Derivations in Classical Electrodynamics

Transcription

1 Deivation in Claical Electodynamic Andew Foete Januay 8, 009 Content Explanation Idea 3 Integation 3. A Special Integation by Pat Anothe Special Integation by Pat Dipole Popetie 5 4. Real Electic Dipole Ideal Electic Dipole Electic Field Dipole-Dipole Foce Dipole-Dipole Toque Radiation 3 5. Localized Souce and Multipole Expanion Finding the Electic Dipole Field Sub Poblem Moe Detail: Without the Cetain Appoximation Above Reult Explanation The Note and Outline povide definition and tatement egading the topic. The Deivation povide poof fo the tatement. Idea Fundamental Media Inteface: Reflection and Refaction Law, Suface Wave exponential decay Reonato, Waveguide Radiation Law: Paticle, Multipole, Black-Body like eonato?,... Lamo Powe Fomula Pactical Atmopheic Phenomena 3 Integation It tun out that thee pecial integation by pat deivation ae imple uing the potential athe than the field themelve.

2 F G F + F + F 3 3 G F G + F G + F 3 3 G b wait, I actually wanted a volume integal a i F i G F i i G + G i F i F i i G d i F i G b a b a G i F i d i

3 3. A Special Integation by Pat Giffith doe thi a bette way. See all of the integation-by-pat ule in Integation Deivation and Analyi Note By definition, p p : xρp dv and p p : P dv. Uing integation by pat, we can olve fo a geneal expeion of ρ p in tem of P. In geneal, we have F G dv da F G + G F dv G F dv + F G dv. S Subtituting x fo F and P fo G, we get x P dv da x P + P x dv P x dv + x P dv S da x P + P3 dv P x x + P y y + P z z x dv + x P dv S da x P + 3 P dv P dv + x P dv S da x P + P dv + x P dv. S Reaanging, we then have P dv x P dv da x P x P dv. S If i all pace and P 0 at infinity, then S da x P 0 and x P dv x x + y y + z z P dv x x P dx dy dz + y y P dx dy dz + z z P dx dy dz x x P dx dy dz + y y P dy dx dz + z z P dz dx dy S yz S xz S xy Px P x x dx dy dz + Py P y y dy dx dz S yz + Pz P z z dz dx dy S xy P dv P dv P dv 3 P dv which i pobably alo tue fo P 0 on the uface of a finite egion o P dv x P dv da x S P 3 P dv which mean P dv x P dv xρ p dv and o finally ρ p P unde thee aumption. S xz 3

4 3. Anothe Special Integation by Pat By definition, m m : x Jm dv and m m : M dv. Uing integation by pat, we can olve fo a geneal expeion of J m in tem of M. In geneal, we have F G dv F G da F G dv G F dv G F dv Subtituting x fo F and M fo G, we get x M dv x M da S x M da Reaanging, we then have M dv S S S x M dv x M dv M x dv M dv x M da x M dv x M dv M x dv And, a with P in the ection above, if i all pace and M 0 at infinity, then S x M da 0 and x M dv 3 M dv which i pobably alo tue fo M 0 on the uface of a finite egion o M dv S x M da 3 M dv x M dv which mean M dv x M dv x J m dv and o finally J m M unde thee aumption. 4

5 4 Dipole Popetie 4. Real Electic Dipole The electic potential Φ of an electic dipole made of a chage q located at d/ and a chage q located at d/ i q q Φ K e d/ + d/ q K e x d / + y d / + z d 3 / q x + d / + y + d / + z + d 3 / q K e x + y + z xd + yd + zd 3 + d + d + d 3 /4 q x + y + z + xd + yd + zd 3 + d + d + d 3 /4 q K e x + y + z xd +yd +zd 3 d +d +d 3 /4 x +y +z q x + y + z + xd +yd +zd 3 +d +d +d 3 /4 x +y +z 4. Ideal Electic Dipole In the limit a d, d, d 3 0 and q, while qd p i held contant, / Φ K e q xd + yd + zd 3 d + d + d 3 /4 x + y + z x + y + z K e / q + xd + yd + zd 3 + d + d + d 3 /4 x + y + z x + y + z q + xd + yd + zd 3 d + d + d 3 /4 x + y + z x + y + z q xd + yd + zd 3 + d + d + d 3 /4 x + y + z x + y + z K e qxd + yd + zd 3 x + y + z 3/ K e qd 3 K e p ˆ 5

6 4.3 Electic Field Note that when finding the electic field, we take the deivative of a ingula expeion, o we hould not be upied to find a Diac delta, and we hould be caeful not to mi it. We ll expand expeion a much a poible to ecognize... Note that p pẑ p Cθ, Sθ, 0 E Φ K e p ˆ K e p ˆ ˆ + p + p ˆ + ˆ p { K e K e p ˆ ˆ ˆ p + p p ˆ } ˆ + { p 4π δ ˆ + p p + p ˆ + ˆ p p Sθ Cθ + Sθ θ Sθ Sθ + Sθ φ0 ˆ ˆθ ˆφ p Cθ Sθ 0 / p + p ˆ + ˆ p Sθ ˆ ˆθ Sθ ˆφ θ φ Cθ Sθ 0 { K e 4π p δ ˆ p Sθ Cθ Sθ Sθ Cθ } Sθ ˆφ + ˆ Sθ Sθ Sθ + Sθ ˆφ K e 4π p δ p ˆ ˆθ Sθ ˆφ Sθ θ φ 0 0 Sθ Sθ K e 4π p δ p Sθ SθCθˆ + S θ ˆθ p K e Cθˆ 3 + Sθ ˆθ 4π p δ K e 3p ˆˆ p 3 4π p δ whee the facto of /3 in the lat tem?! 6

7 I thee a way to do thi moe abtactly? Note that p pẑ p 0, 0, p d p Cθ, Sθ, 0 zx ˆθ x + y + z x + y, zy x + y, x + y x Sθ + y x + y + z d E Φ K e p ˆ K e p ˆ ˆ + p + p ˆ + ˆ p { K e p ˆ ˆ + p + p K e p ˆ p 4π δ ˆ ˆ p + p p ˆ } ˆ + ˆ p 7

8 p ˆ ˆx ŷ ẑ 0 0 p x/ 3 y/ 3 z/ 3 py/ 3ˆx + px/ 3 ŷ ˆx ŷ ẑ x y z py/ 3 px/ 3 0 z px x + y + z ˆx 3/ z py x + y + z ŷ 3/ + x px x + y + z + 3/ y py x + y + z 3/ ẑ 3 pxz x + y + z ˆx + 3 pyz 5/ x + y + z ŷ 5/ + p x + y + z 3/ 3 pxx x + y + z 5/ + p x + y + z 3/ 3 pyy x + y + z 5/ ẑ 3p xzˆx + yzŷ x + y ẑ x + y + z 5/ 3p x + y + z 3/ + 3p 3 Sθ ˆθ + p 3 3pSθ ˆθ 3 + p 3p 3 ˆθˆθ + p 3pSθ ˆθ 3 + pcθˆ psθ ˆθ psθ ˆθ 3 + pcθˆ 3pCθˆ 3 pcθˆ + psθ ˆθ 3 3p ˆˆ p pẑ x + y + z 3/ x + y zx zy x + y x +y x +y x + y + z x + y + z d + p 3 8

9 E K e p ˆ 4π p δ K e 3p ˆˆ p 3 4π p δ p 3p K ˆθˆθ e 3 4π p δ 9

10 4.4 Dipole-Dipole Foce Ideal Dipole Loop F m B m B m B + m µ 0 J + c t E m K m 3m ˆ ˆ m 3 K m 3m ˆ m ˆ m m 3 + 8π 3 m δr 3m m K m + 3m m + 3m 5 5 m m m 3 K m 3m m 5 + 3m m 3K m m m 4 m m 3 ˆ 4 + 3m 5 m 5 ˆ 6 { ˆm ˆm 5 ˆm ˆ ˆm ˆ }ˆ + ˆm ˆ ˆm + ˆm ˆ ˆm Method ued below i too difficult: ue pape by Ka W. Yung, Pete B. Landecke, and Daniel D. illani. Foce between two loop of cuent with magnetic dipole moment m I πr ˆm and m I πr ˆm, whee the loop ae fa enough apat to appoximate the magnetic field of m at m a that of an ideal dipole and to ue a linea appoximation of the field at the cente of loop to find the field along the loop: F I d B pependicula to m I πr z B B ẑ B 3m ˆˆ m nea m K m 3 + 8π 3 m δ 0 4 x y z 3 5 A K m 3m m 5 K m 3m z x, y, z x + y + z m x + y + z 5/ 4 A A A 3 A A A 3 A 3 A 3 A x y z A x + A y + A 3 z A x + A y + A 3 z A 3 x + A 3 y + A 33 z

11 A 4 ˆx ˆx ˆx ŷ ŷ ˆx ŷ ŷ ẑ ˆx ẑ ŷ ˆx ẑ ŷ ẑ ẑ ẑ 4 ˆm ˆm ˆm ˆ ˆm ˆ ˆm ˆ ˆm ˆm ˆ ˆm ˆm ˆ ˆm ˆ ˆm ˆm ˆ ˆm ˆ ˆm ˆ ˆm ˆm ˆm ˆ ˆm ˆ + ˆm ˆ ˆm ˆm ˆm ˆ ˆm ˆm ˆ ˆm ˆ ˆm ˆm ˆm ˆ ˆm ˆm x y z 5 4 A x + A y + A 3 z A x + A y + A 3 z A 3 x + A 3 y + A 33 z B x K m 3m z x x + y + z 5/ 3K mm A 3 x + A 3 y + A 33 z A x + A y + A 3 z ˆA x + A y + A 3 z + A x + A y + A 3 z + A 3 x + A 3 y + A 33 z 5/ 3K mm A 3 x + A 3 y + A 33 z A x + A y + A 3 z D 5/ B y K m 3m z y x + y + z 5/ 3K mm A 3 x + A 3 y + A 33 z A x + A y + A 3 z ˆA x + A y + A 3 z + A x + A y + A 3 z + A 3 x + A 3 y + A 33 z 5/ 3K mm A 3 x + A 3 y + A 33 z A x + A y + A 3 z D 5/ B z K m 3m z x + y + z m x + y + z 5/ K m 3m A 3 x + A 3 y + A 33 z ˆA x + A y + A 3 z + A x + A y + A 3 z + A 3 x + A 3 y + A 33 z m ˆA x + A y + A 3 z + A x + A y + A 3 z + A 3 x + A 3 y + A 33 z 5/ K mm ˆA x + A y + A 3 z + A x + A y + A 3 z A 3 x + A 3 y + A 33 z ˆA x + A y + A 3 z + A x + A y + A 3 z + A 3 x + A 3 y + A 33 z 5/ K mm ˆA x + A y + A 3 z + A x + A y + A 3 z A 3 x + A 3 y + A 33 z D 5/

12 x B x 3K mm D 5/ A 3 A x + A 3 A y + A 3 A 3 z + A 3 A y + A 33 A z A 3 x + A 3 y + A 33 z A x + A y + A 3 z D 3/ y B x z B x D 5 x B y y B y z B y B B x B y B z A T B A A A 3 A A A 3 A 3 A 3 A 33 B x B y B z A B x + A B y + A 3 B z A B x + A B y + A 3 B z A 3 B x + A 3 B y + A 33 B z z B z B xˆx + z B y ŷ A z B x + A z B y + A 3 z B z ˆx + A z B x + A z B y + A 3 z B z ŷ B x B x + y B y A x B x + A x B y + A 3 x B z + A y B x + A y B y + A 3 y B z Magnet Foce between two cube of ide a and b of magnetically polaized mateial with magnetic dipole moment m M a 3 and m M b 3 : 4.5 Dipole-Dipole Toque

13 5 Radiation Electic and Magnetic Multipole Radiation Show that thee i no monopole adiation Imagine a neutal phee that peiodically ha a epaation of chage uch that thee i an exce poitive chage on the uface and exce negative chage in the cente egion both pheically ymmetically ditibuted, then a eveal of polaity. Impotant example Rotation geneal, dipole Show multipole adiation too, not jut dipole. Whee doe the enegy in the n-pole adiation go fo n 3? 5. Localized Souce and Multipole Expanion Let thee be a ouce of adiation with chage ditibution ρx, t and cuent ditibution Jx, t. We make the following aumption: the ouce i in empty pace o D ε 0 E and H µ 0 B late deive moe geneal cae, the ouce i localized within the egion, and 3 the ouce i mall with maximal extent d compaed to the wavelength of the adiation of ubtantial magnitude: d λ whee tand fo ubtantial adiation. We alo make the uual phyical aumption that we can Fouie decompoe the ouce function ρt and Jt and any othe phyical, time-dependent function into hamonic weighted by function of angula fequency: ρx, t Jx, t Bx, t Ex, t Ax, t E + t B 0 B c t E 0 B A π π π π π etc dω ρx, ω e iωt dω Jx, ω e iωt dω Bx, ω e iωt dω Ẽx, ω e iωt dω Ãx, ω e iωt Ẽ iω B B iω c Ẽ B Ã S µ 0 E B S µ 0 Ẽ B 3

14 Uing the continuity equation, we ee π t ρx, t Jx, t dω iω ρx, ω e iωt π iω ρx, ω Jx, ω ick ρx, ω Jx, ω dω Jx, ω e iωt given that λν π ω k π ω k c. Fo convenience, we ll ue k kω ω/c. Uing the etaded eqn fo the vecto potential?, we have Ax, t K m d 3 x dt Jx, t δ t t + R/c R K m d 3 x dt π dω Jx, ω e iωt δ t t R/c R K m d 3 x π dω Jx, ω e iωt R/c K m π Ãx, ω K m K m Fa zone: R ˆ x. Small ouce / keep dipole tem: d λ π/k, o k d o... kd, and ince x d, kˆ x, o e ikˆ x, thu R dω d 3 x Jx, ω e iωt R/c R d 3 x Jx, ω eiωr/c R d 3 x Jx, ω eikr R Ãx, ω K m K m Ã, ω Ã, ω K m d 3 x Jx, ω eik ˆ x d 3 x Jx, ω e ikˆ x e ik e d 3 x Jx ik, ω By integation by pat with J 0 on the bounday of and the continuity equation eult above, d 3 x Jx, ω d 3 x x Jx, ω iω d 3 x x ρx, ω iω ω pω d 3 x x ρx, ω 4

15 o Note that ince pω Ã, ω ik m ω ω eik d 3 x x ρx, ω π π dω d 3 x xρx, t dω pt e iωt d 3 x x dω ρx, t e iωt π e iωt 5. Finding the Electic Dipole Field... in the fa field fo a ouce of adiation that i much malle than any wavelength of the ubtantial adiation d λ. Bx, ω Ãx, ω Ã, ω θ + Cθ à φ Sθ Sθ φ à θ, Sθ φ à + à φ, + à θ θã Cθ Sθ Ãφ, à φ, à θ 0, Ãφ, Ãθ Thi Cetain Aumption, that Cθ Sθ Ãφ i negligible compaed to à and à θ, i ued to obtain below, which matche Jackon Eqn 9.8 fo the magnetic field. ˆ ˆθ ˆφ 0 0 à à θ à φ ˆ Ã, ω e ik ik m ω ˆ ω Cetain Appoximation Negligible in fa zone, but we ll keep it thi time. e ik ik m ω ˆ ik ω e ik ik m ω ˆ ik ω K ˆ m e ik c ω ω 5

16 Ẽ i c ω B i c ω ik m ω K m c ˆ K e ik e ik e ik ˆ ik ω e ik pω ik e ik ˆ ˆ ω ω 5.. Sub Poblem We d like to take the cul of eik eik ˆ and ˆ. Let a eik eik ˆ, b and c ˆ. Note that ince b i not a vecto field, all of it patial, deivative ae zeo. a b a b b a + b a a b b a + b a c b b c + b c Alo, note that in pheical component, whee v v, v θ, v φ, v Sθ Sθ v +, θ + Cθ Sθ, Sθ φ v. 6

17 In decateian component, whee x x, y, z d, e ik a ˆ + e ik ik eik eik + eik eik ik + eik ik + e ik c ˆ + e ik ik eik eik 3 eik ik + 3 eik 3 ik + eik 3 7

18 b a eik ˆ y, z x, d e ik x, y, z d e ik x + y y + z z x e ik x + y y + z z x e ik + x + y y + z z x e ik x + y y + z z x, y, z d x + eik e ik e ik ik eik { eik 3 p + eik p + eik p + ik ik eik p + ik ˆ x + y y + z z x x x + y y + z z p ik } ˆ 8

19 and o b c eik ˆ y, z x, d e ik x, y, z d 3 e ik x + y y + z z x 3 e ik 3 x + y y + z z x e ik + x + y y + z z x 3 e ik 3 x + y y + z z x, y, z d x e ik ˆ + eik 3 e ik 3 e ik 3 ik eik 3 { 3eik 4 p + eik 3 p + eik 3 p + ik 3 ik 3 eik 3 p + ik 3 ˆ a b x + y y + z z x x x + y y + z z p ik 3 } ˆ b a + b a e ik ik + + eik p + ik ˆ eik ik ik ˆ eik ik + ik ˆ eik ik ˆ ˆ 9

20 and e ik ˆ c b b c + b c e ik 3 ik + eik 3 p + ik 3 ˆ eik 3 ik + + ik 3 ˆ eik 3 ik + ik 3 ˆ 0

21 Theefoe Ẽ i c ω B e ik i c ω ik m ω ˆ ik e K m c ˆ ik p ik e K e ik ik ˆ e ik ˆ { e K e ik ik } { e ikˆ ˆ ˆ ik } 3 ik ˆ + 3 ˆ { e K e ik ik } { e ik ˆ ˆ ik } 3 ik + ik 3 ˆ e ik K e k ˆ ik ˆ eik 3 ik + ik 3 ˆ e ik K e K e K e K e K e K e K e k ˆ eik k ˆ eik k ˆ eik k ˆ eik k ˆ eik k ˆ eik k ˆ ˆ eik 3 ˆˆ ik ˆ eik 3 ik + ik 3 ˆ e ik ik + ik 3 ˆ + ik ˆ e ik ik + 3ik 3 ˆ 3 e ik ik + ik 3 ˆ + 3 ˆ ik eik ˆ ik e ik ˆ ˆ + 3 ˆˆ ik e ik D E D E E 3, 0, 0, φ D, θ, φ θ, D E «, 0, 0, φ ˆ θ, E E D0, φ, θ, 0, 0 D0, θ, φ ˆ 3 3 3

22 5.3 Moe Detail: Without the Cetain Appoximation Above Bx, ω Ãx, ω Ã, ω θ + Cθ Ã φ Sθ Sθ φ Ã θ, Sθ φ Ã Cθ Sθ Ãφ, + Ã φ, + Ã θ + 0, Ãφ, Ãθ + Cθ Ãφ, 0, 0 Sθ + ˆ Ã + Cθ Ãφ, 0, 0 Sθ e ik ik m ω + e ik ik m ω ˆ K ˆ m e ik c ω ω ˆ ik + ik m ω ik m ω ik m ω 0, 0 φω, + Ã φ, Cθ 0, 0 φ, Sθ 0, 0 φ, Cθ e ik Sθ e ik Cθ Sθ e ik + Ã θ θã

23 Ẽ i c ω B i c ω K ˆ m e ik c ω ik m ω 0, 0 ω φω, e ik m c ik Cθ e ik k ˆ i 0, 0 φ, Sθ e ik Cθ e ik ik e k ˆ + K e 0, 0 φ, Sθ e ik e k ik ˆ Cθ e + K e ik Sθ 0, 0 φ, K e ik eik Cθ θ, ik θ, ik φ K e Sθ S θ K e k eik Cθ e 0, ik θ, φ + ik Sθ 0, 0 K e θ, S θ K e k eik ˆ ˆ + e ik Sθ ikcθ θ, 0, 0 Sθ K e k ˆ ˆ eik Sθ Cθ Sθ e ik e ik 3 0, 0, φ e ik 3 0, 0, φ e ik 0, 0, φ ik Cθ θ, 0, 0 Sθ Jackon eult Eqn 9.8 K e k ˆ ˆ eik { } e ik + 3ˆˆ ik 0, 0, φ Sθ e ik 3

24 5.4 Reult Fa zone o, Even Fathe Zone: B, ω ik m ω ˆ ik e ik ω Ẽ, ω K e k ˆ ˆ eik + 3 ˆˆ B, ˆ e ω K m ck ik ω ˆ Ẽ, ω K e k ˆ eik K e ck m B ˆ c B ˆ ik e ik Time-aveaged enegy flux denity in empty pace Fouie tanfom modal coefficient function: S, ω t Re Ẽ H c Re B ˆ µ B 0 c Re µ B B ˆ 0 c Re µ B B ˆ + ˆ B B 0 c Re ˆ µ B 0 c B ˆ µ 0 c K m ck ˆ µ 0 ˆ ˆ c µ0 ωk ˆ ˆ p ˆ µ 0 4π ˆ ˆ ω cµ 0 p ˆ λ πν Z 0 ˆ p ˆ ˆ λ Z 0 π ν ˆ ˆ ˆ Z 0 I λ ω A ˆ c Z 0 3π k4 ˆ ˆ ˆ Z 0 ω 3π c ω4 ˆ ˆ ˆ ω ince Z 0 µ0 ε 0 4

25 c µ 0 c 3 K m ck k ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ K m µ 0 c 3 µ 0 4π k4 p ˆ c Z 0 ˆ 4π k4 ˆ ˆ c Z 0 ˆ ˆ ˆ ω 3π k4 OR Z 0 4π ˆ ωk ˆ ω ˆ 4π ZI A ˆ What to do with thoe 4π? but 0 hould not be included... o how doe that get enfoced? k 0. ω Time-aveaged powe Fouie tanfom dp t, ω S S, ω da, ω t t dωˆ S dp dω Ω, ω ˆ, ω t t Re ˆ S Z 0 4π ˆ ωk ˆ ω 4π c Z 0 ˆ ˆ ω 3π k4 Fo a dipole Fouie tanfom pointing in the z-diection dp dω Ω, ω Z 0 t ωk in θ 4π ω 4π c Z 0 ω 3π k4 in θ 5

26 Total Powe Radiated angula-fequency-denity: P t ω dω dp S dω Ω, ω t π π Z 0 in θ dθ dφ ωk in 0 0 4π ω θ 4π Z 0 ωk π in 4π ω 3 θ dθ 0 Z 0 ωk 4 4π ω 3 Z 0 ωk 3 4π ω c Z 0 ω π k4 Total Powe Radiated: P t Z 0 3 4π dω P t ω c Z 0 π dω ωk ω dω k 4 ω might I poibly get quae of Diac delta function hee?! Dicete fom: P t P t ω i i,ω i 0 Z 0 3 4π i,ω i 0 c Z 0 π ω i k i i ω i,ω i 0 k i 4 ω i and emembe that ω i doe not count a the ame a ω i, o they each have thei own quae-magnitude tem 6