Using the Green s Function to find the Solution to the Wave. Equation:

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1 Using the Green s Funtion to find the Soution to the Wave Exampe 1: t 2 Equation: r,t q 0 e it r aẑ r aẑ r,t r 1 r ; r r,t r 1 r 2 The Green s funtion soution is given by r,t G R r r,t t Fr,t d 3 r dt Θt t Θt t 4 r r r r t t Fr,t d 3 r dt 4 r r F r,t r r 1 t t r r Fr,t d 3 r dt 1 4 r r d3 r A7 r,t q 4 0 q 4 0 e it expit r r r aẑ r aẑ 1 r r d3 r r aẑ 1 r aẑ expit r aẑ 1 r aẑ Normay the Green s funtion soution woud have surfae integra terms evauated at r and at t. They woud be of the form: dt G G ds r dv t G G t t. The surfae integras do not ontribute sine both the soution and the Green s funtion vanish as r.the time derivatives at t. do not appear beause of the Θt t in the Green s funtion and at t one assumes that there is no time derivative of the soure signa in the Green s funtion, and no time derivative of the wave funtion,. Thet boundary ondition has been taken are of by our hoie of ontour. In most appiations of the Green s funtion the disturbane is assumed to take pae near t 0 and to turn off at t In this ase one an see that the boundary onditions in time are automatiay taken are of.

2 Maxwe s Equations Generay one wants to find Er,t and Br,t, but in pratie it is easier to find Ar,t and r,t.first and then determine Er,t and Br,t from the foowing: B A E 1 A t Sine the above defines A uniquey, one has to suppy another ondition on A. There are two ommony used hoies or gauges: Lorentz Gauge Ar,t 1 t r,t 0 A8 2 r,t t r,t r,t/ Ar,t t Ar,t 0Jr,t 2 S.I. units S.I. units Couomb/Radiation Gauge Ar,t 0 A9 2 r,t r,t/ 0 2 Ar,t t Ar,t 0J 2 t r,t J J Jt Jt 0;.J 0 In the Lorentz gauge: Ar,t G R r r,t t 0 Jr,t d 3 r dt 0 Θt t 0 Θt t 4 r r r r t t Jr,t d 3 r dt 4 r r 0 J r,t r r 1 t t r r Jr,t d 3 r dt 1 4 r r d3 r A10

3 Exampe: A singe frequeny urrent density soure J t r,t Jrexp i 0 t Ar,t 0 4 J r,t r r 1 r r d3 r 0 4 Jr exp i 0 t r r 1 r r d3 r 0 4 e i 0t Jr exp i 0 r r 1 r r d3 r 0 4 e i 0t Jr expik 0 r r d 3 r ; k r r 0 0 Ar,t 0 4 e i 0t Jr expik 0 r r d 3 r ; k r r 0 0 A11 Using the expansion for the Hemhotz Equation Green s funtion we have e ik r r r r 0 4ikj kr h kr Y m, Y m, m and finay we have the mutipoe expansion of the vetor potentia, A: Ar,t ik 0 e i 0t 0 Jr j kr h kr Y m, Y m,d 3 r A12 m Sine the soure urrent is oaized near r 0,the j kr an be used in the integra and the soution for r r is given by: A 0 r ik 0 0 Ar,t A 0 re i 0t j kr in kry m, J 0 r j kr Y m, r 2 d dr m where k k 0 0.andJr J 0 r Approximating j kr with the form near r 0:

4 j kr 1 2 1!! kr A 0 r k 0 0 ij kr n kry m, J 0 r k 1 m 2 1!! r 2 Y m, d dr. kr 0r r /2f r 2 r In the near fied region, kr 1 r, anda 0 r is approximated by etting: A 0 r k 0 0 k 0 0 ij kr n kr n kr 2 1!! kr 1 Note: r (soure) is ose to 0 and ess than r and r m m 2 1!! kr 1 Y m, J 0 r k 1 2 1!! r 2 Y m, d dr 1 r Y 1 m, J 0r r 2 Y m, d dr. A13 In the far fied region, kr 1and r, the radiation form for A 0 r is approximated using A 0 r k 0 ij kr n kr exp i kr 2 kr Note: r (soure) is ose to 0 and ess than r and r 0 m k 2 1!! exp i kr 2 kr Y m, J 0 r r 2 Y m, d dr. A14 This an be evauated term by term. When kd 1 the series is generay dominated by the owest non-zero term. It is aso a usefu expression if J 0 r is desribed by a superposition of one or two spheria harmonis.

5 More genera time dependene In this setion we onsider the potentias generated by a moving harge oated at Rt. The harge and urrent densities are r,t qr Rt Jr,t qr Rt d Rt dt These an be used to obtain the potentias, at r and time t (Gaussian units) from: Ar,t 1 t r,t 0 A8b : Ar,t q Θt t r,t q Θt t 2 r,t t 2 r,t 4r,t/ 0 2 Ar,t t 2 Ar,t 4 Jr,t r r r r t t d dt Rt r Rt d 3 r dt r r r r t t r Rt d 3 r dt. The r integrations an be done first, using r Rt : Ar,t q d r Rt dt Rt r Rt t t dt ; r,t q r Rt r Rt t t dt Finay, the t integration for r,t an be done with the r Rt t t : r,t q t t o dt r Rt d r Rt t t dt t t o q 1, r Rt o d r Rt dt t t o where t o represents the vaues of t t for whih Simiary r Rt t t 0att t 0 r Rt r Rt 2 t t 2.

6 Ar,t q r Rt o d Rt dt t t o d r Rt dt t t o As a speia ase, if Rt v o t one obtains t o as foows: r v o t r v o t 2 t t 2 W 2 R o bt t R o bt t 2 t t 2 W ; b v o R o bw R o bw W 2, R o r v o t with soution W 2 2 R b R o 1 1 o b R o 2 2 1/2 t t o, where b. 2 Ony one vaue of W 0 satisfies ausaity. The soution for Rt v o t is (after some agebra) q r,t 1 r v o t 1 b 2 sin 2 where b R o os; br o q Ar,t v o r v o t 1 b 2 sin 2. In order to auate the fieds in the genera ase for Rt one needs a reationship between the derivatives with respet to r and t and the derivative with respet to t.webegin with the equaity t t r Rt. d dt r Rt r Rt ut, r Rt where ut dr. dt

7 Reativisti four vetor notation: Covariant form for Maxwe s Equations Under a Lorentz transformation, L, x L x with the foowing hoie of metri tensor g L T L for, 1,2,3 0 0 otherwise. x t,r 0,1,2,3 x t, r 1 t, 1 t, t 2 2 The vetor potentia is a four vetor (transforms under Lorentz transformation as a four vetor), A r,t,a A r,t, A; A A r,t 1,A t r,t,a A A r,t 1, A t Eq. 31 (the Lorentz gauge ondition) beomes A A 1 t r,t A 0 Note that the genera ondition on whih ensures the Lorentz gauge is 2 r,t t r,t Ar,t 1 2 t r,t, Finay, A 4 J where Conservation of harge is given by J,J R5 R6a R6b

8 J t J 0 R7