Liénard-Wiehert,et Maxwell Maxwell MKS E x, t + B x, t t B x, t B x, t E x, t t µ j x, t 3 E x, t ε ρ x, t 4 ε µ ε µ E B ρ j A x, t φ x, t A x, t E x, t φ x, t 5 t B x, t A x, t 6 B A A A E t A φ A x, t + φ x, t t 7
Maxwell t A x, t A A t A B + E t + t φ B + E t + t φ t µ j x, t φ x, t φ φ t E t A + A t ε ρ x, t φ t t t A x, t µ j x, t 8 φ x, t ε ρ x, t 9 3 Green A x, t Fouroer A x, ω A x, t A x, ω π Eq.8 Eq.9 Fourier A x, ω e iωt dω A x, t e iωt dt A x, ω + ω A x, ω µ j x, ω φ x, ω + ω φ x, ω ρ x, ω ε 3 j ρ Eq. Eq.3 A φ Green G x, ω G x, ω + ω G x, ω δ3 x δxδyδz 4 δ 3 x δ δx π e ikx dk, δ 3 x π 3 e ik x d 3 k, f x f x δx x d 3 x
4 Maxwell Green A x, ω µ φ x, ω ε d 3 x Gx x, ω j x, ω 5 d 3 x Gx x, ω ρ x, ω 6 Eq.4 x, ω x x, ω µ j x, ω d 3 x µ d 3 x Gx x j x, ω + µ d 3 x ω Gx x j x, ω µ d 3 x δ 3 x x j x, ω [ ] ] µ d 3 x Gx x j x, ω + [µ ω d 3 x Gx x j x, ω µ j x, ω A x, ω + ω A x, ω µ j x, ω Eq. Eq.4 x, ω x x, ω ρ x, ω/ε d 3 x d 3 x Gx x ρ x, ω + d 3 x ω ε ε Gx x ρ x, ω d 3 x δ 3 x x ρ x, ω ε [ ] [ ] d 3 x Gx x ρ x, ω + ω d 3 x Gx x ρ ε x, ω ρ x, ω ε φ x, ω + ω φ x, ω ρ x, ω ε Eq.3 Eq.4 Green G x, ω x Fourier Ĝ k, ω d 3 x e k x G x, ω, G x, ω d 3 k e ik x Ĝ k, ω π 3 ε Eq.4 d 3 k e ik x k Ĝ k, ω + ω d 3 k e ik x Ĝ k, ω δ 3 x e k x d 3 x Ĝ k, ω π 3 k ω π 3 k µ 7 G x, ω π 3 dk eik x k µ π 3 k dk + k k µ ω / r x k x krν ν os θ dν eikrν k µ ke ikr 4iπ r k µ dk k ±µ k
exp [ik R k l r] exp ik R r exp k i r exp k l r ke ikr k µ dk C ke ikr k µ dk [ C Cauhy e ikr k + µ dk + C e ikr ] k µ dk 4. k lim ɛ + e ikr dk lim k ± µ iɛ ɛ + C e ikr dk πie iµr k ± µ iɛ lim ɛ + e i kr dk lim k ± µ + iɛ ɛ + C e i kr k ± µ + iɛ dk lim ɛ + lim ɛ + e i kr + k ± µ iɛ dk P e i kr + k ± µ + iɛ dk P P e ikr dk + lim k ± µ ɛ + π e ikr dk + lim k ± µ ɛ + π f k µ ɛ dk lim k ± µ ɛ + e µ+ɛeiφ r ɛe iφ e µ+ɛeiφ r ɛe iφ f k + k ± µ dk + µ+ɛ iɛe iφ e ikr dφ P dk + iπe iµr k ± µ iɛe iφ e ikr dφ P dk iπe iµr k ± µ f k k ± µ dk Cauhy s Prinipal Values e ikr P dk iπe iµr k ± µ f x lim ɛ + f x + x x ± iɛ dx P f x x x dx iπ f x x x ± iɛ P iπδx x x x
4. running wave G ± x, ω 4iπ r lim ke ikr ɛ + k µ ± iɛ dk 4iπ r lim ɛ + ke ikr k µ ± iɛ dk ±iɛ ɛ > ɛ + G ± x, ω 4iπ r lim ke ikr ɛ + k µ ± iɛ dk [ 4iπ r lim πi ke ikr ] ɛ + k ± µ ± iɛ k±µ±iɛ 4iπ r lim πi ± µ ± iɛ ei[±µ±iɛ]r [ ] ±µe i±µr ɛ + ± µ ± iɛ ± µ ± iɛ 4iπ r πi ±µ ± µ e±iµr 4πr e ±i ω x 4π x G x, ω k G ± x, ω 4iπ r lim ke ikr ɛ + k iɛ µ dk { 4iπ r lim e ikr ɛ + k µ ± iɛ + e ikr k µ ± iɛ } dk 8 G + x, ω G + x, ω { 4iπ r lim e ikr ɛ + k µ + iɛ + os µr 4πr e ikr } dk k µ + iɛ 4iπ r iπ e iµr + e iµr 9 G x, ω G x, ω 4.3 G k, µ π 3 k + µ Fourier G x, µ d 3 k G k, µ e ik x d 3 eik x k π 3 k + µ π 3 k sinkr dk ξ sin ξ π k + µ kr π r ξ + r µ dξ π dφ k dk dξ kr ξ, dk r π dθ sin θ eikr os θ k + µ ξ sin ξ ξ + r µ ξ sin ξ ξ + r µ dξ ξ sin ξ ξ + r µ dξ
ξ sin ξ ξ + r µ dξ ze CR iz z + dz ze CR iz z + dz z e iz R sin θ C R z + CR dz Re R dz z Re iθ θ π, dz z θ dθ ire iθ dθ Rdθ π R R r sin θ Re < R R R R Rdθ R π/ π R < e R sin θ dθ + π π/ R R π R R e R sin θ dθ R e R sin ψ dψ R R πr R R π/ π/ π e R sin θ dθ + e R sin θ dθ π/ e R sin θ dθ ξ sin ξ ξ + r µ dξ Im [ πir+irµ ] { ξ sin ξ ze dξ Im πi iz } z irµ ξ + r µ z + irµ z irµ z+irµ ] Im [πi irµe rµ Im irµ [πi e rµ ] πe rµ G x, ω π r ξ sin ξ ξ + r µ dξ π r πe µr e µr 4π r t ht ht π dx eixt x iɛ, ɛ > t > ht > π dx eixt x iɛ πi πie t < ht < 4.4 t t t t ± x x
Eq.8 G ± x, ω Eq.8 ] A x, t [µ d 3 x Gx x, ω j x, ω e iωt dω µ dω e iωt d V V 3 x + G ± x, ω j x, t e iωt dt π µ d 3 x + 4π V x x dt j x, t + dω e iω ± x x +t t π µ d 3 x dt δ t t x x j x, t 4π V x x 3 µ d 3 x j x, t 4π V x x 4 Eq.9 [ ] + φ x, t d 3 x Gx x, ω ρ x, ω e iωt dω ε dω e iωt d ε V V 3 x G ± x, ω π d 3 x + 4πε V x x dt ρ x, t + dω e iω ± x x +t t π d 3 x dt δ t t x x ρ x, t 4πε V x x d 3 x ρ x, t 4πε V x x Eq.3 Eq.5 Eq. δ t t ρ x, t e iωt dt 5 6 4.5 Liénard-Wiehert ρ x, t q δ x x t, j x, t qut δ x x t 7 t h x, t, t t t + x x t Rx, t x x t, nx, t x x t Rx, t, ut dx t dt βt Eq.3 Eq.5 δ dt δ f t gt d f dt d f δ f g t f i d f t i dt gt i f ti 8
A x, t µ dt δ hx, t, t qut 4π x x t µ 4π d dt t t + x x t qut Rx, t µ qut 4π nt βt Rx, t hx,t,t φ x, t dt δ hx, t, t q 4πε x x t q 4πε d dt t t + x x t Rx, t q 4πε nt βt Rx, t hx,t,t hx,t,t hx,t,t 9 3 Eq.9 Eq.3 Liénard-Wiehert 5 Eq.5 E x, t q β t nt βt 4πε κ 3 t R x, t + q nt [ nt βt βt ] 4πε κ 3 t Rx, t 3 κt nt βt 3 5. φ x, t x 4πε dt [ δ hx, t, t ] q x Rx, t + 4πε dt δ hx, t, t [ ] q x Rx, t dt δ hx, t, t [ ] q 4πε x Rx, t dt δ hx, t, t qn x 4πε R x, t q n x 4πε d dt t t + x x t R x, t q n x hx,t,t 4πε nt βt R x, t q n x 4πε κt R x, t hx,t,t hx,t,t
4πε hx,t,t dt [ δ hx, t, t ] q x Rx, t q dt 4πε q + dy dt n x 4πε dy Rx, t y δt y, y t + x x t q + dy dt n x 4πε y dy Rx, t δt y q [ ] dt n x 4πε t dt Rx, t q t [ dt hx,t,t 4πε t t dt q [ ] n x 4πε κt t κt Rx, t dt t dt t κt [ δ t t + x x t ] x ] n x Rx, t hx,t,t Rx, t φ x, t q [ ] n x x 4πε κt t κt Rx, t hx,t,t q 4πε κt n x R x, t hx,t,t 33 5. nt t t nt ṅt x x t t Rx, t ut Rx, t + x x t βt + nt nt βt Rx, t ẋt Rx, t + x x t drx, t d dt drx, t x x t ẋt R x, t Rx, t Rx, t 34 n n β n n β β n n t nt nt nt βt Rx, t 35
5.3 A x, t t t A x, t qµ 4π q 4πε { δ t t + Rx, t ut } Rx, t { δ dt t dt ut Rx, t Rx, t t t + Rx, } t q dt βt { 4πε Rx, t Rx, t δ t t + Rx, } t q + dy dt βt 4πε dy Rx, t y δt y, y t + x x t q + dy dt βt 4πε y dt Rx, t δt y q [ dt βt ] 4πε t dt Rx, t q t [ dt hx,t,t 4πε t t dt q [ βt ] 4πε κt t κt Rx, t hx,t,t βt ] Rx, t hx,y,t 6 Eq.5 A x, t E x, t φ x, t t Eq.6 B x, t A x, t µ q 4π q 4πε q 4πε V d 3 x dt nt ut q [ nt 4πε R x, t κt + κt t dt δ x x t nt ut Rx, t dt nt βt [ δ t t Rx,t nt βt ] κt Rx, t Rx, t δ t t Rx,t Rx, t + R x, t Rx, t Rx, t δ E x, t B x, t q [ βt nt 4πε R x, t κt + βt nt ] κt t Rx, t κt E x, t Eq.35 E x, t q [ nt 4πε R x, t κt + nt nt βt + nt R x, t κ t κt t κt Rx, t δ t t Rx,t Rx, t t t Rx, ] t κt t βt κt Rx, t ]
t nt R x, t κt + nt nt βt nt κt R x, t κ t R x, t κ t + nt nt βt R x, t κ t nt κ t R x, t nt nt βt nt nt βt κ t R x, t nt κ t R x, t βt κ t R x, t E x, t q [ nt 4πε R x, t κ t + nt κt t κt Rx, t βt R x, t κ t κt t βt κt Rx, t ] 36 Eq.36 d dt βt βt d κt dt Rx, t dκt dt Rx, t + κt drx, t dt nt nt ut Rx, t { d dt nt βt ut + nt ut nt nt ut ut Rx, t β t nt βt Rx, t nt βt R } Rx, t κt nt ut nt nt ut ut ut + nt ut R nt nt ut ut E x, t t E x, t q [ n 4πε R κ + n κ t β κr R κ ] β κ t κr t q [ n 4πε R κ + n β n β R n β κ κ R β R κ { β κ κr + β β t nt βt Rx, t nt βt } ] κ R q 4πε q 4πε q 4πε q 4πε [ n β n nβ + n β n n β β + ββ n β β κ 3 R + n n β β n β β n β ] κ 3 R t ṅ β R + κn β n β + R κṅ + κ n κ 3 R + q [ ] n β n β κ β 37 4πε κ 3 R t t n β β n β n β β n n β + κ 3 R κ 3 R t β n β κ 3 R + q n { n β β } 4πε 38 κ 3 R t t Eq.3 /R t
7 Eq.36 B x, t q [ βt 4πε R x, t κ t + κt Eq.36 Eq.39 t βt Rx, t κt ] nt 39 B x, t nt E x, t 4 Eq.4 8 Poynting Vetor Larmor Poynting Vetor P x, t E x, t B x, t µ 4 4πR Eq.3 Eq.4 /R Poynting Vetor /R [ E rad x, t q nx, t nx, t βx, t βx, t ] 4πε κ 3 x, t Rx, t 4 B x, t nx, t E rad x, t 43 Poynting Vetor P x, t E rad x, t B rad x, t µ E {n E rad} µ E rad n n E rad E rad µ β κ E rad µ n, E rad n E n 44 P x, t [ q n n β ] n [ q n β n β ] n µ 4πµ R µ 4πµ R [ q β os Θ n β ] n q β os Θ + β β os Θ n µ 4πµ R µ 4πµ R µ q 4πµ β sin Θ R n 45
Θ n β dω ds ndω n n R dω q P ds P x, t R 6π 3 µ ε q u π dθ sin 3 Θ q u 8πε 3 8πε 3 lim R π π dφ π dθ R sin Θ u sin Θ R 3 sin Θ sin 3Θ dθ q u 4 4 8πε 3 3 q u 6πε q u 46 3 3 4πε 3 Eq.46 Larmor 9 MKS gs Gauss MKS gs Gauss MKS Gauusian B MKS B Gaussian ε 4π µ 4π µ ε Thomson k ω E E exp [i k x ωt], B B exp [i k x ωt] 47 Eq. k k, ω k E + B t i k E iωb B k E k 48 Eq.4 ρ E i k E k E 49 w E D + B H
w E D + B H ε E + B ε E + µ µ E ε E 5 m q a qe inident m n dω Eq.45 β a/ P sattered x, t q β sin Θ n q a sin Θ n µ 4πµ r µ 4πµ r q ε 4πε m E inident sin Θ n r r ε E inident sin Θ n 5 r Θ n a r q e q 4πε m 5. Eq.5 P x, t P x, t n dσ Ω j sattered r dω P sattered r dω r j inident w sin Θ dω 53 dσ Ω dω r sin Θ 54 σ T dω dσ Ω π πr dθ sin 3 Θ 8π 3 r 55 q e σ T Thomson λ m σ T 8 3 α e 4πε e π 8 4πε m 3 α πλ 56 α
z κ n θ x o φ ϕ Θ E y Thomson. κ k/k z z x y E B xy E x ϕ n z θ n xy x φ n sin θ os φ, sin θ sin φ, os θ, E E os ϕ, E sin ϕ, os Θ n E sin θ os φ os ϕ + sin θ sin φ sin ϕ sin θ os φ os ϕ + sin φ sin ϕ n E sin θ os φ ϕ ϕ sin Θ sin θ os φ ϕ 57 π sin Θ sin θ os φ ϕ { sin θ os φ ϕ } dϕ π π { sin + os [ φ ϕ ] } dϕ π π π sin θ π + os θ { π + sin φ sin φ [ π sin θ sin { φ ϕ} π + } sin θ ] π
dσ Ω dω r sin Θ r + os θ 58 unpolarized θ.3 Eq.4 E sattered x, t q nx, t [ nx, t βx, t βx, t ] 4πε κ 3 x, t Rx, t q nx, t 4πε nx, t qe inidentx, t m κ 3 x, t Rx, t q nx, t nx, t βx, t 4πε κ 3 x, t Rx, t β β 59 E sattered E sattered n, E sattered E inident Bremsstrahlung Ze v b a Ze mb t oll b v Larmor Ze Wb P t oll e Ze b 6πε 3 mb v e 6 Z 6πε 3 m v b 3 6 N e v N Z dp dv N zn e v e6 Z N Z N e 6πε 3 m π e 6 Wb πb db b min b db e6 Z [ N Z N e b min 6πε 3 m π ] b b min N 3 ε m 3 e N Z Z 6 b min b min b min mv
dp dv N 3 ε m 3 e N Z Z v b min 3 ε m 3 e 6 e 6 N en Z Z 8π 3 4πε e 4πε 3 mv mv vn e N Z Z 8π 3 4πε α 3 λ rd WvN en Z Z 6 W mv, λ ed de Broglie mv Gaunt fator g B g B dp dv 6π 3 3 4πε g B α 3 λ ed WvN en Z Z 63