features of purely aesthetic nature to heighten the effect, and finally succeeded in getting a reproduction of a' solar eclipse w h i c h could h a r d l y be distinguished from t h e r e a l i t y , except t h a t the polar streamers are straight, instead of being curved, as a l l t h e recent photographs show t h e m . The curious greenish-blue * Nature,
January 10, 1901, pp. 250-251.
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141
colour of t h e sky, and the peculiar pearly lustre and m i s t y appearance are f a i t h f u l l y reproduced. For lecture purposes a n artificial eclipse of t h i s sort w o u l d be a d m i r a b l y a d a p t e d , a n d I k n o w of no other w a y i n which an audience could be given so v i v i d idea of the beauty of the p h e n o m e n o n . D r a w i n g s and photographs are w h o l l y i n a d e q u a t e in g i v i n g any n o t i o n of the actual appearance of the sun's surroundings, a n d I feel sure t h a t a n y one w i l l feel a m p l y repaid for the s m a l l a m o u n t of trouble necessary i n fitting u p the arrangement w h i c h I shall describe. A rectangular glass t a n k about a 30 X 30 cm square on the front a n d 12 or 15 centimeters w i d e , a n d a six candle-power incandescent l a m p are all t h a t is necessary. The dimensions of the t a n k are not of m u c h i m p o r t a n c e , a s m a l l a q u a r i u m being a d m i r a b l y adapted to t h e purpose. The t a n k should be nearly filled w i t h clean water, and a spoonful or two of an alcoholic solution of m a s t i c added. The mastic is at once t h r o w n down as an exceedingly fine precipitate, g i v i n g the water a m i l k y appearance. The wires leading t o t h e l a m p should be passed t h r o u g h a short glass t u b e , and t h e l a m p fastened to t h e end of the t u b e w i t h sealing w a x , t a k i n g care to m a k e a t i g h t j o i n t t o prevent the water from entering the t u b e (Fig. 107). F i v e or six strips of t i n f o i l are now fastened w i t h shellac along the sides of the l a m p , leaving a space of from 0.5 t o 1 m m between t h e m . The strips should be of about the same w i d t h as the clear spaces. They are t o be m o u n t e d i n two groups on opposite sides of t h e l a m p , and the rays passing between t h e m produce t h e polar streamers.
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The proper n u m b e r , w i d t h and d i s t r i b u t i o n of the strips necessary to produce the most realistic effect can be easily determined b y experiment. A circular disc of m e t a l a trifle larger t h a n the l a m p , should be fastened to the t i p of the l a m p w i t h sealing-wax, or any soft, water-resisting cement; this cuts off the direct l i g h t of the l a m p
Fig. 107
Fig. 108
and represents the dark disc of the m o o n . The whole is t o be immersed i n the t a n k w i t h the l a m p i n a h o r i z o n t a l position and the m e t a l disc close against the front glass plate (Fig. 108). I t is a good p l a n t o h a v e a rheostat i n circuit w i t h the l a m p t o regulate the i n t e n s i t y of t h e i l l u m i n a t i o n . O n t u r n i n g on the current and seating ourselves i n front of the t a n k , we s h a l l see a most b e a u t i f u l corona, caused b y the scattering of the l i g h t of the l a m p by the s m a l l particles of m a s t i c suspended i n the water. I f we look at it through a Nicol prism we shall find t h a t it is r a d i a l l y polarized, a dark area appearing on each side of the l a m p , w h i c h turns as we t u r n the
143
An Artificial Solar Eclipse
N i c o l . The i l l u m i n a t i o n is not u n i f o r m around the l a m p , owing to unsymmetrical distribution of the candle-power, and t h i s heightens the effect. I f the polar streamers are found to be too sharply defined or too w i d e , the defect can be easily remedied by altering the t i n f o i l strips. The eclipse is not yet perfect, however, the
Fig. 109
i l l u m i n a t i o n of the sky background being too w h i t e a n d too b r i l l i a n t i n comparison. B y a d d i n g a s o l u t i o n of some bluish-green a n i l i n e dye (I used malachite-green), the sky can be given its weird colour a n d the corona b r o u g h t out m u c h more d i s t i n c t l y . If t h e proper a m o u n t of the dye be a d d e d , the sky can be strongly coloured w i t h o u t a p p a r e n t l y changing t h e colour of the corona i n t h e slightest degree, a rather surprising circumstance since both are produced b y the same means. W e s h o u l d h a v e now a most b e a u t i f u l a n d perfect reproduction of the wonderful atmosphere a r o u n d the sun, a corona of pure golden w h i t e l i g h t , w i t h pearly lustre and exquisite texture,
44
A. Kosourov
the m i s t y streamers stretching out u n t i l lost on the bluish-green background of the sky. The rifts or darker areas due to the unequal i l l u m i n a t i o n are present as well as the polar streamers. The effect is heightened if the eyes are p a r t i a l l y closed. A photograph of one of this artificial eclipses is reproduced in F i g . 109. Much of the fine detail present i n t h e negative is lost in the p r i n t .
Believe It or Not by G.
Kosourov
V i s i o n is our m a i n source of i n f o r m a t i o n about the e n v i r o n m e n t , and we are used t o t r u s t i n g our eyes. The expression ' I c a n ' t believe m y eyes', for example, indicates extreme surprise, and n o r m a l l y , our reliance on our eyes is justified. The eyes, w i t h the appropriate parts of the b r a i n , are a sophisticated a n a l y t i c a l apparatus w h i c h serves our purposes under diverse c o n d i t i o n s — i n a b r i g h t s u n l i g h t , or darkness, w i t h slow or r a p i d m o v e m e n t s . The image t h a t reaches the retina appears free of defects. The i m a g e seems q u i t e sharp, a n d the perspective is correct. S t r a i g h t lines seem straight. Objects lack iridescence, i . e . , chromatic aberration. O u r eyes are not ideal i n s t r u m e n t s , however. Objective studies show t h a t the eye possesses all the drawbacks of a lens. O u r b r a i n , however, constructs a correct image from t h e incorrect picture of the environment on the retina of the eye. For e x a m p l e , a m a n , who develops shortsightedness gets a very distorted perspective
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145
when he puts on glasses for the first t i m e in his life. S t r a i g h t lines seem curved; planes are irregular and sloping. Sometimes this causes slight giddiness. B u t as t i m e passes, t h e m a n begins t o perceive perspective and straight lines correctly. The world a g a i n appears undistorted, even t h o u g h its picture on t h e retina remains askew. U n d e r u n u s u a l c o n d i t i o n s — w h e n the eyes get conflicting i n f o r m a t i o n , when contrasts are great, when correct perception of distances, dimensions, a n d ratios is difficult, or w h e n certain parts of t h e retina are tired from constant s t i m u l a t i o n — our b r a i n falters, a n d various o p t i c a l illusions can occur. W e shall give you some i l l u s t r a t i o n s of h o w our eyes can be m i s t a k e n . W e do not w a n t t o u n d e r m i n e your trust i n your eyes b u t t o show you the i m p o r t a n c e of the synthesis performed b y the b r a i n i n f o r m i n g images. For t h e first experiment, w h i c h is u s u a l l y used as proof t h a t the i m a g e on t h e r e t i n a , l i k e t h a t on a camera, is upside d o w n , we need two pieces of cardboard. T w o postcards, for example, w i l l do. M a k e an opening a b o u t 0 . 5 m m i n d i a m e t e r i n one of the cards w i t h a large needle, and h o l d i t about 2-3 cm from your eye. L o o k t h r o u g h t h e h o l e at a b r i g h t landscape, sky, or l a m p . N o w g r a d u a l l y shade the p u p i l b y slowly m o v i n g t h e edge of t h e second postcard u p w a r d s . The shadow of t h e edge of the postcard w i l l appear t o m o v e downwards from above i n t o t h e field of v i s i o n . Let us discuss the o p t i c a l o u t l a y of the experim e n t i n more d e t a i l . As l o n g as t h e postcard w i t h t h e opening is not held i n front of t h e eye, a l l p o i n t s i n the field of v i s i o n send their rays over 10-01544
146
ft, Kosourov
the entire surface of the p u p i l i n t o the eye (Fig. 110a). A n d the l i g h t from every p o i n t of the p u p i l is d i s t r i b u t e d over the entire surface of the r e t i n a . W h e n we place t h e first postcard i n front of t h e eye, every p o i n t i n t h e held of vision is represented by rays passing t h r o u g h a s m a l l p o r t i o n of the p u p i l (Fig. 1106). The upper p o i n t s are t r a n s m i t t e d b y rays passing
Fig. 110
t h r o u g h t h e lower part of the p u p i l , whereas lower p o i n t s are t r a n s m i t t e d by rays passing t h r o u g h the u p p e r p o r t i o n of the p u p i l . B y s h a d o w i n g the lower p o r t i o n of the p u p i l w i t h the edge of the postcard, we block the u p p e r field of v i s i o n , a n d we see t h e edge of the card descending from above. This u n u s u a l experiment is obviously not dependent on t h e p a t h rays t a k e t o the eye a n d , therefore, cannot be used as, proof t h a t the image i n the retina is u p t u r n e d . The field of vision is formed before t h e l i g h t rays enter the p u p i l . This can easily be proved if the i m a g e is projected onto a m a t glass p l a t e instead of t h e h u m a n eye. O u r second experiment w i l l show h o w t h e eyes h a n d l e conflicting i n f o r m a t i o n . Place a paper t u b e a b o u t 2 cm i n diameter over one eye, and
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147
look t h r o u g h it at objects i n front of you. N o w hold t h e p a l m of your h a n d i n front of your other eye about 10-15 cm a w a y from your face and close to the tube. Y o u w i l l clearly see a hole in t h e p a l m , t h r o u g h w h i c h objects can be seen. The i m a g e of the centre of t h e p a l m is completely suppressed b y the images seen through the tube. A more refined experiment can be performed w i t h t h e different i n f o r m a t i o n the right and left eyes receive. Tie a s m a l l w h i t e object to a w h i t e thread. N o w start this p e n d u l u m swinging i n one p l a n e , and then step back 2-3 m . H o l d a l i g h t filter of any density a n d colour i n front of one eye, and w a t c h t h e p e n d u l u m . Y o u w i l l see t h a t i t is not swinging i n one p l a n e b u t m a k i n g an ellipse. I f you m o v e t h e l i g h t filter i n front of t h e other eye, the m o t i o n of the p e n d u l u m w i l l reverse. The o p t i c a l i l l u s i o n i n F i g . 111a is well k n o w n : the straight l i n e seems t o break when i t intersects t h e b l a c k strip. N o t m a n y people k n o w , however, t h a t if t h e figure is completed b y d r a w i n g a w i n c h a n d a load (Fig. 1116), p r o m p t i n g the b r a i n t o believe t h a t t h e l i n e is a t a u t w i n c h l i n e , the i l l u s i o n of a broken l i n e disappears. O u r last experiment shows h o w a n i m a g e is formed w h e n the b r a i n is given a choice of alternatives. Figure 112 shows t w o pictures of t h e l u n a r landscape i n the area of the Mare H u m o r u m a n d A p e n n i n e R i d g e . I n one of the pictures you see circular l u n a r m o u n t a i n s and i n the other circular l u n a r craters, i . e . , a n inverse landscape. T u r n t h e pictures, and the landscape w i l l reverse. These pictures are absolutely i d e n t i c a l , b u t one of t h e m is upside d o w n . The effect of inverse 10*
ft, Kosourov
148
relief is often observed when t h e Moon is viewed through a telescope. The astronauts w h o visited the M o o n had difficulty correctly perceiving the landscape w h i c h lacked atmospheric perspective in the h i g h l y contrastive surroundings. The effects of reverse m o t i o n can be observed b y
X XI (a)
(6)
Fig. Ill
w a t c h i n g the silhouette of a dish-shaped radar aerial as i t rotates. Y o u w i l l notice t h a t the aerial b l u n t l y reverses the direction of its rotat i o n at certain moments. S t u d e n t s used t o be advised t o s t u d y this phenomenon b y observing the silhouette of a w i n d m i l l . M a n y interesting illusions are related t o colour perception. O p t i c a l illusions are not s i m p l y a m u s i n g tricks, since studies of the organs of sight under u n u s u a l conditions can h e l p e x p l a i n the complex processes i n the; eye a n d b r a i n d u r i n g the synthesis of images of the e n v i r o n m e n t . Readers who wish to learn more about physiological optics are referred to Experiments in Visual Science b y J . Gregg.* The book contains a * Gregg J a m e s R . Experiments Home and School. New Y o r k .
in Visual Science. For R o n a l d , 1966, 158 p.
Believe It or Not
149
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B. Kogan
n u m b e r of s i m p l e experiments w i t h v i s u a l perception. A l l of t h e m are q u i t e m a n a g e a b l e b y school c h i l d r e n , and m a n y are p a r t i c u l a r l y instructive. U n f o r t u n a t e l y , t h e language used t o describe p h y s i c a l optics is far from scientific.
Colour Shadows by B.
Kogan
A Green Shadow I n a room l i t by normal w h i t e l i g h t , t u r n on a desk l a m p w i t h a red b u l b . Place a sheet of w h i t e paper on the desk a n d t h e n h o l d a s m a l l object, a pencil, for example, between t h e l a m p and the desk. The paper w i l l cast a s h a d o w , w h i c h w i l l not be black or grey b u t green. T h i s effect seems to relate more to physiology and psychology t h a n t o physics. The shadow of the object appears green because it contrasts w i t h the b a c k g r o u n d , w h i c h , a l t h o u g h a c t u a l l y reddish, we perceive as w h i t e since we know the paper is w h i t e . The absence of the colour red in the area covered b y the shadow is apparently interpreted b y our brain as the colour green. B u t w h y green? R e d and green are complementary colours, i . e . , when c o m b i n e d , they produce w h i t e . W h a t does this mean? As early as the seventeenth century, Newton found t h a t w h i t e s u n l i g h t is complex and combines the p r i m a r y colours v i o l e t , b l u e , green-blue, green, yellow, orange, and red.. This can be illustrated w i t h a glass prism. If a narrow beam of s u n l i g h t is passed t h r o u g h the p r i s m , a coloured image of the beam w i l l appear. Newtop
Colour Shadows
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also used a lens to combine a l l these colours a n d o b t a i n the colour w h i t e a g a i n . I t was found t h a t when one of the colours, green, for example, is 'intercepted', the beam becomes coloured, i n t h i s case, red. W h e n yellow is intercepted, the beam becomes blue, and so on. Thus, green and red, yellow and b l u e , a n d s i m i l a r pairs of colours are complementary. This interesting experiment can also be carried out w i t h l i g h t bulbs of other colours. I f the l i g h t b u l b is green, for example, the shadow w i l l be red. I f t h e b u l b is blue, t h e shadow w i l l be y e l l o w , and if the b u l b is yellow, the shadow w i l l be b l u e . Generally, the colour of the shadow w i l l always be c o m p l e m e n t a r y to the colour of t h e b u l b . The above p h e n o m e n o n can easily be observed i n winter t i m e near neon advertisements i n t h e c i t y . Shadows from the neon, w h i c h are c o m p l e m e n t a r y to t h e colours of t h e advertisem e n t itself, should show u p clearly when t h e ground is covered w i t h snow. R e d Leaves T u r n oft the l i g h t i n your r o o m , and switch on a l a m p w i t h a blue b u l b . L o o k at the leaves of p l a n t s i n the room: t h e green leaves look red, rather t h a n green or b l u e , i n the b l u e l i g h t . W h a t causes this? A c t u a l l y , the glass of the blue l i g h t b u l b passes a certain a m o u n t of red l i g h t along w i t h the blue. A t t h e same t i m e , p l a n t leaves reflect not only green b u t red l i g h t to some extent as well, w h i l e absorbing other colours. Therefore, when the leaves are i l l u m i n a t e d w i t h blue l i g h t , they reflect only red a n d , therefore, appear as
E. Pal'chikov
152
such i n our eyes. This same effect can be obtained in a different w a y by looking at the leaves through blue spectacles or a b l u e light filter. The famous Soviet scientist K . T i m i r y a z e v was speaking about such eye-glasses when he wrote: " Y o u have only t o p u t t h e m on, and the w h o l e world looks rosy for y o u . Under a clear blue sky a fantastic landscape of coral-red meadows and forests rolls out ..."
W h a t Colour is Brilliant Green? by E.
Pal'chikov
W h a t colour is t h e ' b r i l l i a n t green' often used as an antiseptic for m i n o r bruises a n d wounds? M a n y w o u l d p r o b a b l y answer t h a t i t is green (and they w o u l d be right). B u t look through a bottle of the b r i l l i a n t green at a bright light source, the sun, the filament i n an electric b u l b , or an arc discharge, for example. Y o u w i l l see that the b r i l l i a n t green transmits only the colour red. So, is the 'green' tincture red? Pour a b r i l l i a n t green s o l u t i o n * i n t o several developing trays of various depth or thin-wall glass beaker, and examine it i n the l i g h t . T h i n layers of the s o l u t i o n are, indeed, green, b u t thicker layers h a v e a grayish t i n t ( w i t h purple hues), whereas the thickest layers appear reddish. I n other words, the colour depends on the thickness of the layer of solution. H o w this can be explained? T w o transparency b a n d s — a broad * The tincture is not diluted, the vessels must he extremely shallow.
What Colour is Brilliant Green?
153
blue-green b a n d and a narrow red b a n d (Fig. 113) — are visible i n the transmission band of a t h i n layer of b r i l l i a n t green. I n r e a l i t y , however, the red b a n d is not narrow: it extends i n t o the infrared range, a l t h o u g h the h u m a n eye can detect o n l y a s m a l l fraction of the b a n d . The a b s o r p t i o n
Ultraviolet light
--
O
V i s i b l e liulit
^ Fig. 113
in the red b a n d is lower t h a n t h a t in the bluegreen b a n d (the transmission factor for t h e red b a n d is s u b s t a n t i a l l y greater t h a n t h a t for the blue-green). B u t the blue-green band is wider t h a n the red, and it is situated i n the part of the spectrum where the eye is most sensitive. Therefore, a solution of b r i l l i a n t green in a t h i n layer w i l l appear green. N o w let us double the thickness of the layer or place two layers over one another, w h i c h
ft, Kosourov
154
w i l l h a v e the same effect. O b v i o u s l y , the transmission factor w i l l decrease. To get the v a l u e of this new transmission, we m u l t i p l y the factors of the first a n d second layers. I n other words, the transmission factor for the t o t a l layer should be squared. I n this case, the transmission factor for the blue-green band w i l l decrease very signifi-
c a n t l y , whereas it w i l l remain almost the same for the red b a n d . Figure 114 shows changes in the transmission factors for the blue-green and red bands w i t h increases in the thickness of the b r i l l i a n t green. The proportion of blue-green t o red o b v i o u s l y decreases. A t a certain thickness, the solution w i l l t r a n s m i t o n l y red l i g h t . N o w answer t h e question again. W h a t is the real colour of b r i l l i a n t green?
An Orange Sky by G.
Kosourov
A n u m b e r of interesting experiments concern colour perception. Those we suggest here i n v o l v e
An Orange Sky
155
various optical illusions caused b y u n u s u a l v i s u a l conditions or eye fatigue. Colour perception is a very complex m e c h a n i s m , w h i c h has not yet been studied adequately. The retina of the eye contains two types of coloursensitive cells called rods and cones. The rods contain p h o t o c h e m i c a l l y sensitive p i g m e n t , i . e . , p u r p u r a or rhodopsin. W h e n acted u p o n b y l i g h t , rhodopsin decolourizes and reacts w i t h the nerve fibres, w h i c h t r a n s m i t signals to the b r a i n . I n very bright l i g h t , t h e p i g m e n t decolourizes completely, and the rods are b l i n d e d . The process is reversed i n the d a r k , i . e . , p u r p u r a is recovered. R o d or t w i l i g h t v i s i o n is very sensitive b u t achromatic, since t h e rods cannot d i s t i n g u i s h colours. I n fairly b r i g h t l i g h t , cone vision, w h i c h is sensitive to colour, takes over. M a n y c o n v i n c i n g experiments i n d i c a t e t h a t the cones contain three k i n d s of p h o t o c h e m i c a l l y sensitive p i g m e n t s which are m a x i m a l l y sensitive i n the red, green, and blue bands of the spectrum. The v a r i a b l e degree of their decolourization produces the sensation of colour i n the b r a i n and allows us to see the world in different colours, t i n t s , halftones, and hues. This p r i n c i p l e of t r i c h r o m a t i c vision is used in m o t i o n pictures, colour television, photography, and p r i n t i n g . Methods for measuring colours q u a n t i t a t i v e l y are also hased on the t r i c h r o m a t i c p r i n c i p l e . Colour perception can be generated not only b y colour itself b u t b y i n t e r m i t t e n t i l l u m i n a t i o n , for example. To test this, draw the black-andw h i t e circles shown i n F i g . 115a-d w i t h I n d i a i n k . Y o u r circles should be a p p r o x i m a t e l y 812 cm in diameter. Cut out the discs, and spin
ft, Kosourov
156
t h e m s l o w l y , e.g., 1-3 revolutions per second, on t h e axis of a film projector, record p l a y e r , tape recorder, or a c h i l d ' s t o p . I n s t e a d of b l a c k arcs you w i l l see coloured circles. The colour
(«)
(c)
(b)
id)
Fig. 115
depends on the velocity of the revolutions, the i l l u m i n a t i o n , and the design on the circle itself. O n the disc i n Fig. 115a, for example, the arcs t h a t follow the black sectors (in the direction of r o t a t i o n ) appear red when poorly l i g h t e d , and yellow u n d e r d a z z l i n g l i g h t . A t a certain speed and brightness, the black sectors appear blue.
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This phenomenon is s t i l l i n c o m p l e t e l y understood. Colours are distinguished not only by shade or variety b u t also b y s a t u r a t i o n . I f we slowly add w h i t e p a i n t to red, t h e red w i l l g r a d u a l l y become pinker. I n p a i n t i n g s a n d , p a r t i c u l a r l y , i n p r i n t e d copies of p a i n t i n g s , i t is very difficult t o o b t a i n well saturated tones a n d a broad spectrum of brightness. The brightness r a t i o of the brightest w h i t e p a i n t t o the deepest b l a c k barely reaches one h u n d r e d , whereas i n nature the ratios reach m a n y thousands. R e p r o d u c t i o n s of p a i n t i n g s , therefore, often appear either washed out or too d a r k , a n d one of t h e most i m p o r t a n t elements of steric perception—atmospheric perspective—is, t h u s , lost. The i m a g e of a landscape or a genre scene i n such p a i n t i n g s seem two-dimensional. The range of brightness i n projections of slides o n t o a screen is m u c h broader. T h a t is w h y photographs on colour reversible film are so expressive a n d h a v e such w o n d e r f u l perspective. The perception of a p a i n t i n g can be i m p r o v e d considerably b y i l l u m i n a t i n g i t i n a special w a y . M a k e a negative f r o m a colour picture out of a m a g a z i n e a n d t h e n use a contact p r i n t i n g process t o m a k e a black-and-white slide of t h e negative. N o w project the slide onto the o r i g i n a l using a projector w i t h a powerful l i g h t source. M a k e sure t h a t the projected slide lines u p exactly w i t h the o r i g i n a l . The result w i l l m a k e y o u glad you m a d e t h e effort. The picture w i l l seem livelier; i t w i l l seem to g a i n d i m e n s i o n a n d a special charm. N o w t u r n off t h e projector, a n d you w i l l see h o w d u l l a n d inexpressive the o r i g i n a l is w i t h u n i f o r m l i g h t i n g .
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ft, Kosourov
The f o l l o w i n g experiments deal w i t h so-called successive colour images. Complete recovery of colour-sensitive p i g m e n t is a rather slow process. I f you look at a m o n o c h r o m a t i c picture for a long time and then shift your eyes t o a piece of w h i t e paper or a w h i t e w a l l or ceiling, the w h i t e w i l l appear to lack the colour t h a t has tired t h e eyes. The same picture w i l l appear on the w h i t e surface b u t i t w i l l be i n t h e complement a r y colour. Cut out red, orange, y e l l o w , green, blue, a n d violet paper squares 2 b y 2 cm i n size. P u t one of these coloured squares on a piece of w h i t e paper i n front of you a n d look at i t , w i t h o u t s t r a i n i n g your eyes, for about 30 seconds. Stare fixedly at one p o i n t , and do not let t h e i m a g e shift on t h e r e t i n a . N o w shift your gaze t o a field of w h i t e , a n d after a second y o u w i l l see a clear afterimage of the square i n a c o m p l e m e n t a r y colour. T h i s shows t h a t the c o m p l e m e n t a r y t o red is green, t o blue—orange, a n d t o y e l l o w — violet. E a c h p a i r of complementaries, if m i x e d , should produce achromatic w h i t e or grey. To m i x complementaries place t w o 'complement a r y ' squares (red and green, for example) close together, a n d p u t a glass p l a t e between t h e m u p r i g h t (Fig. 116). N o w position y o u r eye so t h a t one of the squares is visible t h r o u g h the glass and the other reflected i n i t . B y v a r y i n g the angle of the p l a t e a n d t h u s changing t h e r a t i o of t h e l i g h t fluxes from t h e squares, you can almost completely decolourize the superimposed images. To achieve complete a c h r o m a t i z a t i o n of the image, t h e colours m u s t m a t c h perfectly. A d u l l brown colour is most often o b t a i n e d . B u t if the colours are absolutely u n c o m p l e m e n t a r y , green
159
An Orange Sky
a n d yellow or red and violet, for example, t h e r e s u l t i n g colour w i l l always be b r i g h t . The images appear even brighter if the squares are placed against a c o m p l e m e n t a r y , rather t h a n w h i t e background. The most s t r i k i n g a n d inexplicable colour i l l u s i o n is illustrated b y our last experiment. W e k n o w t h a t colour reproduction is based on
Fig. 116
t h e p r i n c i p l e of t r i c h r o m a t i s m . I f we p h o t o g r a p h t h e same scene three t i m e s using three different light filters—red, green, a n d b l u e — a n d t h e n project the pictures from three different projectors onto t h e same screen, the resulting picture w i l l have realistic colours. The l i g h t filters should produce t h e colour w h i t e when c o m b i n e d . Try this experiment w i t h o n l y t w o c o m p l e m e n t a r y l i g h t filters, red a n d green, for e x a m p l e . The transmission of colour should be good i n t h i s case too, a l t h o u g h not as perfect as i n the threecolour projection. E x p e r i m e n t s show t h a t even one filter is enough for projection. P h o t o g r a p h the same scene twice on panchro m a t i c film w i t h o u t m o v i n g t h e camera. Use a red
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light filter for the first picture a n d a green'filter for t h e second. The filters from a school k i t w i l l suit t h i s purpose since they need not be caref u l l y m a t c h e d . Use a contact p r i n t i n g t e c h n i q u e to m a k e positives, and then project t h e two slides Red li.'iit fi!tc-1-
Fig. 117 from t w o projectors onto a • single screen. L i n e the images u p exactly. N o w place a red filter i n front of t h e projector w i t h t h e slide taken w i t h a "red filter. Leave t h e picture i n t h e second projector b l a c k and w h i t e (Fig. 117). The result w i l l be a colour picture f u l l of tones a n d hues even t h o u g h y o u are projecting o n l y red a n d b l a c k a n d w h i t e pictures i n w h i c h the d i s t r i b u t i o n of
The Green Red Lamp
161
l i g h t and shade differs. Objective investigations of t h e l i g h t reflected from t h e various places on t h e screen show o n l y t h e colour red, a l t h o u g h w i t h different degrees of c l a r i t y a n d s a t u r a t i o n . Colour perception i n t h i s p a r t i c u l a r case is entirely subjective. The projectors should be powered b y separate autotransformers so t h a t t h e i l l u m i n a t i o n from each can be controlled i n d e p e n d e n t l y . N o r m a l l y , t h e colour seems n a t u r a l when t h e screen is d i m l y l i t . Thus, phenomena t h a t seem s i m p l e and o b v i o u s are a c t u a l l y f u l l of secrets and mystery.
The Green Red L a m p by V.
Mayer
In his excellent book Ihe Universe of Light, W. Bragg describes a n elegant experiment to demonstrate a peculiar property of the h u m a n eye. The experiment is s i m p l e enough to be reproduced at h o m e w i t h o u t m u c h difficulty. The necessary e q u i p m e n t can be assembled from a t o y constructor k i t (Fig. 118). A t t a c h a m i c r o m o t o r (2) b y a n a l u m i n i u m or t i n p l a t e m o u n t t o a n a l u m i n i u m baseplate (1) 15 X 60 X 110 m m i n size. Insert a shaft (4) w i t h a p u l l e y (5) whose inner diameter is 1525 m m i n t o the openings of t w o risers (3) a b o u t 60 m m in h e i g h t . A t t a c h a cardboard disc (6) 100-140 m m i n diameter to one end of t h e shaft before install i n g (if the shaft is threaded, t h e disc can be * The Universe of London, Dover, 1950.
SI—01544
Light.
By
Sir
William
Bragg,
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V. Mayer
The Green Reel Lamp
163
fastened w i t h nuts). Fix another riser (7) t o the same base plate w i t h an opening for a s m a l l l i g h t b u l b (#). Connect the shaft of t h e m i c r o m o t o r a n d t h e p u l l e y w i t h a rubber ring (9). The m o t o r s h o u l d be connected to one or two flashlight batteries connected i n series. According to Bragg, the disc should rotate at a speed of 2 or 3 revol u t i o n s per second. The speed can be controlled t o a certain extent b y slowing the p u l l e y w i t h one finger. Test the setup before b e g i n n i n g the experim e n t . Then cut a sector i n the cardboard disc whose arc is about 45°. G l u e w h i t e paper t o one h a l f of the r e m a i n i n g sector and black paper t o the other half (Fig. 119). P a i n t the b u l b of the flashlight, w h i c h should be rated for 3.5 V , w i t h red nitrocellulose enamel (fingernail polish w i l l do). W h e n the enamel has dried, insert t h e l i g h t b u l b i n t o the opening in the riser, and s u p p l y it w i t h wires to the batteries. Place a desk l a m p 20-50 cm i n front of the disc to i l l u m i n a t e i t squarely. N o w connect the b u l b to the batteries, a n d switch on the m o t o r . I f the disc rotates so t h a t on each r e v o l u t i o n the b u l b is first shaded b y the black sector, the b u l b w i l l appear red regardless of the strength of i l l u m i n a t i o n or r o t a t i o n speed. I f we change directions b y c h a n g i n g t h e p o l a r i t y of t h e batteries so t h a t t h e b u l b is shaded first b y the w h i t e sector, the b u l b w i l l appear green or blue-green! I f the conditions of the experiment (the speed of r o t a t i o n or t h e colour of the enamel) h a v e not been carefully fulfilled, the l a m p w i l l appear l i g h t blue w i t h a w h i t i s h tinge rather t h a n blue-green. 11*
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N. Rostovtsev
N o w let us t r y t o explain t h e results of the experiment. The disk spins r a p i d l y , and each t i m e t h e red b u l b is revealed t h r o u g h the cut-out sector, a brief red i m a g e reaches t h e retina of the eye. I f t h e w h i t e sector shades t h e b u l b as t h e r o t a t i o n continues, the retina receives a reflection of the scattered w h i t e l i g h t from t h e desk l a m p . This w h i t e l i g h t acts on the retina for a longer t i m e t h a n the red. After the b l a c k sector passess before t h e eye, the process starts a g a i n , and the l a m p appears blue-green since t h e eye perceives the colour c o m p l e m e n t a r y to t h e red b u l b . The retina a p p a r e n t l y becomes more sensitive to t h e other spectral components of w h i t e after brief i l l u m i n a t i o n w i t h red l i g h t . W h e n the eye, w h i c h 'tires' of the colour red, is i l l u m i n a t e d b y w h i t e l i g h t , i t perceives t h e w h i t e w i t h o u t a 'red c o m p o n e n t ' . The retina has become more sensitive t o the colour c o m p l e m e n t a r y t o red— blue-green. Since the retina is exposed t o the white l i g h t m u c h longer t h a n t o t h e red, the l i g h t b u l b appears blue-green rather t h a n red. This hypothesis is supported b y t h e result of r o t a t i o n i n t h e opposite direction so when t h e b u l b is shaded first b y the b l a c k sector. D u r i n g exposure t o the colour b l a c k t h e p a r t of the retina on w h i c h the red image of t h e b u l b appears is able t o recover. Therefore, w h e n the w h i t e h a l f of t h e disc appears, the eye perceives a l l the colours t h a t compose w h i t e e q u a l l y . Since the red l i g h t acts on the retina for a longer t i m e t h a n a l l other components (first t h e red component of the w h i t e colour appears a n d then the red colour of the b u l b itself), the b u l b looks red.
Measuring Light Wavelength with a Wire
165
Measuring Light Wavelength with a W i r e by N.
Rostovtsev
Stretch a t h i n wire {W) 0.05-0.12 m m i n diametre vertically a b o u t 2-3 m m i n front of a n i m a g i n a r y eye (E). N o w direct a beam of l i g h t from a p o i n t source towards the eye (Fig. 120). To the right and left of the p o i n t source, we w i l l
see a b r i g h t t h i n b a n d . The b a n d , w h i c h appears as a result of the d i S r a c t i o n of l i g h t is called a diffraction fringe. W e w i l l use an ordinary l i g h t b u l b or a b u l b from a flashlight as a l i g h t source i n our observations. Place the l i g h t b e h i n d a s m a l l opening (0) in a screen (.A) 1-1.5 m a w a y from t h e observation p o i n t . The wire (W) can be replaced w i t h a t h i n filament or h a i r . I f we examine t h e centre of t h e diffraction fringe carefully, we can detect a w h i t e b a n d w i t h reddish edges. This b a n d is called the central m a x i m u m . I t is bounded on either side b y darker bands, w h i c h are called first m i n i m a . Colour bands follow next, w h i c h , as we m o v e from t h e centre t o the edges, change g r a d u a l l y from green-
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ish-blue to red. Darker bands, called second m i n i m a , reappear at the edge of t h e red. This p a t t e r n then repeats, a l t h o u g h t h e m i n i m a become paler, a n d t h e l i g h t bands finally merge i n t o a c o n t i n u o u s b a n d . Observations w i t h wires of different diametres show t h a t the smaller the wire diametre, the greater the distance between adjacent m i n i m a . To perform interesting experiment place the wire used for diffraction observations between the jaws of a vernier caliper. T i g h t e n the caliper s l i g h t l y , a n d t h e n carefully remove t h e wire. The w i d t h of t h e slit between the jaws w i l l equal the diametre of the wire exactly. N o w look t h r o u g h this slit from the same distance from w h i c h y o u m a d e your diffraction observations, a n d a l i g n the slit w i t h the l i g h t source 0 . O n either side of the source, you should see a diffraction fringe whose m i n i m a and m a x i m a are exactly t h e same distance apart as those i n t h e diffraction of the wire. This observation is an excellent i l l u s t r a t i o n of the B a b i n e t principle, according t o w h i c h diffraction patterns from a screen a n d a n opening of the same w i d t h are i d e n t i c a l outside t h e area of a direct ray. N o w coil t h e piece of wire we h a v e been using for our observations of the diffraction fringe i n t o a disc whose diametre is abou,t h a l f the diametre of a d i m e . For this we w i l l need a wire 2-3 m l o n g . H o l d t h i s disc i n front of one eye, and look at a p o i n t source. Y o u should see a n u m b e r of haloes: a central w h i t e circle w i t h reddish fringe, surrounded b y coloured circles. The haloes are separated from one another by narrow d a r k circles, i.e. the m i n i m a . Each such m i n i m u m
Measuring Light Wavelength with a Wire
167
follows the red fringe of t h e preceding h a l o . I f t h e observation is m a d e from the same distance as i n the experiment w i t h t h e diffraction of t h e wire, the diametres of t h e d a r k circles w i l l e q u a l the distances between the respective m i n i m a of the diffraction fringe. The finer the d i a m e t r e of the wire, the more v i s i b l e t h e haloes w i l l be. W h y do such haloes appear if a coil of w i r e is placed i n the p a t h of rays from a p o i n t source? E a c h s m a l l section of t h e wire i n front of t h e eye produces its own diffraction fringe, w h i c h i s symmetrical w i t h respect t o the l i g h t source. Since every section i n t h e coil is oriented differe n t l y , the resulting diffraction fringes are t i l t e d differently around a single p o i n t , w h i c h coincides w i t h the l i g h t source. Since the thickness of t h e wire is u n i f o r m , m i n i m a of t h e same order are located the same distance from the l i g h t scurce i n a l l diffraction fringes and merge t o p r o d u c e dark circles. The coloured sections between t h e m i n i m a also merge t o produce coloured circles. N o w let us determine the conditions i n w h i c h m i n i m a appear i n t h e diffraction p a t t e r n produced b y a wire w i t h d diametre and a s l i t of t h e same w i d t h . Since t h e distance between t h e m i n i m a is the same i n b o t h cases, either t h e w i r e or the slit can be analysed. To s i m p l i f y t h e c a l c u l a t i o n , we shall select the s l i t . Let us examine the waves t h a t do not change direction after passing through the s l i t (in F i g . 121 t h e y are represented by dotted lines). The eye converges them on the retina at p o i n t 0 . W a v e s from a l l points of the slit enhance each other at t h i s spot because they reach the eye regardless of t h e distance travelled, and at p o i n t O they are i n
168
N. Rostovtsev
the same phase. Therefore, t h e central m a x i m u m is formed i n the neighbourhood of t h e p o i n t 0 . The eye converges waves diffracted at angle
the s l i t . D r a w a section BC perpendicular to the ray e m a n a t i n g from p o i n t A. The new intercept AC equals t h e difference between t h e distances travelled b y t h e t w o extreme rays. I t follows from F i g . 121 t h a t AC — d sin
(1)
where ^ is t h e l i g h t wavelength and k is the n u m b e r (order) of a m i n i m u m (k = 1, 2, 3, . . .). N o w we s h a l l test the v a l i d i t y of f o r m u l a (1) for t h e first m i n i m u m , i.e. k — 1. Let the second-
Measuring Light Wavelength with a Wire
169
ary waves t h a t e m a n a t e from a l l p o i n t s of t h e slit travel at an angle
(2)
D i v i d e the slit i n t o t w o i m a g i n a r y rectangular strips (zones) AD and DB, b o t h of w i d t h d!2. A c c o r d i n g t o d e f i n i t i o n (2) the difference i n t h e distance travelled b y rays from p o i n t s A a n d D is X/2. The difference between rays from a n y t w o p o i n t s dl2 apart on t h e slit w i l l be t h e same. The waves t h a t t r a v e l X/2 suppress one another b y superposition a n d , therefore, if c o n d i t i o n (2) is observed, t h e waves from zone AD w i l l suppress t h e waves from DB. A s a result t h e first m i n i m u m w i l l appear at p o i n t K. S i m i l a r l y , we can show t h a t the next (second) m i n i m u m w i l l appear i f d sin
dsinJ K
t
(3)
Measurements of X can be simplified considerably b y using a p r i m i t i v e metre called an eriometre. Y o u can m a k e an eriometre from a square piece of cardboard whose sides are 10-15 cm long.
N. Rostovtsev
170
D r a w a circle w i t h a radius of 20-30 m m i n t h e m i d d l e of t h e square. M a k e a n o p e n i n g of 2-3 m m i n diametre i n the centre a n d 6-8 openings of smaller diametre around t h e circumference. Place the eriometre A directly i n front of a n electric b u l b . N o w stand 1-2 m a w a y from t h e i n s t r u m e n t so t h a t rays pass d i r e c t l y from a sector of the incandescent filament t h r o u g h openi n g 0 t o the eye. H o l d a coil of wire i n front of one eye, and m o v e it perpendicular t o the rays u n t i l t h e haloes are clearly v i s i b l e . V a r y t h e distance between the i n s t r u m e n t a n d your eye t o find a position from w h i c h the perforated circumference of the eriometre coincides w i t h the m i d d l e of dark r i n g of k order (in F i g . 120, k = 2). As is obvious from the figure, the tangent of the diffraction angle cp for a dark r i n g is calculated from t a n cp = r!l, where r is the r a d i u s of t h e circumference of the eriometre a n d I is the distance from the i n s t r u m e n t t o the coil of wire. A t low diffraction angles, w h i c h are c o m m o n i n such measurements, the f o l l o w i n g r e l a t i o n s h i p is true sin
t a n
y
B y s u b s t i t u t i n g t h e v a l u e of sin tp i n expression (1), we get a formula for w a v e l e n g t h
<
4
>
W e already know t h e r a d i u s of t h e eriometre r. The distance I can be easily measured. The order of a d a r k r i n g k is determined b y obsarving the haloes. The wire diametre d is measured w i t h a
Measuring Light Wavelength with a Wire
171
micrometre. I f the measurements are t a k e n i n w h i t e l i g h t , we can determine the effective l i g h t wavelength to w h i c h t h e h u m a n eye is most sensitive w i t h f o r m u l a (4). This wave is approxi m a t e l y 0.56 fim l o n g , and waves of this length correspond to the green section of the colour spectrum. Haloes m a y appear i n diffraction p a t t e r n s caused b y round obstacles. They can be observed b y spreading a s m a l l a m o u n t of l y c o p o d i u m powder (composed of t h e spores of a club moss, i t can be obtained i n a n y drugstore) on a glass p l a t e . G e n t l y t a p the edge of t h e p l a t e against your desk to remove excess powder. N o w , look t h r o u g h the p l a t e at a l i g h t source. Y o u should see haloes formed b y the r o u n d spores, w h i c h act as obstacles. Especially b r i g h t haloes appear around a drop of blood pressed between t w o glass plates. I n this case the diffraction is caused by red blood cells called erythrocytes. The haloes produced b y r o u n d and rectangular obstacles differ s l i g h t l y . The m i n i m u m c o n d i t i o n for haloes from rectangular obstacles is described by formula (1). The m i n i m u m c o n d i t i o n for haloes from r o u n d obstacles is d sin cp =
1.22*,; 2.23X; . . .
(5)
Here d is the diametre of a round screen. W i t h the help of an eriometre a n d f o r m u l a (5), we can determine the average diametre of ciub moss spores and erythrocytes w i t h o u t a microscope! H a l o e s ' c a n be observed around the S u n , t h e M o o n , and even other planets. These haloes appear when l i g h t passes t h r o u g h clusters of water drops or ice crystals suspended i n the
A. Bondar
172
atmosphere (through a t h i n c l o u d , for example). Clear haloes appear only if t h e cloud is m a d e of drops of equal diametre or crystals of the same thickness. I f the drops or crystals v a r y i n size, however, rings of different colours superimpose to produce a w h i t i s h corona. This is w h y a h a l o appears around the Moon, p a r t i c u l a r l y at t w i l i g h t on a clear d a y . The water v a p o u r i n the atmosphere condenses s l i g h t l y on such nights and produces s m a l l drops or crystals of the same size. Haloes sometimes occur w h e n the l i g h t from a d i s t a n t l a m p passes t h r o u g h a fog or a w i n d o w p a n e covered w i t h a t h i n layer of ice crystals or condensed v a p o u r .
EXERCISES 1. The effective light wavelength is approximately 0.56 (j,m. Using an eriometre, determine the diametre of the strands in nylon stockings and ribbons. 2. How does the appearance of the halo indicate whether the cloud contains water droplets or ice crystals? 3. The angular diametre of the Moon is 32 minutes. Determine the diametre of the drops in a cloud, if the angular radius of the central circle of its halo is four times the angular diametre of the Moon.
Measuring Light with a Phonograph Record by A.
Bondar
O n e of t h e most accurate ways of determining the spectral composition of r a d i a t i o n is the m e t h o d based on diffraction. A diffraction g r a t i n g is a good spectral i n s t r u m e n t . W e can observe
Measuring Light with a Phonograph Record
173
diffraction and even measure the wavelength of visible l i g h t w i t h a standard phonograph record. I n acoustic recording evenly spaced grooves are cut on the surface of a d i s k . These grooves scatter l i g h t , whereas the i n t e r v a l s between t h e m reflect i t . I n t h i s w a y the disk becomes a reflecting diffraction g r a t i n g . I f t h e w i d t h of reflecting strips is a and t h e w i d t h of scattering strips is b, then the value d = a + b is the period of t h e grating. Consider a p l a n e m o n o c h r o m a t i c wave of l e n g t h X w h i c h is i n c i d e n t at angle 8 to a g r a t i n g w i t h period d. According t o t h e Huygen-Fresnel principle, every p o i n t of a reflecting surface becomes an i n d i v i d u a l p o i n t source sending out l i g h t i n a l l possible directions. Consider the waves travell i n g at an angle cp t o t h e g r a t i n g (see F i g . 122). These waves can be collected at one p o i n t w i t h a condensing lens (the crystalline lens of the eye, for example). Let us determine under w h a t comb i n a t i o n of conditions the waves w i l l enhance one another. The difference between distances travelled by rays 1 a n d 2 issued by p o i n t s A and B from neighbouring reflecting areas (Fig. 123) is | AK
| =
| NB
|=
d sin cp — d sin 9 = = d (sin cp — sin 6)
(KB is the front of t h e reflected wave directed at angle cp, AN is the front of the i n c i d e n t wave). I f t h e difference is a c o m m o n m u l t i p l e of the wavelength, the phases of oscillations t r a v e l l i n g from p o i n t s A and B w i l l be equivalent, and w i l l enhance each other. A l l other reflecting areas of the g r a t i n g behave s i m i l a r l y . Therefore, the
A. Bondar
174
c o n d i t i o n of central m a x i m u m is described as d ( s i n cp — s i n 6) =
kX
(1)
where k = 0, 1, 2, . . .. Hence we can t h e wavelength k. For this we need to g r a t i n g period d, the incidence angle wave w i t h respect to the g r a t i n g , and
Fig. 122
determine know the 0 for the the angle
Fig. 123
of its direction to a corresponding m a x i m u m (p. N o r m a l l y , the g r a t i n g period is m u c h larger t h a n the wavelength (d is m u c h larger t h a n X), and angles cp are, therefore, s m a l l . This means t h a t the central m a x i m a are s i t u a t e d very close t o one another, and the diffraction p a t t e r n is rather h a z y . The larger the incidence angle (9), however, the larger the cp angles a n d , consequently, the more convenient the measurements. Thus, the rays s h o u l d be directed towards the grating at an angle. So far we have been discussing m o n o c h r o m a t i c l i g h t . W h a t if complex w h i t e l i g h t strikes such a g r a t i n g ? I t is clear from e q u a t i o n (1) t h a t the l o c a t i o n of every central m a x i m u m depends on w a v e l e n g t h . The shorter the wavelength, the s m a l l e r the angle cp corresponding t o the maxi-
Measuring Light with a Phonograph Record
175
m u m . Thus, all m a x i m a (except for the central) stretch out i n a spectrum whose violet end is directed towards the centre of the diffraction p a t t e r n and whose red end is directed o u t w a r d . Two spectra of t h e first order, followed b y t w o spectra of the second order, a n d so on lie on either side of the central (zero) m a x i m u m . The distance between corresponding lines of spectra increases w i t h an increase i n the order of t h e spectrum. As a result, spectra m a y overlap. I n the spectrum of t h e S u n , for example, secondarid third-order spectra overlap p a r t i a l l y . N o w let us t u r n to the experiment itself. To measure the w a v e l e n g t h of a specific colour, we need to determine t h e period of grating (d), t h e sine of the incidence angle of the ray w i t h respect to the g r a t i n g (sin 0), a n d the sine of the angle t h a t determines t h e direction towards a m a x i m u m , for example, the m a x i m u m of the first order (sin cp]). The period of the lattice can easily be determined b y p l a y i n g the record:
Here A R is the absolute displacement of the stylus along t h e r a d i u s of t h e record i n At t i m e , a n d n is the n u m b e r of revolutions per u n i t of t i m e . U s u a l l y , d is a p p r o x i m a t e l y 0.01 c m . A desk l a m p can be used as a l i g h t source. M a k e a screen w i t h a slit out of cardboard, and cover the l a m p t o a v o i d l i g h t interference i n diffraction p a t t e r n analysis. The filament of t h e b u l b should be v i s i b l e t h r o u g h the s l i t . Place the l a m p close t o one w a l l of the room. Place t h e record h o r i z o n t a l l y near the opposite w a l l . N o w
ft, Kosourov
176
find the i m a g e of the slit (see F i g . 124). Y o u s h o u l d see the diffused colour bands t h a t indicate the spectrum of the first order (k =• 1) simultaneously. I t is easy to prove t h a t t h e greater the angle 0, t h e wider the colour i m a g e of the slit a n d t h e more accurate measurements of the angle at w h i c h the ray i n question is diffracted w i l l be.
Fig. 124
To determine sin cpx, have a friend h o l d a pencil (or some other object) over the slit so t h a t its image coincides w i t h the selected spectrum b a n d i n the l i g h t reflected from t h e g r a t i n g (as from a flat mirror) (see Fig. 124). Once we have measured a, b, a n d h w i t h a ruler, we can determine sin (f! and sin 8 a
sin
These expressions could be somewhat simplified. Since b
/af +
1
ft"
/ l + 62/«2
2 a*
A Ball for a Lens
177
and a 2
^ 2
]/a + A
1
! _ J _ 2
2
2 a2
] / l + fc /a ~
A n d finally A, = d (sin (p t — sin
/j a —
ja
I t is interesting t o compare t h e estimated wavelengths of various colours w i t h the values i n the reference tables. I n our experiments the error, w i t h very careful measurements, was on the order of 10~8 m . This level of accuracy is q u i t e acceptable for w a v e l e n g t h i n the v i s i b l e b a n d (A, is a p p r o x i m a t e l y 10 -7 m ) .
A Ball for a Lens by G.
Kosourov
Geometrical optics is based on the idea t h a t l i g h t rays m o v e i n straight lines. Y o u can prove this for yourself e x p e r i m e n t a l l y . R e p l a c e t h e objective lens of your camera w i t h a sheet of b l a c k paper w i t h a very s m a l l opening i n i t . B r i g h t l y i l l u m i n a t e d objects can be photographed w i t h such a device, called a camera obscura. The picture i n Fig. 125, for example, was t a k e n w i t h a n ordinary camera whose objective lens was replaced w i t h a sheet of b l a c k paper i n w h i c h a n opening 0.22 m m i n diametre had been m a d e . The A S A 80 film was exposed for 5 seconds. The i m a g e on the film coincides exactly w i t h t h e cent r a l projection of the p o i n t s of the object by straight 12-01544
178
ft, Kosourov
lines passing through the opening. The image is a hard evidence that light rays travel in straight lines. A shadow on a white screen cast b y an opaque object is explained as a projection of the contour of the object on the plane of the screen by rays
Fig. 125
from every p o i n t of the l i g h t source. Since the l i g h t source is u s u a l l y rather large, the dark centre of the shadow called the u m b r a is bounded by a diffused semisliadow - or p e n u m b r a . W o m i g h t expect to reduce or even e l i m i n a t e the p e n u m b r a by reducing the size of the light source. E x p e r i m e n t s show q u i t e the opposite, however. W h e n the light source is fairly s m a l l , it reveals phenomena t h a t were earlier masked by the p e n u m b r a . For e x a m p l e , the straight edge
A Ball for a t.i'ns
179
of an opaque plate casts the shadow shown i n Fig. 120. The shadow was photographed at a distance of 0.5 m from the screen i n Avhite l i g h t through a red lilter, a n d the picture was t h e n enlarged. The distance between the first t w o dark bands is 0.0 m m . The edge of the shadow is diffused, and dark a n d bright bands of d i m i n i s h i n g contrast, lie parallel to the edge. I f the light source is w h i t e , the bands are all the colours of the spectrum. A shadow cast b y a t h i n wire (Fig. 127) also has a complex structure. The edges are fringed w i t h bands s i m i l a r to those of the opaque p l a t e , b u t there are dark a n d bright bands w i t h i n the umbra whose w i d t h reduces w i t h the thickness of the wire. (This picture was taken in w h i t e light through a red filter. The wire diameter is 1.2 m m , and the distance from the wire t o the camera is 0.5 m . ) The shadow cast b y a b a l l or a s m a l l opaque disk is q u i t e u n u s u a l (Fig. 128). I n a d d i t i o n to the f a m i l i a r dark and b r i g h t circles surrounding t h e shadow, a bright spot appears in the centre of the u m b r a as t h o u g h there were a s m a l l o p e n i n g in the centre of the disk. (For this picture we used a b a l l 2.5 m m i n diametre and a red filter. R 1 = R 2 = 0.5 m . ) Diffraction refers to the phenomena t h a t result when light does not propagate i n strict accordance w i t h the principles of geometrical optics. These phenomena can be explained by the wave nature of l i g h t . W e can get a more exact description of light p r o p a g a t i o n not from the structure of the rays but from the patterns of light wave propagation themselves.
12*
180
ft, Kosourov
Fig. 120
A Ball for a Lens
181
Consider the circular waves t h a t appear w h e n a stone disturbs t h e calm surface of a p o n d . I f the waves reach a log floating on the surface, a clearly defined shadow appears i n the w a k e of the log. The shadow is b o u n d e d b y the rays d r a w n from the p o i n t at w h i c h the stone hits the surface of the water t h r o u g h t h e ends of the log. W i t h i n the shadow, diffraction creates less noticeable waves, w h i c h scarcely d i s t u r b the p a t t e r n of t h e geometrical shadow. I f the waves r u n i n t o a pile, t h e wave p a t t e r n for a very short distance b e h i n d the p i l e does not resemble a geometrical shadow. F i n a l l y , if t h e waves h i t a t h i n pole s t i c k i n g out of t h e water, no shadow appears at a l l since waves m o v e freely around s m a l l obstacles. I n this case o n l y a weak circular wave caused b y t h e pole is v i s i b l e on t h e surface. Thus, sometimes straight rays accurately describe the patterns of wave p r o p a g a t i o n , and sometimes diffraction patterns d o m i n a t e . The p a t t e r n depends on t h e r e l a t i o n s h i p between t h e wavelength, the dimensions of the obstacle (or opening) t h a t l i m i t w a v e p r o p a g a t i o n , a n d the distance to the p l a n e of observation. This relationship can be f o r m u l a t e d as follows: if the obstacle or o p e n i n g can be seen from the p o i n t s of t h e screen on w h i c h we observe the shadow at a n angle greater t h a n t h e angle at w h i c h t h e entire wavelength can be seen from the distance e q u a l t o the w i d t h of t h e obstacle, t h e n the diffraction does not strongly distort the picture of t h e rays. In
the resulting f o r m u l a ^ »
— , a is t h e size
of the opening, R is the distance t o the screen on w h i c h the shadow is observed, a n d X is t h e wave-
182
ft, Kosourov
l e n g t h . I f the angles are comparable, however, or if the first angle is less t h a n the second, i.e., jjr
A Ball for a Lens Figure 129 shows the setup, w i t h which diffraction patterns reproduced i n this were obtained. W e used an optical bench experiments. This is not m a n d a t o r y , of although the diffraction patterns can lie
183
all the article i n the course, moved
Fig. 129: (1) l a m p ; ( 2 ) J p r o j e c t i n g l e n s , (-J) p o i n t d i a p h r a g m w i t h a l i g h t s c r e e n , (4) c l a m p for d i f f r a c t i n g o b j e c t s , (5) l i g h t p r o o f t u b e , (6) p h o t o g r a p h i c camera without objective
more easily i n t o the centre of the field of vision if you have supports w i t h screw displacements. The shadow from a straight-edged, opaque screen can be provided w i t h the b l a d e of a safety razor. Use two such razor blades, to m a k e a slit 0.3-1 m m wide. A piece of wire u p to 1 m m i n diametre w i l l give you the diffraction p a t t e r n
184
ft, Kosourov
of a narrow screen. The t i p of a needle also produces an interesting p a t t e r n . To produce a b r i g h t spot i n t h e centre of a shadow cast b y a r o u n d object use a steel b a l l bearing 2-4 m m i n diametre. A t t a c h the b a l l to a glass p l a t e w i t h a drop of glue. (The glue should not be visible beyond t h e contour of the b a l l , a n d t h e surfaces of the p l a t e a n d t h e b a l l should be clean.) W h e n the glue is d r y , fix t h e plate i n the setup. A d j u s t the p l a t e u n t i l t h e shadow of t h e b a l l is i n the centre of t h e field of v i s i o n . I n t h i s position both the outer diffraction circles and t h e spot i n t h e centre surrounded b y d a r k and b r i g h t circles w i l l be clearly visible. The diffraction patterns can easily be photographed w i t h a camera from w h i c h t h e objective lens has been removed. The shadow of the object s h o u l d be projected directly onto the film i n the camera. To increase the resolution a n d the number of diffraction fringes, t h e opening, w h i c h serves as a l i g h t source, s h o u l d be covered w i t h a l i g h t filter. A red filter works best since there are m a n y red rays i n t h e spectrum of l i g h t from a c o n v e n t i o n a l electric b u l b . N o r m a l l y , an exposure of 5 or 10 seconds is sufficient. To a v o i d overexposure the camera a n d setup should be connected b y a t u b e w i t h a b l a c k l i n i n g . D i s p l a c e m e n t of the l i g h t source w o u l d cause displacement of the shadow, a n d hence the spot itself. Therefore, if you use a slide instead of a p o i n t source i n your setup, every transparent p o i n t on the slide w i l l cast its own, slightly displaced shadow of the b a l l w i t h its own bright spot. A s a result, the outer diffraction circles w i l l diffuse, a n d a n i m a g e of t h e slide w i l l
A Ball for a Lens
185
appear i n the centre of the shadow. The b a l l w i l l act as a lens. The i m a g e of P l a n c k ' s constant i n F i g . 130 was o b t a i n e d i n t h i s w a y . The photograph was m a d e w i t h a b a l l 4 m m in d i a m e t r e . The height of the s y m b o l K is 1 m m . The o r i g i n a l slide was m a d e w i t h contrast film b y photo-
Fig.
130
g r a p h i n g a letter d r a w n i n I n d i a i n k on w h i t e paper. I f y o u d r i l l a n opening about 2 m m i n d i a m e t r e i n a t h i n t i n p l a t e , y o u can see how the diffraction pattern of the o p e n i n g changes at various distances. Cover the o p e n i n g w i t h a l i g h t filter, a n d move closer t o i t . F r o m a distance of 1-2 m , you should be able t o see b l a c k circles i n t h e centre of the pattern w i t h a m a g n i f y i n g glass. A s you move closer, the circles change to d a r k rings, diffusing towards t h e b o u n d a r y of the shadow. The n u m b e r of d a r k rings, i n c l u d i n g t h e d a r k spot i n t h e centre, is determined b y t h e difference
186
ft, Kosourov
in the distances travelled by the central ray and the ray from the edge of the opening. This fact can be used t o determine t h e l i g h t wavelength if the opening diametre, the distance from the light source to the opening, a n d the distance from the opening to the image are known. The plane of the image can be determined by placing
Fig. 131 (») 1 trim in d i a m e t r e . R j - - H 2 = - 0 . 5 m. lied f i l t e r . T h e o p e n i n g shows t w o zones. Black s p o t in t h e c e n t r e ; (b) 1 m m in dia m e t r e . K^—H2=0.5 m . Blue f i l t e r . T h e o p e n i n g shows a l m o s t t h r e e zones; (c) 1.5 m m in d i a m e t r e . Jii=U2=().f> m . R e d f i l t e r . T h e o p e n i n g s h o w s s l i g h t l y m o r e t h a n f o u r zones
a needle in front of the m a g n i f y i n g glass and m o v i n g it u n t i l it resolves sharply against the background of the diffraction pattern. W e w i l l discuss the derivation of the calculation formula later. Y o u can observe the b e a u t i f u l v a r i a t i o n in the colour of diffraction circles in w h i t e l i g h t . These colours, w h i c h do not resemble spectral colours, are called complementaries. They can be observed when one spectral band is missing from the complete spectrum of w h i t e l i g h t . I n this case, for example, when green is represented b y a dark centre spot, the r e m a i n i n g parts of the spectrum, i.e. red-orange and violet, m a k e the centre of
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the picture look purple. The absence of red produces the complementary green-blue colour, and so on. Figure 131 shows examples of diffraction patterns from a round opening. W h y do we have a black spot in the centre of the pattern, which the light rays seem to reach without interference? Let us return to our observation of waves on the surface of a pond. Consider two stones thrown simultaneously into the pond and the resulting two systems of waves. I m a g i n e points on the surface reached by the crests of the
Time
Fig. 132
Fig. 133
two wave systems simultaneously. W i t h t i m e the same points w i l l be reached by the troughs, and the waves w i l l become larger (Fig. 132). This enhancement w i l l occur at points that lie at various distances from the wave sources. The waves w i l l also be enhanced at points that lie an entire wavelength, two wavelength, and so on from the source. W h e n the crests of one wave system meet the troughs of another, the waves dampen one another (Fig. 133). This interference
ft, Kosourov
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plays the decisive role i n f o r m i n g diffraction patterns. E v e r y p o i n t i n space t h a t is pierced b y a l i g h t wave can also be regarded as a source of a secondary spherical wave (Fig. 134). I f t h e l i g h t passes through a round opening, we can replace the l i g h t source w i t h secondary l i g h t sources distrib u t e d over the area of the o p e n i n g . A l l these sources w i l l fluctuate i n concord w i t h the first wave to reach the opening. The a m p l i t u d e of the
Fig. 134
fluctuations at a p o i n t b e h i n d the screen is calculated as the sum of the fluctuations caused at the observation p o i n t b y each secondary source. W a v e s from different sources travel different distances and can enhance or d a m p e n one another when c o m b i n e d . Let us observe changes i n t h e a m p l i t u d e of oscillations around the axis of a r o u n d opening i l l u m i n a t e d b y a p o i n t source. W h e n t h e distance t o a n observation p o i n t is very great i n comparison w i t h the diametre of the opening, t h e waves from all secondary sources travel almost the same distance a n d enhance one another when
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they reach the observation p o i n t . As we m o v e closer to the opening, the secondary waves from the sources at t h e edge w i l l lag significantly behind waves t r a v e l l i n g from the centre, and the resulting a m p l i t u d e w i l l decrease. W h e n the ray from the edge to the observation p o i n t becomes an entire wavelength longer t h a n the centra] ray, the oscillations are completely dampened (compensated), and a b l a c k spot appears i n the centre of the diffraction p a t t e r n . I f we m o v e even closer to the opening, we d i s t u r b the d a m p e n i n g of oscillations on the axis, and the centre becomes b r i g h t again. This t i m e t h e d a m p e n i n g w i l l t a k e place at a distance from the axis, and the centre of the diffraction p a t t e r n w i l l be surrounded b y a dark ring. W h e n the ray on the edge lags t w o wavelengths b e h i n d the central r a y , t h e d a m p e n i n g at t h e axis reoccurs. The diffraction p a t t e r n i n this case w i l l be a b r i g h t spot w i t h a dark centre and one dark r i n g . The dark spot i n the centre w i l l appear periodically as we m o v e closer t o the opening. W e can t e l l h o w m a n y oscillations occurred at t h e axis b y c o u n t i n g the n u m b e r of dark rings. The same phenomenon can be seen if we change t h e radius of the opening rather t h a n the distance t o i t . These explanations are enough for you t o derive a formula to determine wavelength. Good
luck!
This selection of interesting articles from the popular science journal Kvant is a collection of fairly simple but challenging and instructive experiments that require little space and the simplest possible equipment. Designed to illustrate the laws of physics, the book includes lessons on growing crystals, studying the oscillations of a pendulum and experimenting with light using a gramophone record or a ball bearing. Suitable for secondary school students and teachers.
