Vector Spherical Harmonics and Spherical Waves

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1 DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH5020 Eectomagnetic Theoy Mach 2017 by Suesh Govinaajan, Depatment of Physics, IIT Maas Vecto Spheica Hamonics an Spheica Waves Let us sove the time-epenent Maxwe s equations ME when the time-epenence is of the fom e iωt. Abitay time-epenence can be ecovee by supeposing iffeent fequencies. We wite Ex, t = Re Êx, ω e iωt, Bx, t = Re Bx, ω e iωt. Futhe, we assume that Êx, ω an Bx, ω ae compex vaue vecto fies. The soucefee ME in this basis takes the fom Êx, ω = 0, 2 + k 2 Êx, ω = 0, 1 whee k = ω/c. The secon equation is the vecto Hemhotz equation. The magnetic fie is then etemine by the equation Bx, ω = Êx, ω iω. 2 Hee ou stategy is to fist etemine the eectic fie an then obtain the magnetic fie using the above equation. We cou equay we have fist etemine the magnetic fie an obtaine the eectic fie fom it. We eave it as an execise fo the inteeste stuent. Vecto spheica hamonics Reca that the we use the basis of spheica hamonics to convet the soution to Lapace s equation to an oinay iffeentia equation fo the aia pat of the potentia. We wote Φx = a Y θ, ϕ. Can we use a simia stategy to sove fo the eectic fie? The answe is an affimative one an eas us to a vectoia vesion of spheica hamonics which we iscuss beow. 1

2 Consie the foowing basis 1 : Y θ, ϕ = ê Y θ, ϕ, Ψ θ, ϕ = Y θ, ϕ, Φ θ, ϕ = ê Ψ x Remak: Fo = 0, ony Y 0,0 θ, ϕ = ê is non-vanishing. Execise: Veify that Y θ, ϕ Ψ θ, ϕ = 0, Y x Φ θ, ϕ = 0, Ψ θ, ϕ Φ θ, ϕ. Using the ientity show that 2 2 Y θ, ϕ = + 1 Y θ, ϕ, Y θ, ϕ Y,m θ, ϕ Ω = δ δ mm Ψ θ, ϕ Ψ,m θ, ϕ Ω = + 1 δ δ mm Φ θ, ϕ Φ,m θ, ϕ Ω = + 1 δ δ mm These estabish the othogonaity of the vecto spheica hamonics. We sha assume thei competeness without poviing a poof. An abitay vecto fie can be expane in a basis of vecto spheica hamonics. We expan the eectic fie as foows: Êx, ω = =0 m= E Y θ, ϕ + E 1 Φ θ, ϕ + E 2 Ψ θ, ϕ. Execise Show that [ + 2 Ê = Ê = ] E +1 E 1 Y θ, ϕ +1 E 2 Y + [ + ] 1 2 E Φ + 1 E + [ + ] 1 1 E Ψ 1 We foow the notation use in Wikipeia hamonics hee. This is the notation use by Baea et a., Vecto spheica hamonics an thei appication to magnetostatics, Eu. J. Phys

3 Spheica waves We fist consie the case of = 0 this is impotant as the foowing theoem impies that thee ae no spheicay symmetic waves. Theoem 1 Bikhoff. The ony spheicay symmetic soutions of the time-epenent Maxwe s equations ae static. Poof. We wi assume the existence of a spheicay symmetic time-epenent soution. The eectic fie wi be of the fom Ex, ω = E ê. The two equations in Eq. 1 impy + 2 E = k2 E = 0 The fist equation is sove by E 1/ 2 which oes not sove the secon equation uness k = 0. Thus, thee is no soution when k 0. k = 0 impies ω = 0 which is the static soution. Now we consie the > 0 situation whee we fin spheica waves. Imposing the conition Ê = 0 equies fo > 0 imposes the foowing conition that expesses E1 in tems of E. + 1E 1 = [ + 2 ] E, 3 whie E 2 is unconstaine. Thus, thee ae two istinct soutions to Ê = 0 an these coespon to the two possibe poaisations fo the spheica wave. Soution 1 Let us fist wite out the soution with E 2 0 fo a paticua vaue of > 0, m an zeo othewise. Futhe, E = E1 = 0. Thus, the eectic fie has the fom Using the foowing ientity E = 2 E = Ex, ω = E 2 Ψ θ, ϕ E 2 2 Ψ, 3

4 the secon equation in Eq. 1 becomes the scaa Hemhotz equation which is an oinay iffeentia equation fo E 2 with > 0: k 2 E = 0 4 The two soutions to the above iffeentia equation ae given by spheica Hanke functions, h i k fo i = 1, 2. The fist soution coespons to an outgoing wave whie the secon soution coespons to an incoming wave. Execise: Detemine the magnetic fie coesponing to the above soution. Soution 2 It is simpe to specify the magnetic fie fo the secon soution. Let Bx, ω = B 2 Ψ θ, ϕ. Then, B 2 has to satisfy the scaa Hemhotz equation k 2 B = 0, 5 which is again expessibe in tems of spheica Hanke functions. Execise: Detemine the eectic fie coespon to the above soution by obtaining a fomua anaogous to Eq. 2. Veify that it has E 0 an that E1 0 being etemine by Eq. 3 Appenix on spheica Besse an Hanke functions The Digita Libay of Mathematica Functions DLMF is a wonefu onine esouce fo specia functions. Spheica Besse Functions ae iscusse at the URL: nist.gov/ Ou iscussion is base on this esouce. We ae inteeste in soutions to the equation f = 0. 6 Fo = 0, the two ineay inepenent soutions ae given in tems of the spheica Besse functions. j 0 = sin, y 0 = cos. Fo > 0, the soutions ae given by Raeigh s fomuae: j = 1 j 0, y = 1 4 j 0.

5 One efines the spheica Hanke functions as inea combinations of the two soutions. h 1 := j + iy. h 2 := j iy = h 1 We thus see that an hence h 1 0 = i ei, h 1 = 1 h 1 0, = ei poynomia of egee in 1. Execise: Expicity compute the spheica Hanke functions fo = 1, 2 an show that h 1 = ei 1 i sin = cos + i cos sin, 2 2 h 2 = iei 1 + 3i sin = 3 cos sin + i 3 cos 3 sin + cos