Radiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t)

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1 Radiation from moving harges# Liénard-Wiehert potential Yuji Chinone Maxwell Maxwell MKS E x, t + B x, t = B x, t = B x, t E x, 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 B x, t = A x, t 6 B = A = A A E = A φ A x, t + φ x, t = 7

2 Maxwell A x, t = A A = A B + E + = φ B + E + φ = µ j x, t φ x, t = φ φ = E A + A = ε ρ x, t φ 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 E.8 E.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 E. E.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

3 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 E.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 E. E.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 E.3 E.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 E.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 3

4 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

5 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 = 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 + r µ ξ + r µ dξ = ξ + r µ dξ 5

6 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 + r µ dξ = Im [ πir+irµ ] { 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 + 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 6

7 E.8 G ± x, ω E.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 E.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 E.3 E.5 E. δ t t ρ x, t e iωt dt Liénard-Wiehert ρ x, t = δ x x t, j x, t = ut 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 E.3 E.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 7

8 A x, t = µ dt δ hx, t, t ut 4π x x t = µ 4π d dt t t + x x t ut Rx, t = µ ut 4π nt βt Rx, t hx,t,t = φ x, t = dt δ hx, t, t 4πε x x t = 4πε d dt t t + x x t Rx, t = 4πε nt βt Rx, t hx,t,t = hx,t,t = hx,t,t = 9 3 E.9 E.3 Liénard-Wiehert 5 E.5 E x, t = β t nt βt 4πε κ 3 t R x, t + 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 ] x Rx, t + 4πε dt δ hx, t, t [ ] x Rx, t = dt δ hx, t, t [ ] 4πε x Rx, t = dt δ hx, t, t n x 4πε R x, t = n x 4πε d dt t t + x x t R x, t = n x hx,t,t 4πε = nt βt R x, t = n x 4πε κt R x, t hx,t,t = hx,t,t = 8

9 = 4πε hx,t,t = dt [ δ hx, t, t ] x Rx, t = dt 4πε = + dy dt n x 4πε dy Rx, t y δt y, y = t + x x t = + dy dt n x 4πε y dy Rx, t δt y = [ ] dt n x 4πε dt Rx, t = [ dt hx,t,t = 4πε dt = [ ] n x 4πε κt κt Rx, t dt = = dt κt [ δ t t + x x t ] x ] n x Rx, t hx,t,t = Rx, t φ x, t = [ ] n x x 4πε κt κt Rx, t hx,t,t = 4πε κt n x R x, t hx,t,t = nt t nt = ṅt = x x 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 nt = nt nt βt Rx, t 35 9

10 5.3 A x, t t A x, t = µ 4π = 4πε { δ t t + Rx, t ut } Rx, t { δ dt dt ut Rx, t Rx, t t t + Rx, } t = dt βt { 4πε Rx, t Rx, t δ t t + Rx, } t = + dy dt βt 4πε dy Rx, t y δt y, y = t + x x t = + dy dt βt 4πε y dt Rx, t δt y = [ dt βt ] 4πε dt Rx, t = [ dt hx,t,t = 4πε dt = [ βt ] 4πε κt κt Rx, t hx,t,t = βt ] Rx, t hx,y,t = 6 E.5 A x, t E x, t = φ x, t = E.6 B x, t = A x, t = µ 4π = 4πε = 4πε V d 3 x dt nt ut [ nt 4πε R x, t κt + κt 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 = [ βt nt 4πε R x, t κt + βt nt ] κt Rx, t κt E x, t E.35 E x, t = [ nt 4πε R x, t κt + nt nt βt + nt R x, t κ t κt κt Rx, t δ t t Rx,t Rx, t t t Rx, ] t κt βt κt Rx, t ]

11 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 = [ nt 4πε R x, t κ t + nt κt κt Rx, t βt R x, t κ t κt βt κt Rx, t ] 36 E.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 = [ n 4πε R κ + n κ β κr R κ ] β κ κr t = [ 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 = 4πε = 4πε = 4πε = 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 + [ ] 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 + n { n β β } 4πε 38 κ 3 R t t E.3 /R t

12 7 E.36 B x, t = [ βt 4πε R x, t κ t + κt E.36 E.39 βt Rx, t κt ] nt 39 B x, t = nt E x, t 4 E.4 8 Poynting Vetor Larmor Poynting Vetor P x, t E x, t B x, t µ 4 4πR E.3 E.4 /R Poynting Vetor /R [ E rad x, t = 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 = [ n n β ] n = [ n β n β ] n µ 4πµ R µ 4πµ R = [ β os Θ n β ] n = β os Θ + β β os Θ n µ 4πµ R µ 4πµ R = µ 4πµ β sin Θ R n 45

13 Θ n β dω ds ndω n n = R dω P ds P x, t = R 6π 3 µ ε = u π dθ sin 3 Θ = 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 = u 4 4 8πε 3 3 = u 6πε = u πε 3 E.46 Larmor 9 MKS gs Gauss MKS gs Gauss MKS Gauusian B MKS B Gaussian ε 4π µ 4π µ ε 3

Kroneher Levi-Civita 1 i = j δ i j = i j 1 if i jk is an even permutation of 1,2,3. ε i jk = 1 if i jk is an odd permutation of 1,2,3. otherwise. 3 4

[2642 ] Yuji Chinone 1 1-1 ρ t + j = 1 1-1 V S ds ds Eq.1 ρ t + j dv = ρ t dv = t V V V ρdv = Q t Q V jdv = j ds V ds V I Q t + j ds = ; S S [ Q t ] + I = Eq.1 2 2 Kroneher Levi-Civita 1 i = j δ i j =