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Classical Electrodynamics for the First Few Values of I the Explicit Forms Are

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Classical Electrodynamics for the First Few Values of I the Explicit Forms Are 1 6 Multipole Fields In Chapters 3 and 4 on electrostatics the spherical harmonic expansion of the scalar potential was used extensively for problems possessing some symmetry property with respect to an origin of coordinates . Not only was it useful in handling boundary-value problems in spherical coordinates, but with a source present it provided a systematic way of expanding the potential in terms of multipole moments of the charge density. For time-varying electromagnetic fields the scalar spherical harmonic expansion can be generalized to an expansion in vector spherical waves. These vector spherical waves are convenient for electromagnetic boundary-value problems possessing spherical symmetry properties and for the discussion of multipole radiation from a localized source distribution . In Chapter 9 we have already considered the simplest radiating multipole systems . In the present chapter we present a systematic development . 16.1 Basic Spherical Wave Solutions of the Scalar Wave Equatio n As a prelude to the vector spherical wave problem, we consider the scalar wave equation. A scalar field V(x, t' satisfying the source-free wave equation, 2 (16. 1 ) °2 V - C 2 at2 = o can be Fourier-analyzed in time a s V(X' t) = . V(X, c ))e-iwt do) (1 6.2) E with each Fourier component satisfying the Helmholtz wave equation , (V2 + kl)V(X, co) = 0 (16.3) 538 [Sect. 16.1] Multipole Fields 539 with V = C021c2. For problems possessing symmetry properties about some origin it is convenient to have fundamental solutions appropriate to spherical coordinates. The representation of the Laplacian operator in spherical coordinates is given in equation (3 .1) . The separation of the angular and radial variables follows the well-known expansion , V(X, a)) = I .fi(r) Yl,,JB , (16.4) I'm where the spherical harmonics Ylm are defined by (3 .53). The radial functions fi(r) satisfy the radial equation , 2 A (16. 5) dr2 r dr r2 1 With the substitution, fa(r) = rq ul(r) (16.6) equation (1 6. 5) is transformed into (1 + d ~ + 1 d + k2 ~)2 C _ ]ut(r)=O (16.7) dr r dr r2 This equation is just Bessel's equation (3.75) with v = l + 2. Thus the solutions for fi(r) are AM "' r~ Ja+q(kr), ~ N,+q(kr) (16.8) It is customary to define spherical Bessel and Hankelfunctions, denoted by jt(x), nz(x), hi1,2)(x), as follows : .la x} a x) 2rrx ),J !,~ ( na(x) - Na+ ij(x) ( 16 ,9) (27r)" h G x For real x, h~2)(x) is the complex conjugate of hl"(x). From the series expansions (3 .82) and (3 .83) one can show that ~z ~sl x xl Y x) = {--x}l ~x x _d x (16.10) cos X n,(x) -(-x)' x CIxI J 540 Classical Electrodynamics For the first few values of I the explicit forms are ix = sin x ~ nQ~x} - ~ cos x ~ hl) x~ e Jo~x) ( = X X Ix sin x cos x cos x sin x = - - 11 ) = - 2 - Ji(x) ni(x X2 x2 x X BoxBo l hll'(x) _ - ~1--+- x x 3 1 3 cos x ( ) 3 1 sin x j2(X) 3 -~ since - 2 n2 x = -~ --- cosx-3 ~x x, x X3 x x2 ix + 3 i _ h2i)~x~ = €e 3 X2 X . X (16.11) From the asymptotic forms (3 .89)-(3 .91) it is evident that the small argument limits are X < 1 xi Yx) (2 1 + 1) ! r (16.12) (21 - ) nz(~' a ~ xa+ i where (21 + 1)! ! = (21 + 1)(21 - 1)(21 - 3) • • • (5) • (3) • (1). Similarly the large argument limits are x >~> l 1 ~ l r 1 a(x) --~- - sin x - - x 21 ) T ni(x) 1 cos x - d' ~ (16.13) x 2 ix x The spherical Bessel functions satisfy the recursion formulas , 2 1 + 1 ~ za(x) = z1 - ~GO + za+ i(x) x ( 16. 14) 21 + I. j [Sect. 16.1 1 Multipole Fields 541 where zl(x) is any one of the functions jl(x), ni(x), h(z)(x), h ,9( )(x) . The Wronskians of the various pairs ar e W{1,, nr) = i W{ji, hii~} _ -W(ni, hzi)) = 2 (16.15) X The general solution of (16.3) in spherical coordinates can be writte n ) TI'(x) [Aimh (')(kr) + Ai,2,~)hi2 ( kr)] ~a~ .z(8, (16.16) 1 . m ► where the coefficients AI") and Aims will be determined by the boundary conditions . For reference purposes we present the spherical wave expansion for the outgoing wave Green's function G(x, x'), which is appropriate to the equation, (V2+ k2)G(x, x) = -6(x - x') (16.17) in the infinite domain . The closed form for this Green's function, as was shown in Chapter 9, is e2kl z - Y' I G(x, x') = ( 1 b.18) 47r I x - x' The spherical wave expansion for G(x, x') can be obtained in exact ly the same way as was done in Sections 3 .8 and 3 .10 for Poisson's equation [see especially equation (3 .117) and below, and (3 .138) and below. An expansion of the form, ( i6.i9) l . m substituted into (16 .17) leads to an equation for g,(r, r') : + + k2 _ W 1)1 2 gz = --• (16.20) dr r r2 1dr' r The solution which satisfies the boundary conditions of finiteness at the origin and outgoing waves at infinity i s gl(r, r') = AJa(kr<)hz"(kr,) (16.21 ) The correct discontinuity in slope is assured if A = ik, Thus the expansion of the Green's function is ikl a -g' °r' Z e ik Ja( kr <)h~~~(kr>) ~ Yirn(O~, 0`)Yim(B, 0) (16.22) 4~r ix - x 'j 1 = 0 m=-l 542 Classical Electrodynamics Our emphasis so far has been on the radial functions appropriate to the scalar wave equation . We now re-examine the angular functions in order to introduce some concepts of use in considering the vector wave equation . The basic angular functions are the spherical harmonics YIJB, 0) (3 .53), which are solutions of the equation , 2 ] ( I a (sin B 9 + sin1 2 Yim. = l l + )~'i~,a. ( 16.23) sin 0 c30 a is well known in quantum mechanics,0 As thi s equation can be written in the form : LI- Yj m = l(t "'F 1) Yim (16.24) The differential operator L2 = L.,2 + L,,2 + Lz2, where L (r x V) (16.25) is the orbital angular-momentum operator of wave mechanics. The components of L can be written conveniently in the combinations, 2'k L + =Lx+iLy --e ~aae + icot a~B ~ X - 45 (16.26 L_ ` L 1L?l e-i l _ a + i cot e ~ ) a o I Lz= -i ~ We note that L operates only on angular variables and is independent of r . From definition (16 .25) it is evident that r•L=0 (16 .27) holds as an operator equation . From the explicit forms (16 .26) it is easy to verify that L2 is equal to the operator on the left side of (16.23). From the explicit forms (16 .26) and recursion relations for Yzm the following useful relations can be established : L+Yzm = 1/ (1- m)(1 + m + 1) Yi .m+1 ( L_ Y"~m = 1/(l + m)(l - m + 1) Yl, m 16.28) Lz Yl m = yY! Yl m [Sect. 16 .2] Multipole Fields 543 Finally we note the following operator equations concerning the com- mutation properties of L, L2, and V2 : L2L = LL2 L x L =iL (16.29) L;v2 = v2L j where 2 (16.30) ° r are r2 16.2 Multipole Expansion of the Electromagnetic Field s In a source-free region Maxwell's equations ar e V xE = V xB - C at =cat (16 .31) D•E=0, Q•B =0 With the assumption of a time dependence, e", these equations become D x E= ikB, D x B = -- ikE (16 .32) V •E=0, V • B = 0 If E is eliminated between the two curl equations, we obtain the following equations, (V2 ) 0, 0 + k2 B = 0 B = and the defining relation, (16.33) E= l V x B k Alternatively B can be eliminated to yield plus (16.34) B =-~V xE k Either (16.33) or (16 .34) is a set of three equations which is equivalent to Maxwell's equations (16 .32) . We now wish to determine multipole solutions for E and B . From (16 .33) it is evident that each rectangular component of B satisfies the Helmholtz wave equation (16.3). Hence each component of B can be 544 Classical Electrodynamics represented by the general solution ( 1 6. 1 6) . These can be combined to yield the vectorial result : R EAmhi')(kr) + Ai~hl2)(kr)] YIm.EB, (16.35) a,m where Arm are arbitrary constant vectors. The coefficients Alm in (16.35) are not completely arbitrary . The divergence condition V • B = 0 must be satisfied. Since the radial functions are linearly independent, the condition V • B = 0 must hold for the two sets of terms in (16.35) separately. Thus we require the coefficients Az,,,z to be so chosen that (16.36) d, m The gradient operator can be written in the form : r a i D= --- „ r x L (16.37) r ar r" where L is the operator (16.25) . When this is applied in (16.36), we obtain the requirement, lyl~ :E [~h-' AzmYtm - L X A lm ylln 0 ( 1 6 .
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