Classical Electrodynamics

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1 A First Look t Qutum Phsics Clssicl Electrodmics Chpter Boudr-Vlue Prolems i Electrosttics: I 11 Clssicl Electrodmics Prof. Y. F. Che

2 Cotets A First Look t Qutum Phsics.1 Poit Chrge i the Presece of Grouded Coductig Sphere Imge Chrge Method. Poit Chrge i the Presece of Chrged, Isulted, Coductig Sphere.3 Solve Poisso Equtio with Sphericl Boudr.4 Gree Fuctio of Two-Dimesiol (D) Rectgulr Sstem 11 Clssicl Electrodmics Prof. Y. F. Che

3 .1 Poit Chrge i the Presece of Grouded Coductig Sphere Imge Chrge Method ˆ p q ˆ q q : imge chrge : imge positio 1 q q 4 1 q q cos 4 cos 1 q q 4 cos cos q q 1 11 Clssicl Electrodmics Prof. Y. F. Che

4 .1 Poit Chrge i the Presece of Grouded Coductig Sphere Imge Chrge Method (1) Potetil t 1 q 4 cos q cos () iduced chrge desit o the surfce of the sphere cos q cos cos cos q 4 cos 3 11 Clssicl Electrodmics Prof. Y. F. Che

5 .1 Poit Chrge i the Presece of Grouded Coductig Sphere Imge Chrge Method (3) totl iduced chrge tot q d d S 3 4 cos Assume q is o the z-is, u q 1 d sidd S 3 4 cos cos q 1 q 1 1 d dud 3 S 1 4 u u u1 u1 q q 11 Clssicl Electrodmics Prof. Y. F. Che

6 (4) totl force ctig o the surfce q cos 3 cos F cosd d 3 Assume q is o the z-is, q cos 3 cos F si d d 3 q u 3 u F dud 3 q u du 3 16 u u1 q u du 16 4 u 4 u u1 q q 1 q Clssicl Electrodmics Prof. Y. F. Che

7 q q p ˆ ˆ Q q Use the cocept of lier superpositio 1 4 q q Q q 1 4 q Q q q () Electric force ctig o the chrge is 1 4 ˆ q Q q q F q (1) Potetil t 11 Clssicl Electrodmics Prof. Y. F. Che. Poit Chrge i the Presece of Chrged, Isulted, Coductig Sphere

8 .3 Solve Poisso Equtio with Sphericl Boudr ˆ p ˆ mes of Gree fuctio (1) How to oti Gree fuctio? GD,, S, 4 o S G D with Aove Equtios re equivlet, i.e. G D,,, q 4 Hece, we c oti Gree fuctio from eq. (1) G D q o S 1 1 cos cos 11 Clssicl Electrodmics Prof. Y. F. Che with

9 () Geerl solutio of the potetil i terms of Gree fuctio,, P ˆ ˆ P ẑ P P ŷ 1 4 S G D, ˆ d cos GD, GD, 1 cos 1 cos cos 3 3 cos 3 1,, d 3 4 S cos cos coscos sisicos 11 Clssicl Electrodmics Prof. Y. F. Che

10 .4 Gree Fuctio of Two-Dimesiol (D) Rectgulr Sstem Oe Dimesiol Helmholtz Equtio d d k with Geerl solutio: AsikBcos k B k Asi fuctios d 1 re orthoorml fuctios which c e used to epd ritrr f C Epsio of Delt fuctio: C A C si si 11 Clssicl Electrodmics Prof. Y. F. Che

11 (1) Gree fuctio: Eigefuctio epsio method m m,, si si km, m,, m km,, G, ;, 4 D m GD, ;, Fm, m,, Fm, si si m, m, let m m m,, m,, si si si si m, m, G, ;, GD, ;, m 4 Fm, m,, m D Fm, m,, 4 m,, m,, m, m, 16 1 m m GD, ;, si si si si m m 11 Clssicl Electrodmics Prof. Y. F. Che

12 () Solutio of Lplce Equtio i D Rectgulr Sstem Lplce Equtio:, V, let X Y 1 d 1 d, X d Y d Geerl solutio V V V V m X si, m 1, m Y Am sih, Am si sih m1 m m 11 Clssicl Electrodmics Prof. Y. F. Che

13 Appl oudr coditio for, m m, V Am si sih m1 Multipl si i oth side d itegrte m m m V si d A si si sih d A sih m m m m1 1 V m Am V si d 1 1 m m m sih sih Fill, we oti 4V 1 1 m m, si sih modd m m sih Specil cses if the sih e e 4V 1 1 m, si e m modd m e m 11 Clssicl Electrodmics Prof. Y. F. Che

14 (3) Gree Fuctio: Divisio of regio method V V, V t : ΙΙ Ι V, ;, 4 G D regio Ι : regio ΙΙ : G, ;, G, ;, Ι Accordig to the geerl solutio i the previous ΙΙ 3 m m m GΙ, ;, Am si si sih m m m m GΙΙ, ;, Bm si si sih m m m Amsih Bmsih We c oti the geerl solutio m m m m G, ;, Fm si si sih sih m Sustitute eq. (4) ito eq. (3) d itegrte m d m m Fm sih sih 8 m d Clssicl Electrodmics Prof. Y. F. Che

15 Upper limit: d m m m m m Fm sih sih Fm sih cosh d lower limit: d m m m m m Fm sih sih Fm cosh sih d 6 7 Sustitute eq. (6) d eq. (7) ito eq. (5) m m m m m 8 Fm sih cosh cosh sih F m 8 m msih Fill, we oti 8 m m m m G, ;, si si sih sih m m msih mi, where m, 8 11 Clssicl Electrodmics Prof. Y. F. Che

16 (4) Gree fuctio i eq. () d eq. (8) re equivlet 16 1 m m G, ;, si si si si m m eq. (): D 8 m m m m m msih eq. (8): G, ;, si si sih sih m m m si si sih sih m m m sih 11 Clssicl Electrodmics Prof. Y. F. Che

CHAPTER 2: Boundary-Value Problems in Electrostatics: I. Applications of Green s theorem

CHAPTER 2: Boundary-Value Problems in Electrostatics: I. Applications of Green s theorem CHAPTER : Boudr-Vlue Problems i Electrosttics: I Applictios of Gree s theorem .6 Gree Fuctio for the Sphere; Geerl Solutio for the Potetil The geerl electrosttic problem (upper figure): ( ) ( ) with b.c.