Theoretische Physik 2: Elektrodynamik und spezielle Relativitätstheorie (Springer-Lehrbuch)

Springer-Lehrbuch

Reiner M. Dreizler · Cora S. Lüdde

Theoretische Physik 2 Elektrodynamik und spezielle Relativitätstheorie

Mit 208 Abbildungen und einer CD-ROM

Professor Dr. Reiner M. Dreizler Cora S. Lüdde Johann Wolfgang Goethe Universität Frankfurt /M Fachbereich Physik, Institut für Theoretische Physik Robert-Mayer-Str. 8–10 60054 Frankfurt [email protected] [email protected]

Bibliografische Information Der Deutschen Bibliothek Die Deutsche Bibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über abrufbar. ISBN 3-540-20200-5 Springer Berlin Heidelberg New York DiesesWerk ist urheberrechtlich geschützt. Die dadurch begründeten Rechte, insbesondere die der Übersetzung, des Nachdrucks, des Vortrags, der Entnahme von Abbildungen und Tabellen, der Funksendung, der Mikroverfilmung oder der Vervielfältigung auf anderenWegen und der Speicherung in Datenverarbeitungsanlagen, bleiben, auch bei nur auszugsweiser Verwertung, vorbehalten. Eine Vervielfältigung diesesWerkes oder von Teilen dieses Werkes ist auch im Einzelfall nur in den Grenzen der gesetzlichen Bestimmungen des Urheberrechtsgesetzes der Bundesrepublik Deutschland vom 9. September 1965 in der jeweils geltenden Fassung zulässig. Sie ist grundsätzlich vergütungspflichtig. Zuwiderhandlungen unterliegen den Strafbestimmungen des Urheberrechtsgesetzes. Springer ist ein Unternehmen von Springer Science+Business Media springer.de © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, dass solche Namen im Sinne der Warenzeichen und Markenschutz-Gesetzgebung als frei zu betrachten wären und daher von jedermann benutzt werden dürfen. Sollte in diesem Werk direkt oder indirekt auf Gesetze, Vorschriften oder Richtlinien (z.B. DIN, VDI, VDE) Bezug genommen oder aus ihnen zitiert worden sein, so kann der Verlag keine Gewähr für die Richtigkeit, Vollständigkeit oder Aktualität übernehmen. Es empfiehlt sich, gegebenenfalls für die eigenen Arbeiten die vollständigen Vorschriften oder Richtlinien in der jeweils gültigen Fassung hinzuzuziehen. Satz: Digitale Druckvorlagen der Autoren Umschlaggestaltung: design & production, Heidelberg Herstellung: PTP-Berlin Protago-TEX-Production GmbH, Germany Gedruckt auf säurefreiem Papier 56/3141/Yu – 5 4 3 2 1 0

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(Z) I\ldXee,MXe[Y\[`e^le^9 6D ( ) `jk Xl] ^\jZ_cfjj\e\e Ac!XZ_\e mfi^\^\Y\e 1

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6R

?`\ ^\jlZ_k\ G!fjle^ [\i Kf`jjfe^c\`Z_le^ bXee `e [\i Afid

D (,) : D@EC(,) *

EEE

=(,2 ,$)+(,$ ) [D $

(3-1/)

Xe^\j\kqk n\i[\e- ?Xj KXik`blc!Xi`ek\^iXc [\i `e_fdf^\e\e ?`!\i\ek`Xc^c\`, Z_le^ `jk [liZ_ [`\ *E::BF8=: )HB?G>CB =( 2 $ ) (le[ [`\ GX[le^jm\ik\`, cle^) [Xi^\jk\cck- D@EC `jk \`e\ Xcc^\d\`e\ G!fjle^ [\i ql^\_!fi`^\e _fdf^\e\e ?`!\i\ek`Xc^c\`Z_le^+ Xcjf [\i GXgcXZ\^c\`Z_le^- R`ibk dXe d`k [\d GXgcX, Z\fg\iXkfi Xl] [\e <ejXkq (3-1/) \`e+ jf ]fc^k

,,

EEE

!D (,) : !D@EC(,) * !! =(,2 ,$)+(,$ ) [D $ 1 ?X D \`e\ G!fjle^ [\i Kf`jjfe^c\`Z_le^ (3-08) le[ D@EC GXgcXZ\^c\`Z_le^ `jk+ ^`ck

\`e\ G!fjle^ [\i

0*/ >aTT]bRWT =d]ZcX^]T]

!3*P4 +(,) :

EEE

--/

,2 ,$)+(,$) 1

[D $ !! =(

<e [`\j\i M\cXk`fe bXee dXe XYc\j\e+ [Xjj [`\ *E::BF8=: )HB?G>CB [`\ ?`!\i\ek`Xc^c\`Z_le^

!! =(,2 ,$) : !3* P4 Æ(, ! ,$ )

(3-10)

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!! =(,2 ,$) : !3*P4 Æ(, ! ,$) !

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(3-11)

?`\ Ac!XZ_\ J(8 ) `jk [`\ JY\i#!XZ_\ [\j =\i\`Z_j 8 - Q\ikXljZ_k dXe [`\ Mfcc\ [\i Alebk`fe\e - le[ . le[ bfdY`e`\ik [`\ i\jlck`\i\e[\ M\cXk`fe d`k [\d \ijk\e Dek\^iXcjXkq (3-11)+ jf \i_!Xck dXe [\e KI:>G:B *E::BF8=:B +BG:# <E6@F6GK (XlZ_ *E::BF 3=:CE:A ^\eXeek) "

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(3-12)

De [`\j\ q\ekiXc\+ i\Z_k Xcc^\d\`e\ M\cXk`fe Yi`e^k dXe [`\ Bi\\ejZ_\ Alebk`fe \`e- =\elkqk dXe `e (3-12) [`\ Alebk`fe\e

-(,) : D (,)

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le[ Y\XZ_k\k [`\ ?`!\i\ek`Xc^c\`Z_le^\e

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le[

!! =(,2 ,$) : !3*P4 Æ(, ! ,$) 2

jf \i_!Xck dXe eXZ_ \`e]XZ_\d Nfik`\i\e [`\ Bi\\ejZ_\ G!fjle^j]fid\c

D (,) :

EEE

) *

=(,$2 ,)+(,$)[D $

(3-13)

6D (, ) ! D (,$ ) 6=(, 2 ,) [L $ 1 % = (, $ 2 ,) 3*P4 6") # 6R$ 6R$ EE

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0*/ >aTT]bRWT =d]ZcX^]T] 4

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;#% =(,$ 2 ,) !!! !!,

/

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=(,$ 2 ,) : 0, !P4 ,$ 0 : =(,2 ,$) 2

(3-14)

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(3-16)

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D (,) :

(, 2 ,) $ D (,$ ) 6=6R $ [L 1

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EE

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=(,$ 2 ,)+(,$)[D $ ! 3*0P % 4

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6=(,$2 ,) [L $ : E%E !! =(,$2 ,) " [L $ 6R$ 6") # 6") #

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:

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!! =(,$2 ,) [D $ !

: !3*P4

EEE

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Æ(, ! ,$ ) [D $ : !3*P4 (, . 8) 1

@`e\ df!^c`Z_\ (XY\i kifkq[\d \`e]XZ_\) RX_c [\i MXe[Y\[`e^le^ `jk 6=( $2 ) : Zfejk1 ] l!i $ : . J(8) 1 $

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D (,) :

EEE

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=(,$2 ,)+(,$)[D $

0 * < 6 (8 )

EE

%

(3-17)

6D (, ) [L $ 1 % = (, $ 2 ,) 3*P4 6") # 6R$

D (,$)[L $ * 0

6") #

EE

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0*/ >aTT]bRWT =d]ZcX^]T]

--3

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,,

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M7(T2 T$ ) : G7T7 * T7H$&7

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T

M'7(T2 T$ ) : G'7 (T$ )T7 * H'7(T$ ) T70$&

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M&7(T2 T$ ) : G&7 (T$ ) T7 ! AT7$& (

M'7(T2 T$ ) : H'7 (T$ ) T70$&

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--5

?`\ Npdd\ki`\Y\[`e^le^ [\i Bi\\ej Alebk`fe \i]fi[\ik ]l!i [`\ MX[`Xc, Xek\`c\ M&7 (T2 T$ ) : M'7 (T$ 2 T)- ?`\j\ =\[`e^le^+ [`\ [`\ Alebk`fe\e `e [\e Y\`[\e B\Y`\k\e m\ibel!g]k+ cXlk\k `d ?\kX`c

'7$& G&7 (T$ ) T7 ! AT7$& : H'7 (T) (T$ )07$& 1 Q\i^c\`Z_ [\i T , le[ [\i T$ ,
(

qn\` MX[`Xc]lebk`fe\e

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M'7(T2 T$ ) : I7

.

T 7 ! A'7$& T7$& (TT$ )7$& !

/

T 3 T$ T$ 3 T 1

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&

M&7(T2 T) : M'7 (T2 T)

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T$ : T 1

@`e\ n\`k\i\ =\[`e^le^ qli A\jkc\^le^ [\j efZ_ f!\e\e AXbkfij I7 `jk [Xj Q\i_Xck\e [\i \ijk\e
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M'7(T2 T$ ) : 3*P4

.

T 7 ! A'7$& T7$& (TT$ )7$&

(1Q * 0) (1Q * 0)

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T 3 T$

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T$ 3 T 1

De]fc^\ [\i Npdd\ki`\Y\[`e^le^ bXee dXe [`\ mfccjk!Xe[`^\ Bi\\ejZ_\ Aleb, k`fe Y\i\Z_e\e+ `e[\d dXe M&7 f[\i M'7 Y\elkqk+ jf q-=( 7 D A'7$& ) F7'8 (#)F " (#$ ) =( 2 $) : (13Q*P*40) (T$T)7$& ! (TT 7'8 $ )7$& 7'8

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: P4

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7

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4.5 Polarisation von Dielektrika

1 = C

n

1 . Cn

127

(4.37)

Die Gesamtspannung an dem System ist die Summe der Einzelspannungen U = n Un . Benutzt man die Aussage, dass an den einzelnen Kondensatoren alt man f¨ ur die Gesamtkapazit¨at die Spannung Un = Q/Cn liegt, so erh¨ C=

Q = U

n

1 1 1 + + ... C1 C2

(a)

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oder

n

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U

U

Reihenschaltung

Parallelschaltung

Abb. 4.17. Schaltkreise mit Kondensatoren

4.5 Polarisation von Dielektrika In dem folgenden einfachen ‘Experiment‘ wird die Modifikation der Eigenschaften eines Plattenkondensators untersucht, wenn man in dem Raum zwischen den Platten ein Dielektrikum einbringt. Die auf diese Weise gewonneur die notwendige Erweiterung der Theorie. nen Resultate liefern das Muster f¨ Man bringt (durch Anschluss an eine Batterie) die Ladung +Q0 bzw. −Q0 auf die Platten (Abb. 4.18a), entfernt die Batterie und misst die Spannung U0 zwischen den Platten U0 =

Q0 C0

C0 =

F 4πke d

.

In den Zwischenraum bringt man nun eine Glasplatte oder ein anderes nichtleitendes Material (Abb. 4.18b). Die Platte soll den gesamten Zwischenraum (b)

(a)

Q0 -Q0 ohne Material

U0

Q0 -Q0 mit Material

Abb. 4.18. Versuche mit Dielektrika (konstante Ladung)

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a

Br (r, θ) −−−−−−→

2mkm cos θ kf r3

r

a

Bθ (r, θ) −−−−−−→

181

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(2n − 1)!! 1 ∂(rA(r, θ)) = iπ a2 km (−1)n n r ∂r 2 (n + 1)! n=0   2n  r         (2n + 2) a2n+3   ra  −(2n + 1)  r2n+3

Bθ = −

und 1 ∂ sin θA(r, θ) r sin θ ∂θ ∞ a (2n + 1)!! (−1)n = 2iπkm r n=0 2n n!

Br =

2n+1 r< 2n+2 r>

P2n+1 (cos θ) ,

die f¨ ur alle Werte von r = a anwendbar sind (es bleibt die Frage nach langsamer oder schneller Konvergenz der Reihenentwicklungen). Aus den Ergebnissen kann man die folgenden qualitativen Aussagen gewinnen: Das Vektorpotential A ist immer tangential an Kreise um die z Achse und hat f¨ ur alle Tangentialpunkte eines Kreises den gleichen Betrag. ¨ Dieser variiert mit dem Abstand vom Koordinatenursprung und dem Offnungswinkel des Kegels (Abb. 5.18a). Das entsprechende Feldlinienbild des Magnetfeldes B in der x - z Ebene sieht folgendermaßen aus: Die Feldlinien sind geschlossene Kurven um die Durchstoßpunkte des Ringes. Sie sind fast kreisf¨ ormig in der N¨ ahe des Leiters und passen sich allm¨ahlich an die z Achse (ebenfalls eine Feldlinie) an (Abb. 5.18b). Die Feldrichtung entspricht (a)

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A x

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Eine einfache N¨ aherung dieses Ergebnisses f¨ ur das Abstrahlungsmuster gewinnt man durch direkte Entwicklung der Kosinusfunktionen in (7.82) nach Potenzen von a = kd/2 i2 dP = 0 a4 sin2 θ 1 − dΩ 8πc 1 4 a + 720

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Kn dem folgenden Xerzeichnis "ndet man die in dem Vext zitierten Nitera3 turstellen2 eine nicht zu lange Niste der im Jandel und in den Dibliothe3 ken verfu!gbaren Nehrbu!cher der Glektrodynamik und der .speziellen/ Telati3 vit!atstheorie2 sowie der relevanten mathematischen Nehrbu!cher und Hormel3 werke4 Gine ausfu!hrlichere Fokumentation der mathematischen Niteratur ist in den `Oathematischen Grg!anzungen` auf der zugeh!origen enthalten4 Fie Nehrbu!cher sind alphabetisch aufgefu!hrt2 die Teihenfolge nimmt also keinen Dezug auf das Piveau oder die Uchwierigkeit der Farstellung4 Yer3 ke2 die .soweit den Knternet3Ueiten der Xerlage entnehmbar/ nicht mehr im Jandel erha!ltlich sind2 sind durch .0/ markiert4

:CLC?JL? 2CL?J;LMJKL?EE?G$ :?CLK=BJC@L?G %

C4C4 Oichelson and G4Y4 Oorley2 Rhil4 Oag4 &' .7>>=/2 ::?

%

C4 Ginstein2 Cnn4 der Rhys4 %* .7?6;/2 >?7

%

U4L4 Rlimpton and Y4G4 Nawton2 Rhys4 Tev4 ($ .7?9
%

C4 Utewart2 Ucient4 Cm4 &%$ .7?<:/2 766

%

L4 Daily et al42 Pature &)+ .7?=7/2 967

%

T4 Rerrin2 Cm4 L4 Rhys4 '* .7?=?/2 98=

:CLC?JL? 2CL?J;LMJKL?EE?G$ ,MS=B?J % %

P4 Cshcroft and F4 Oermin@ `Uolid Utate Rhysics` .Uaunders Rublications2 Rhiladelphia2 7?=
497

% % % %

: H]gYeUgheiYemY]W\a]f

.0/ L4E4 Oaxwell@ `Vreatise on Glectricity and Oagnetism` .Elarendon Rress2 Qxford2 7>=9/ L4E4 Oaxwell@ `Nehrbuch der Glectricit!at und des Oagnetismus` .8 Dde/2 .Xerlag Lulius Upringer2 Derlin2 7>>9/ .0/ R4 Ooon2 F4 Gberle@ `Hield Vheory Jandbook` .Upringer Xerlag2 Jei3 delberg2 7?<7/ R4O4 Oorse and J4 Heshbach@ `Oethods of Vheoretical Rhysics` .8 Dde/2 .OcIraw Jill2 Pew Zork2 7?;9/ O4G4 Tose@ `Glementary Vheory of Cngular Oomentum` .Fover Rublica3 tions2 Pew Zork2 7??;/

/CG@MSBJ?G>? 8?PL? % % %

T4 Heynman2 T4D4 Neighton2 O Uands@ `Heynman Xorlesungen u!ber Rhysik` Dand 8 .Xerlag Qldenbourg2 Ou!nchen2 8667/ G4O4 Rurcell@ `Derkeley Rhysikkurs` Dand 8 .Upringer Xerlag2 Derlin2 7??:/ N4 Oarder@ `Teisen durch die Taum3[eit` .H4 Xieweg2 Draunschweig2 7?>8/

/E?DLJH>QG;FCD % % % % % % % % % % % %

X4F4 Darger and O4I4 Qlson@ `Elassical Glectricity and Oagnetism` .Cllyn and Dacon2 Doston2 7?>=/ V4 Hlie(bach@ `Glektrodynamik` .Upektrum Ckademischer Xerlag2 Jeidel3 berg2 8666/ Y4 Ireiner@ `Mlassische Glektrodynamik` .Xerlag J4 Feutsch2 Hrankfurt2 8668/ F4L4 IriÆths@ `Kntroduction to Glectrodynamics` .Rrentice Jall2 Gngle3 wood Eli!s2 7?>?/ L4F4 Lackson@ `Mlassische Glektrodynamik` .de Iruyter2 Derlin2 8668/ I4 Nehner@ `Glektromagnetische Heldtheorie` .Upringer Xerlag2 Jeidelberg2 7??6/ Y4 Polting@ `Glektrodynamik` .Upringer Xerlag2 Jeidelberg2 866:/ I4N4 Rollack and F4T4 Utump@ `Glectromagnetism` .Cddison Yesley2 Uan Hranzisko2 8668/ L4 Uchwinger2 N4N4 FeTaad jr42 M4C4 Oilton and Y4 Vsai@ `Elassical Glec3 trodynamics` .Rerseus Dooks2 Teading .Oass4/2 7??>/ Y4T4 Umyth@ `Utatic and Fynamic Glectricity` .OcIraw Jill2 Pew Zork2 7?<>/ C4 Uommerfeld@ `Glektrodynamik` .Xerlag J4 Feutsch2 Hrankfurt2 8667/

: H]gYeUgheiYemY]W\a]f

498

7I?RC?EE? 6?E;LCNCLS;LKLB?HJC? % % % % %

J4 Ioenner@ `Ginfu!hrung in die spezielle und allgemeine Telativit!atstheo3 rie` .Upektrum Ckademischer Xerlag2 Jeidelberg2 7??
3;LB?F;LCD % % % % % % % % % % %

X4K4 Crnold@ `Iew!ohnliche Fi!erentialgleichungen` .Upringer Xerlag2 Jei3 delberg2 866:/ X4K4 Crnold@ `Xorlesungen u!ber partielle Fi!erentialgleichungen` .Upringer Xerlag2 Jeidelberg2 866:/ J4 Dehnke und H4 Uommer@ `Vheorie der analytischen Hunktionen einer komplexen Xer!anderlichen` .Upringer Xerlag2 Jeidelberg2 866:/ J4 Jarro@ `Iew!ohnliche Fi!erentialgleichungen` .Veubner Xerlag2 Yiesba3 den2 866:/ M4 L!anich@ `Hunktionentheorie` .Upringer Xerlag2 Jeidelberg2 7?=8/ .0/ M4 Mnopp@ `Glemente der Hunktionentheorie`2 `Hunktionentheorie K`2 `Hunktionentheorie KK` .de Iruyter2 Derlin2 7?>7/ C4 Myrala@ `Cpplied Hunctions of a Eomplex Xariable` .Yiley Knterscience2 Pew Zork2 7?=8/ I4 Ooretti@ `Hunctions of a Eomplex Xariable` .Rrentice Jall2 Gnglewood Eli!s2 7?<:/ C4 Uommerfeld@ `Rartielle Fi!erentialgleichungen in der Rhysik` .Xerlag J4 Feutsch2 Hrankfurt2 7??8/ Y4 Yalter@ `Iew!ohnliche Fi!erentialgleichungen` .Upringer Xerlag2 Jei3 delberg2 8666/ C4 Vveito und T4 Yinther@ `Ginfu!hrung in partielle Fi!erentialgleichungen` .Upringer Xerlag2 Jeidelberg2 8668/

8; 0HJF?EK;FFEMGA?G ,??;9@9=A9 0BD@9?E6@@?GA;9A

%

J43L4 Dartsch@ `Mleine Hormelsammlung Oathematik` .Janser Xerlag2 Neip3 zig2 7??;/

499

% % %

: H]gYeUgheiYemY]W\a]f

K4 Dronstein2 K4 Uemendjajew2 I4 Ousiol2 J4 Ou!hlig@ `Vaschenbuch der Oa3 thematik` .Xerlag J4 Feutsch2 Hrankfurt2 8666/ J4 Ut!ocker@ `Oathematische Hormeln und Ooderne Xerfahren` .Xerlag J4 Feutsch2 Hrankfurt2 7??;/ G4 Jering2 T4 Oartin2 O4 Utohrer@ `Rhysikalisch3Vechnisches Vaschenbuch` .XFK Xerlag2 Fu!sseldorf2 7??:/

4C9I=9??9 0GA>F=BA9A

% % % %

O4 Cbramovitz2 K4 Utegun@ `Jandbook of Oathematical Hunctions` .Fover Rublications2 Pew Zork2 7?=:/ H4 N!osch ed4@ `Lahnke2 Gmde2 N!osch2 Vafeln J!oherer Hunktionen` .Veubner2 Ututtgart2 7?</ .0/ K4P4 Uneddon@ `Upezielle Hunktionen der Oathematischen Rhysik` .Di3 bliographisches Knstitut2 Oannheim2 7?<7/

1AF9;D6?F6:9?A

% % %

K4 Iradstein2 K4 Tyshik@ `Uummen32 Rrodukt3 und Kntegraltafeln` Dand K und KK .Xerlag J4 Feutsch2 Hrankfurt2 7?>7/ Y4 Ir!obner2 P4 Jofreiter@ `Kntegraltafel` Dand K und KK .Upringer Xerlag2 Yien2 7?=; und 7?=9/ sowie die entsprechenden Cbschnitte der Hormelsammlungen

# )0-08;/,<08

Ciry2 Uir Ieorge Diddell

engl4 Cstronom $ 8=46=47>67 in Clnwick .ID/ . 6:46747>?8 in Nondon .ID/

d-Clembert2 Lean Ne Tond frz4 Rhilosoph2 Oathematiker und Niterat $ 7=47747=7= in Raris .H/ . 8?47647=>9 in Raris .H/ Cmp%ere2 Cndr&e Oarie

frz4 Oathematiker und Rhysiker $ 8646747==; in Nyon .H/ . 7646<47>9< in Oarseille .H/

Dabinet2 Lacques

frz4 Rhysiker $ 6;46947=>: in Nusignan .H/ . 8747647>=8 in Raris .H/

Dessel2 Hriedrich Yilhelm dt4 Oathematiker $ 8846=47=>: in Oinden .F/ . 7=46947>:< in M!onigsberg .F/ Diot2 Lean3Daptiste

frz4 Rhysiker und Cstronom $ 8746:47==: in Raris .H/ . 6946847><8 in Raris .H/

Dohr2 Piels Jenrik Favid d!an4 Rhysiker Pobelpreis 7?88 $ 6=47647>>; in Mopenhagen .FM/ . 7>47747?<8 in Mopenhagen .FM/ Dradley2 Lames

engl4 Cstronom $ B 46947
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engl4 Rhysiker $ 7747847=>7 in Ledburgh .ID/ . 7646847><> in Cllerby .ID/

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Eauchy2 Cugustin Nouis ' Eerenkov2 Ravel2 Clexander Pobelpreis 7?;> de Eoulomb2 Eharles Cugustin

frz4 Oathematiker

$ 8746>47=>? in Raris .H/ . 8946;47>;= in Uceaux .H/ russ4 Rhysiker

$ 8>46=47?6: in Povaia Ehigla .TWU/ . 6<46747??6 in Ooskau .TWU/ frz4 Rhysiker

$ 7:46<47=9< in Cngoul_eme .H/ . 8946>47>6< in Raris .H/

Fescartes2 Ten&e .Tenatus Eartesius/

franz4 Rhilososoph und Oathematiker $ 9746947;?< in Na Jaye .H/ . 7746847<;6 in Utockholm .U/

Firac2 Raul Cdrien Oaurice Pobelpreis 7?99

engl4 Rhysiker $ 6>46>47?68 in Dristol .ID/ . 8646>47?>: in Vallahassee .WUC/

Firichlet2 Lohann Reter I4 Nejeune dt45frz4 Oathematiker $ 7946847>6; in Fu!ren .F/ . 6;46;47>;? in I!ottingen .F/ Ginstein2 Clbert Pobelpreis 7?87

dt4 Rhysiker $ 7:46947>=? in Wlm .F/ . 7>46:47?;; in Rrinceton .WUC/

Haraday2 Oichael

engl4 Rhysiker $ 8846?47=?7 in Pewington Dutts .ID/ . 8;46>47><= in Jampton Eourt .ID/

Hizeau2 Cntoine Jippolyte Nouis

frz4 Rhysiker $ 8946?47>7? in Raris .H/ . 7>46?47>?< in Xenteuil .H/

Hourier2 Lean Daptiste Loseph

Hraunhofer2 Loseph

Hresnel2 Cugustin Lean

frz4 Oathematiker

$ 8746947=<> in Cuxerre .H/ . 7<46;47>96 in Raris .H/ dt4 Qptiker und Rhysiker

$ 6<46947=>= in Utraubing .F/ . 6=46=47>8< in Ou!nchen .F/ frz4 Rhysiker $ 8646;47=>> in Droglie .H/ . 7:46=47>8= in Xille d-Cvray .H/

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dt4 Oathematiker und Cstronom

$ 9646:47=== in Draunschweig .F/ . 8946847>;; in I!ottingen .F/

Ireen2 Ieorge

engl4 Oathematiker und Rhysiker $ 7:46=47=?9 in Pottingham .ID/ . 9746947>:7 in Pottingham .ID/

Jall2 Jerbert

am4 Rhysiker $ 6=47747>;; in Iorlan .Oaine/ .WUC/ . 8647747?9> in Eambridge .Oass4/ .WUC/

Jeaviside2 Qliver

engl4 Rhysiker und Glektrotechniker

$ 7>4;47>;6 in Nondon .ID/ . 6946847?8; in Jome"eld .ID/

von Jelmholtz2 Jermann N4 H4 dt4 Rhysiker und Rhysiologe $ 9746>47>87 in Rotsdam .F/ . 6>46?47>?: in Derlin .F/ Jeisenberg2 Yerner Marl Pobelpreis 7?98

dt4 Rhysiker $ ;47847?67 in Yu!rzburg .F/ . 764847?=< in Ou!nchen .F/

Jenry2 Loseph

am4 Rhysiker $ 7=47847=?= in Clbany .WUC/ . 7946;47>=> in Yashington .WUC/

Jertz2 Jeinrich Tudolf

dt4 Rhysiker

$ 8846847>;= in Jamburg .F/ . 6746747>?: in Donn .F/

Juygens2 Ehristiaan

niederl4 Rhysiker und Oathematiker $ 7:46:47<8? in den Jaag .PN/ . 6>46=47
v4 Lacobi2 Ooritz Jermann

dt4 Rhysiker $ 874?47>67 in Rotsdam .F/ . 764947>=: in Ut4 Retersburg .TWU/

Loule2 Lames Rrescott

engl4 Rhysiker $ 8:47847>7> in Ualford .ID/ . 7747647>>? in Uale .ID/

Mirchho!2 Iustav Tobert

dt4 Rhysiker $ 7846947>8: in M!onigsberg .F/ . 7=47647>>= in Derlin .F/

4:3

= HYVYafXUgYa Mohlrausch2 Tudolf J4C4

dt4 Rhysiker $ 6<4 774 7>6? in I!ottingen .F/ . 6?4 694 7>;> in Grlangen .F/

Naplace2 Rierre Uimon

frz4 Oathematiker und Rhysiker $ 8>46947=:? in Deaumont3en Cuge .H/ . 6;46947>8= in Crceuil .H/

Narmor2 Uir Loseph

ir4 Rhysiker $ 7746=47>;= in Oagheragall .KTN/ . 7?46;47?:8 in Jolywood .KTN/

Negendre2 Cdrien Oarie

frz4 Oathematiker $ 7>46?47=;8 in Raris .H/ . 7646747>99 in Raris .H/

Nenz2 Jeinrich Hriedrich Gmil dt4 Rhysiker $ 7846847>6: in Forpat .F/ . 7646847><; in Tom .K/ Ni&enard2 Clfred3Oarie

frz4 Rhysiker und Oathematiker

$ 6846:47> in Raris .H/

Norentz2 Jendrik Cnton Pobelpreis 7?68

niederl4 Rhysiker $ 7>46=47>;9 in Crnheim .PN/ . 6:46847?8> in Jaarlem .PN/

Oalus2 Gtienne Nouis

frz4 Rhysiker $ 8946<47==; in Raris .H/ . 8946847>78 in Raris .H/

Oaxwell2 Lames Elark

engl4 Rhysiker

$ 7946<47>97 in Gdinburgh .ID/ . 6;47747>=? in Eambrigde .ID/

Oichelson2 Clbert Cbraham am4 Rhysiker Pobelpreis 7?6= $ 7?47847>;8 in Utrelno .RN/ . 6?46;47?97 in Rasadena .WUC/ Oillikan2 Tobert Cndrew Pobelpreis 7?89 Oinkowski2 Jermann

am4 Rhysiker

$ 88469 7><> in Oorrison .WUC/ . 7?47847?;9 in Rasadena .WUC/ dt4 Oathematiker $ 8846<47><: Clexotas .NV/ . 7846747?6? Io!ttingen .F/

= HYVYafXUgYa Oolyneux2 Uamuel

engl4 Cstronom und Rhysiker

$ 7>46=47<>? in Ehester .ID/ . 7946:47=8> in Mew .ID/

Oorley2 Gdward Yilliam am4 Ehemiker $ 8?46747>9> in Pewark .WUC/ . 8:46847?89 in Yest Jartford .WUC/ Peumann2 Earl Iottfried dt4 Oathematiker $ 6=46;47>98 in Mo!nigsberg .F/ . 8=46947?8; in Neipzig .F/ *erstedt2 Jans Ehristian d!an4 Rhysiker und Ehemiker $ 7:46>47=== in Tudk)bing .FM/ . 6?46947>;7 in Mopenhagen .FM/ Qhm2 Ieorg Uimon

dt4 Rhysiker $ 7<46947=>? in Grlangen .F/ . 6<46=47>;: in Ou!nchen .F/

Rlanck2 Oax Pobelpreis 7?7?

dt4 Rhysiker $ 8946:47>;> in Miel .F/ . 6:47647?:= in I!ottingen .F/

Roincar%e2 Jenri Lules

Roisson2 Uim&eon Fenis

frz4 Oathematiker

$ 8?46:47>;: in Pancy .H/ . 7=46=47?78 in Raris .H/ frz4 Oathematiker und Rhysiker

$ 8746<47=>7 Rithiviers .H/ . 8;46:47>:6 Uceaux .H/

Roynting2 Lohn Jenry

engl4 Rhysiker $ 6?46?47>;8 in Oonton .ID/ . 9646947?7: in Dirmingham .ID/

Tutherford2 Grnest Pobelpreis 7?6>

engl4 Rhysiker $ 9646>47>=7 in Drightwater .P[/ . 7647647?9= in Eambridge .ID/

Uavart2 Helix

Uchwartz2 Naurent

frz4 Crzt und Rhysiker

$ 9646<47=?7 in O&ezi%ere .H/ . 7<46947>:7 in Raris .H/ frz4 Oathematiker

$ 6;46947?7; in Raris .H/ . 6:46=48668 Raris .H/

4:4

4:5

= HYVYafXUgYa von Uiemens2 Yerner

dt4 Glektrotechniker $ 7947847>7< in Nenthe .F/ . 6<47847>?8 in Derlin .F/

Unell2 Yillebrord van Toijen niederl4 Rhysiker .Unellius/ $ B 4 B 47;>6 in Neiden .PN/ . 9647647<8< in Neiden .PN/ Utokes2 Uir Ieorge Iabriel engl4 Rhysiker und Oathematiker $ 7946>47>7? in Ukreen .ID/ . 6746847?69 in Eambridge .ID/ Vesla2 Pikola

am4 Rhysiker und Glektrotechniker $ 7646=47>;< in Umiljan .JT/ . 6=46747?:9 in Pew Zork .WUC/

Yeber2 Yilhelm Gduard

dt4 Rhysiker $ 8:47647>6: in Yittenberg .F/ . 8946<47>?7 in I!ottingen .F/

Yiechert2 Lohann Gmil

Zoung2 Vhomas

dt4 Rhysiker und Ieophysiker

$ 8<47847><7 in Vilsit .F/ . 7?46947?8> in I!ottingen .F/ engl4 Rhysiker und Crzt

$ 7946<47==9 in Oilverton .ID/ . 8646;47>8? in Nondon .ID/

# &48304<08;@;<070 /0: &605<:9/@8,745

+&' .C? 3;#KQKL?F? Km Xakuum ben!otigt man zur Hestlegung der Oa(einheiten aller Ir!o(en der Glektrodynamik drei2 im Rrinzip frei w!ahlbare Monstanten4 Furch diese Mon3 stanten werden letztlich die Oa(einheiten des elektrischen Heldes 2 2 der ma3 gnetischen Knduktion 1 und die Xerknu!pfung dieser im Gxperiment zug!ang3 lichen Helder durch das Knduktionsgesetz festgelegt4 Cls zust!andige Ileichun3 gen kann man w!ahlen@ % Fas elektrische Held 2 einer Runktladung F im Cbstand G von dieser Na3 dung

%

2 .G/ A A( GF + &

Fie magnetische Knduktion 1 eines langen2 du!nnen2 geraden Neiters2 der von einem .station!aren/ Utrom @ durch#ossen wird4 Km Cbstand G von der Frahtachse gilt die Cussage

1.G/ A 8A, G@ +

%

Fie Jerausnahme des Haktors 8 ist eine nu!tzliche aber keine zwingende Qption4 Fas Knduktionsgesetz in der di!erentiellen Horm

! # % .1, H/ 1 A /H/ #.1, H/ A 6 + )

Gine Ginschra!nkung ergibt sich aus den Yellengleichungen fu!r die zwei Helder im Xakuum4 Cus den Oaxwellgleichungen gewinnt man in diesem Hall

!% .1, H/ 1 A,A(A) /H/ % .1, H/ A 6 , &

&

und

!#.1, H/ 1 A,A A) /H/ #.1, H/ A 6 + &

(

&

Fa der Haktor vor den [eitableitungen dem Knversen des Suadrates der Cus3 breitungsgeschwindigkeit entspricht und diese fu!r elektromagnetische Yellen im Xakuum mit

4:7

= A]a\Y]gYaflfgY`Y XYe A_Y^gebXlaU`]^

m cm A 8+??=?8;+++ " 76, s s experimentell bestimmt wird2 folgt

; A 8+??=?8;+++ " 76

%$

'

A( ( & A,A)

% &

A;+

Fiese Cussage impliziert2 dass im Gnde!ekt nur zwei der drei Monstanten frei w!ahlbar sind4 Fie explizite2 messtechnische Hestlegung der zwei Monstanten .u!blicher3 weise w!ahlt man A( und A, / erfordert eine etwas l!angere Detrachtung4 Fie Monstante A( wird zun!achst durch das Eoulombsche Mraftgesetz

3

%&

A A(

FF , G % & & %&

das den Detrag der Mraftwirkung zwischen zwei Runktladungen F% und F& im Cbstand G%& beschreibt2 eingefu!hrt4 Furch die Yahl% der dimensionslosen Monstanten

A(%

/02

A7

wird in dem EIU Uystem die Oa(einheit der Nadung mit den mechanischen Oa(einheiten verknu!pft \F ^/02 A

> & cm & % &

' &

A statcoul + s Fie Oa(einheit des elektrischen Heldes ist wegen 2.! A 3%& -F& somit

>&

statvolt + A cm cm%&& s Fie Xorschrift zur Cusmessung von Oagnetfeldern .genauer der magnetischen Knduktion/ lautet \2 ^/02 A

% &

$ A0## +

0

Fiese Telation2 die das magnetische Ooment eines Rrobedipols und die magnetische Knduktion mit einem Frehmoment verknu!pft2 zeigt2 dass mit der Hestlegung der Oa(einheiten der Knduktion die Ginheiten des magne3 tischen Oomentes festgelegt sind4 Km EIU Uystem legt man die Monstante A, auf den Yert A,%/02 A 7; A 6+999;<:+++ " 76!%$ cms fest und es folgt 7 A+ A)%/02 A + A)%/02 A &A(%/02

#

"

; A,%

/02

;

$

Af jYeXYa ahe XUf CUh(fW\Y ?CO OlfgY` haX XUf eUg]baU_]f]YegY IGO= OlfgY` ]` @YgU]_ VYgeUW\gYg/ Bphe mjY] XYe mhfpUgm_]W\Ya IU(flfgY`Y- X]Y ]a XY` Rbejbeg YejpU\ag jheXYa- jYeXYa X]Y eY_YiUagYa GbafgUagYa ]a PUV/ =/2 UhZ[YZph\eg/

=/2 @]Y IU(flfgY`Y

4:8

Fie magnetische Knduktion wird somit in den Ginheiten

>&

% &

A \2 ^/02 , cm%&& s das magnetische Ooment in den Ginheiten \1 ^/02 A

\C^/02 A

> & cm & % &

) &

A \E^/02 s gemessen4 Uowohl fu!r die Helder als auch fu!r die elektrischen und magneti3 schen Fipolmomente sind im EIU Uystem die gleichen Ginheiten zust!andig4 Km rationalisierten OMUC Uystem fu!hrt man als zus!atzliche Ginheit die Ginheit der Utromst!arke "')!%*% .Cbku!rzung C/ ein4 Oit der Fe"nition des elektrischen Utroms kann man alternativ die Nadungseinheit #(+&('$ .Cbku!rzung E/ benutzen @ A ddFH *+ 7 C A 7 Es + Fie messtechnische Fe"nition der Ginheit "')!%*% zeigt2 dass die Hestlegung dieser Ginheit durch die Cussage Hlie(t in zwei langen2 du!nnen2 geraden2 parallelen Neitern .im Xakuum/ , Utrom der gleichen Ut!arke2 so betr!agt die Utromst!arke @ 7 Cmp%ere2 wenn sich die Fr!ahte bei einer Gntfernung < von 7 Oeter mit einer Mraft 3 von 8 " 76!+ P pro Oeter Frahtl!ange anziehen4] die Detrachtung der magnetischen Mraftwirkungen erfordert4 Kn wird gezeigt2 dass die zust!andige Ileichung die Horm

@

&

A

F4tail ;4<

-3 -<

8A) A, B

hat4 Uetzt man die Yerte 3 A 8 " 76!+ P2 @ A 7 C und < A B A 7 m ein2 so erh!alt man A)%21 A,%21 A 76!+ kgE&m + Fie Monstante A)%21 wird u!blicherweise dimensionslos gew!ahlt2 so dass mit

A)%

21

A7

die Hestlegung der Ginheit "')!%*% der Cussage A,%21 A 76!+ kgE&m A :&'$ entspricht4 Fie Monstante A,%21 wird in der Rraxis oft durch die Rermeabilit!at des Xakuums &$ A :' " 76!+ kgE&m A 7+8;<<9+++ " 76!* kgE&m ersetzt4 Cus der einschr!ankenden Dedingung fu!r die Monstanten folgt dann

4:9

= A]a\Y]gYaflfgY`Y XYe A_Y^gebXlaU`]^

' kg m 7 - kg m + A > + ?>=;; ++ " 76 A & & & E E s :')$ Fie Fielektrizita!tskonstante des Xakuums hat den Yert & & )$ A >+>;:7>+++ " 76!%& kgE ms ' + Hu!r die Oa(einheiten der zwei Helder und des magnetischen Ooments "ndet man im UK Uystem kg m Xolt \2 ^21 A A E s& cm kg \1 ^21 A Es E m& + \C^21 A s Gs ist u!blich2 das magnetische Ooment einer ebenen2 stromdurch#ossenen Neiterschleife im UK Uystem in der Horm

A(%

21

C

A 76!+ ;&

23684754%21

A @ " Hl!ache

anzugeben4 Fie Oa(einheit2 die durch diese Fe"nition impliziert wird2 stimmt mit der oben angegebenen Oa(einheit des magnetischen Ooments u!berein4 Km Hall des EIU Uystems tri!t dies nicht zu4 Uchreibt man jedoch

C

23684754

A A) .@ " Hl!ache/ ,

so ist die Fe"nition fu!r beide Ginheitensysteme mit der allgemeinen Fe"nition des magnetischen Oomentes vertr!aglich4 Fie Fiskussion der Heldsituation in Oaterie erfordert die Ginbeziehung der Jilfsfelder und 2 der dielektrischen Xerschiebung und der magne3 tischen Heldst!arke2 sowie der Ir!o(en2 die die gemittelten Gigenschaften des Oaterials wiedergeben2 der Rolarisation und der Oagnetisierung 4 Fas 3Held wird durch die wahren Nadungen erzeugt4 N!asst man die Hrage der Oa(einheiten dieses Heldes o!en2 so impliziert dies die Telation

$

&

$

(

'

! " $ A :'A ( + ' 0

Fie Rolarisation wird durch die Xerteilung der Rolarisationsladungen be3 stimmt4 Kn diesem Hall wird die explizite .jedoch nicht zwingende/ Fe"nition

! " ( A !(

:98

%

benutzt4 Oit der Cussage2 dass das elektrische Held durch die wahren und die Rolarisationsladungen erzeugt wird2 folgt die Telation

$ A AA % ! :'A ( + ' (

'

Km EIU Uystem w!ahlt man die noch verfu!gbare Monstante dimensionslos mit dem Yert 7

=/2 @]Y IU(flfgY`Y

A'%

/02

4::

A A(%/02 A 7 +

Fie resultierende Ileichung

$ A % ! :'(

besagt2 dass alle drei Helder in den gleichen Ginheiten gemessen werden4 Km UK Uystem wird die Monstante ebenfalls dimensionslos gew!ahlt A'%21 A :7' , so dass man die Telation zwischen den drei Heldern explizit in der Horm

$ A) % !( $

schreiben kann4 Fie Oa(einheiten der zwei zus!atzlichen Helder sind gleich2 sie unterscheiden sich jedoch von der des 3Heldes& 4 Detrachtet man magnetische Oaterialien2 so lauten die entsprechenden Cussagen@ Fas 3Held wird durch die wahren Utr!ome erzeugt

%

&

! # & A : 'A / + Fie Oagnetisierung ' entspricht einer Oittelung u!ber die schleifenartigen * 0

Oagnetisierungsstr!ome entlang der Oaterialober#!ache

!#' AA / + ) 0

#

Fie Cussage2 dass das 3Held durch beide Utromdichten bestimmt wird2 fu!hrt somit auf die Xerknu!pfung

& A AA # ! :' AA ' + *

*

,

)

O!ochte man wie im elektrischen Hall sicher stellen2 dass im EIU Uystem alle drei Oagnetfelder in den gleichen Ginheiten angegeben werden2 so setzt man A*%/02 A 7

;

und erh!alt

& A # ! :'' +

Km UK Uystem fu!hrt die -rationalisierte- Yahl A*%21 A :7' auf 7 A ! + #

& &# ' $

@]Y >YmY]W\aha[ (!*$'&!%$)$"(*") IGO= OlfgY` a]``g >Ymh[ UhZ X]Y A_]`]aUg]ba XYe BU^gbeYa 5 # ]a XYa CehaX[_Y]W\ha[Ya/

511

= A]a\Y]gYaflfgY`Y XYe A_Y^gebXlaU`]^

Fie einfachen Oaterialgleichungen fu!r linear reagierende2 isotrope Oedien werden in beiden Ginheitensystemen in der Horm 7 A) A

& &#

$ %

benutzt2 wobei der Xakuumlimes durch )

) &

/02 /02

und

) !) ) 21

$

!)

7

& !) & 21

$

gegeben ist4 Fie relative Fielektrizit!atskonstante und Rermeabilit!at in dem UK Uystem stimmen mit der Fielektrizita!tskonstante und Rermeabilita!t in dem EIU Uystem u!berein %

)& ) $

%

% %

21

A )/02

%

&& & $

21

A &/02 +

Kn den folgenden Vabellen sind die MoeÆzienten

A( , A,, A) , A', A*

fu!r die zwei wichtigsten Ginheitensysteme2 sowie das elektrostatische .elec3 trostatic units 3 esu/ und das elektromagnetische Ginheitensystem .elec3 tromagnetic units 3 emu/ zusammengestellt .Vab4 C47/4 Fie zugeh!origen [ahlenwerte werden in Vab4 C48 aufgefu!hrt4 Fie Vabellen .C49/ und .C4:/ enthalten eine Niste der physikalischen Ir!o(en2 die in der Glektrodynamik eine Tolle spielen4 Peben den Oa(ein3 heiten im UK und im EIU Uystem "ndet man2 zur besseren Qrientierung2 die Hormeln2 anhand derer diese Ir!o(en eingefu!hrt wurden4 Kn den Vabellen .C4;/ und .C4
=/3 PUVY__Ya

+&( 8;YmY]W\aha[ OE ?CO 2 2 3 5# $! 2 3 2 5# 2 "! 3 5# W 2 3 2 W 2 2 3 5# W

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2

2

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2

%

2

2

&

2

2

'

)*+-..- (!#!

TU\_YajYegY mh PUV/ =/2

>YmY]W\aha[

SYeg +OE,

1 < 3%::8:36 " 21)

Y! < 2%7132:3 " 21!"* $! <

9%96529 " 21!"#

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5# " 21!(

2 * 5# $! < 9%:9866 " 21 2 !# 5# < 8%:68858 " 21 "! 5# <

2 " 21!(

Y! +,. < 5%914361 " 21!"! Yfh !

` f ? ?# J`# ^[ ` ?# J `# ?# ^[ ` ?#

512

(

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5

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.

A.BY_X

Y_Y^ge/ B_hff

Y_Y^ge/ @]cb_`b`Yag

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R +Rb_g,

J` < ^[ `# < R ? f# ?

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OE "

?CO

"

fgUgib_g

"

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< fgUgWbh_ W`

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< fgUgWbh_ < Yfh

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HUXha[

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)*+-..- (!$!

513 = A]a\Y]gYaflfgY`Y XYe A_Y^gebXlaU`]^

//

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2 9/0 < 9#3

AaYe[]YX]W\gY 9/0

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Lb_Ue]fUg]ba

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# $ # " ) < 5# 3 6

@]Y_Y^ge/ RYefW\]YVha[

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? `# J` < F `$ `$ ? < = +=`c&YeY, f ? = `# f < `#

?# f# < ? < B +BUeUX, ^[ `# R ? `#

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"

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< fgUgU`c W`#

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< fgUgib_g W`

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(Fortsetzung)

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< KY

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OE Rf ^[ ? f < `# < P +PYf_U,

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`U[aYg]fW\Y EaXh^g]ba

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=/3 PUVY__Ya )*+-..- (!&!

Q`eYW\aha[fZU^gbeYa E >Ym/

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< BU^gbe oW 21 21' oW oW 21

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" fgUgib_g " fgUgib_g " W` " fgUgib_g W`

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oW ]fg XYe TU\_YajYeg XYe H]W\g[YfW\j]aX][^Y]g ]` ?OC OlfgY` oW < 3%::8:36 " 21"!

516

406

A Einheitensysteme der Elektrodynamik

Tabelle A.6. Umrechnungsfaktoren II Name

Bez.

SI kg Cs

1

=

Faktor · CGS 104 · G

magnetische Induktion

B

magn. Moment

m

1 A m2

Vektorpotential

A

1

magn. Fluss

Φm

1Vs

Magnetisierung

M

1

A m

˜c · Oe 103

magn. Feldst¨ arke

H

1

A m

4π · Oe 103

Induktionskoeffizient

L

1

Vs A

magn. Energiedichte

wm

1

J m3

10 ·

erg cm3

Poyntingvektor

S

1

J m2 s

103 ·

erg cm2 s

Widerstand

R

1

V A

= 1Ω

109 statvolt · ˜c2 statamp

Leistung

P

1

J s

= 1W

107 ·

kg m Cs

= 1T

103 · G cm3 = emu = 1Tm

106 · G cm

= 1 Wb

108 · G cm2 = Mx

= 1H

109 statvolt s · ˜c2 statamp

erg s

˜c ist der Zahlenwert der Lichtgeschwindigkeit im CSG System ˜c = 2.997925 · 1010

$ '9:706;,776=82" '9:7068 /0: *05<9:,8,6@;4;

Kn dieser Hormelsammlung sind einige Oehrfachprodukte mit Xektoren .D47/2 ein Uatz von Hormeln fu!r die Cnwendung des 3Qperators auf Rrodukte von Ukalar3 und Xektorfunktionen .in formaler und expliziter Uchreibweise2 D48/2 Cusdru!cke fu!r die Oehrfachanwendung des 3Qperators .D49/ und die vier Fi!erentialoperatoren , ", # und ! in Mugel3 und [ylinderkoordinaten .D4:/ zu "nden4 Yeitere Hormeln fu!r die Farstellung der vier Fi!erentialope3 ratoren in krummlinigen2 orthogonalen Moordinaten kann man anhand der Oathematischen Grga!nzungen2 Oath4Map4 ;482 gewinnen4 Kn Oath4Map4 ;48 wird auch die Jerleitung der zitierten Hormeln fu!r die zwei Moordinatensa!tze erla!utert4 Km Holgenden bezeichnet , , + + + Xektoren *. / eine Ukalarfunktion . /, . / Xektorfunktionen4

!! !

! !

* + 1 "1 #1

,&' 3?BJ@;=BIJH>MDL? NHG 9?DLHJ?G

* " .+ # ,/ A + " ., # */ A , " .* # +/ * # .+ # ,/ A .* " ,/ + ! .* " +/ , .* # +/ " ., # -/ A .* " ,/ .+ " -/ ! .* " -/ .+ " ,/ ,&( 5JH>MDLJ?A?EG @MSJ >C? +GO?G>MGA >?K ! %4I?J;LHJK

! " .*"/ A *.! " "/ 1 !* " " div.*"/ A * div " 1 grad * " " ! # .*"/ A *.! # "/ 1 !* # " rot.*"/ A * rot " 1 grad * # "

519

> Bbe`Y_fU``_ha[; Bbe`Y_a XYe RY^gbeUaU_lf]f

! " . " # # / A # " .! # " / ! " " .! # # / div." # # / A # " rot " ! " " rot # ! # ." # #/ A ".! " #/ ! #.! " "/ 1 .# " !/" ! ." " !/# rot." # # / A ".div # / ! # .div "/ 1 .# " ;D68/" ! ." " ;D68/#

,&) :O?C@;=B? +GO?G>MGA NHG !

! " .! # "/ A div.rot "/ A 6 ! # .!*/ A rot.grad */ A 6 ! # .! # "/ A !.! " "/ ! !" rot.rot "/ A grad.div "/ ! .div grad/" .Fie letzte Hormel gilt nur fu!r eine Farstellung von dinaten+/

" in kartesischen Moor3

,&* .C!?J?GLC;EHI?J;LHJ?G CG 1MA?E% MG> :QECG>?JDHHJ>CG;L?G 2G;9?>BBD8=A6F9A

7 /8 7 /8 . 1 . 1 !8 A /8 /G G /% G sin % /* . /

!

$

! " " A G7 /G/ !G 0 " 1 G sin7 % /%/ .sin %0 / 1 G sin7 % /0 /* &

&

/

$

!

( /0 / ! !#" /% .sin %0$/ ! /* ./ ( ' ' ( 7 / /0 7 /0/ 7 / / 1 G sin % /* ! G /G .G0$/ .! 1 G /G .G0! / ! /% .$

7 A G sin %

'

% & /8 / !8 A G /G G /G 1 G

7

&

&

%

7 / /8 sin % & sin % /% /%

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