Redalyc.Complete Electromagnetic Multipole Expansion Including

Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México

Góngora T., A.; Ley-Koo, E. Complete electromagnetic multipole expansion including toroidal moments Revista Mexicana de Física, vol. 52, núm. 2, diciembre, 2006, pp. 188-197 Sociedad Mexicana de Física A.C. Distrito Federal, México

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Complete electromagnetic multipole expansion including toroidal moments

A. Gongora´ T. Centre for Interdisciplinary Research on Complex Systems, Physics Department, Northeastern University Boston M.A. E. Ley-Koo Instituto de F´ısica, Universidad Nacional Autonoma´ de Mexico,´ Apartado Postal 20-364, 01000 Mexico´ D.F., Mexico.´ Recibido el 3 de marzo de 2006; aceptado el 22 de mayo de 2006

The electromagnetic multipole expansion presented in this paper is complete on two accounts: i) It is valid for all points in space, and ii) it recognizes the existence of toroidal moments. The electromagnetic ﬁeld due to alternating poloidal currents in a toroidal solenoid is evaluated exactly via the solution of the inhomogeneous vector Helmholtz equations, using the outgoing wave Green function technique and the Debye potentials for sources and ﬁelds. The physical meaning of the toroidal moments can be appreciated when they are compared with the familiar electric and magnetic moments; the analysis of the long-wavelength limit of the exact results also explains the previous neglect of the toroidal moments. The magnetostatic limit and the point source limit are also physically and didactically interesting.

Keywords: Electromagnetic radiation; multipole expansion; electric, magnetic and toroidal moments.

El desarrollo multipolar electromagnetico´ que se presenta en este art´ıculo es completo por dos razones: i) es valido´ para todos los puntos del espacio, y ii) reconoce la existencia de momentos toroidales. El campo electromagnetico´ producido por corrientes poloidales alternas en un solenoide toroidal se evalua´ exactamente a traves´ de la solucion´ de ecuaciones de Helmholtz inhomogeneas,´ usando la tecnica´ de la funcion´ de Green de onda saliente y los potenciales de Debye para fuentes y campos. El signiﬁcado f´ısico de los momentos toroidales se destaca al compararlos con los momentos electricos´ y magneticos´ familiares; el analisis´ del l´ımite de longitud de onda grande de los resultados exactos tambien´ explica la ignorancia previa de los momentos toroidales. El l´ımite magnetostatico´ y el l´ımite de fuente puntual son tambien´ interesantes f´ısicamente y didacticamente.´ Descriptores: Radiacion´ electromagnetica;´ desarrollo multipolar; momentos electricos,´ magneticos´ y toroidales.

PACS: 41.20 Jb

1. Introduction the familiar electric and magnetic moments. Section 2 pro- vides the theoretical framework for the study, including as the The standard presentations of the multipole expansions of starting point Maxwell’s equations for harmonic time vary- electrostatic, magnetostatic and electromagnetic fields in the ing sources, and their transformation into inhomogeneous textbooks are usually incomplete since they are restricted to Helmholtz equations. The corresponding solutions are con- the regions outside localized sources [1-6]. Two articles may structed by using the outgoing wave Green function and its be cited from the didactic literature making up for this incom- spherical multipole expansion, as well as the Debye poten- pleteness; one of them deals with the complete vector spheri- tials in order to exhibit the longitudinal and transverse com- cal harmonic expansion for Maxwell’s equations [7], and the ponents of the respective vector sources and fields [14]. Sec- other gives the multipole expansions outside and inside the tion 3 is devoted to the construction of the magnetic field sources for the electrostatic and magnetostatic cases [8]. due to poloidal currents in a toroidal solenoid with a circu- Another gap in the textbooks is the absence of any men- lar ring sector cross section in each meridian plane, including tion about toroidal moments. It will soon be fifty years the multipole expansion, the Debye potentials, the dynamic since the violation of parity was established in the weak in- toroidal multipole moments, the long wavelength limit, the teractions, and the introduction of the anapole moment by magnetostatic limit, and the point source limit. Section 4 Zel’dovich in his note on “Electromagnetic interaction with contains a discussion about the main new results and some parity violation” [9]. The Russian authors have been ac- of their consequences. tive investigating the multipole expansion in classical and 2. Maxwell’s equations and Debye potentials quantum field theory, the electromagnetic fields of toroidal solenoids and correspondingly the existence and importance The charge density ‰, the current density J~, the electric in- of toroidal moments [10-13], in the ensuing period. It is time tensity field E~ and the magnetic induction field B~ are related that the study of such topics should be incorporated into the by Maxwell’s equations [1-6]: advanced courses in electrodynamics. ∇ · E~ = 4…‰ (1) This paper presents a complete electromagnetic multipole expansion valid for all points in space, with emphasis i! ∇ × E~ = B~ (2) on the presence of toroidal moments on the same footing as c COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 189

∇ · B~ = 0 (3) 4… (∇2 + k2)B~ (~r) = − ∇ × J~(~r): (6) 4… i! c ∇ × B~ = J~ − E;~ (4) c c Equation (6) shows that the magnetic induction field is corresponding to Gauss’ law, Faraday’s law, nonexistence of determined by the transverse component of the current den- magnetic monopoles, and Ampere-Maxwell’s´ law, for har- sity. In contrast, Eq. (5) shows that the electric intensity field −iωt monic time variations e with frequency ! for the sources is determined by the gradient of the charge density and both and fields. Here !=c = k is identified as the wave number. longitudinal and transverse components of the current den- Equations (1-4) are a set of linear coupled equations for sity. the fields. They can be decoupled, by taking the curl of The solutions to Eqs. (5), (6) can be constructed with the Eqs. (2) and (4) and using the remaining Eqs. (1) and (3), help of the Green function of the Helmholtz equation to obtain the respective Helmholtz inhomogeneous equations for the electric and magnetic field: 2 2 0 0 (∇ + k )G+(~r; ~r ) = −4…–(~r − ~r ): (7) 4…i! (∇2 + k2)E~ (~r) = 4…∇‰ − J~(~r) (5) c2 The outgoing wave Green function and its multipole expansion are given by

0 eik|~r−~r | X∞ Xl G (~r; ~r 0) = = 4…ik j (kr )h(1)(kr ) (−)mN 2 P m(cos 0)P m(cos )(2 − – ) cos m(’ − ’0): (8) + |~r − ~r 0| l < l > lm l l m0 l=0 m=0

The particular solutions to Eqs. (5) and (6) involve the integrations of the respective sources and the Green function the source densities, Z 0 0 0 0 ~ E~ (~r) = − dv ∇ ‰(~r )G+(~r; ~r ) ∇ · J(~r) − i!‰(~r) = 0; (12) Z i! involves only the longitudinal component of the current den- + dv0J~(~r 0)G (~r; ~r 0) (9) c2 + sity; in terms of the corresponding Debye potential, it be- Z comes 0 0 0 0 B~ (~r) = dv ∇ × J~(~r )G+(~r; ~r ): (10) ∇2iL = i!‰; (13) Use of the multipole expansion of the Green function of showing that this potential and the charge density are related Eq. (8) in Eqs. (9) and (10) leads to the corresponding gen- through Poisson’s equation. On the other hand, the curl of eral and exact expansion of the electromagnetic field. The the current density, particular application to the toroidal solenoid is made in the following section. ∇ × J~(~r) = −i∇ × ~liT (~r) − i∇ × [(∇ × ~l)iP (~r)] The Debye potentials, as has been pointed out by Gray [14], are useful for exhibiting the decomposition of the = −i∇ × ~liT (~r) − i~l[−∇2iP (~r)] (14) source and force fields into their longitudinal and transverse (toroidal and poloidal) components, including the relation- shows that its toroidal and poloidal Debye potentials are ships among them. The gradient of the charge density is a −∇2iP (~r) and iT (~r), respectively. The second line of longitudinal field, since its curl is identically zero. The cur- Eq. (14) is obtained by performing the triple vector prod- rent density field may be written as uct and using the orthogonality of the divergence and angu- ½ ¾ lar momentum operators, as well as the commutability of the J~(~r) = ∇iL(~r) + ∇ × [~riT (~r)] + ∇ × ∇ × [~r × iP (~r)] Laplace and angular momentum operators. The use of Eqs. (11) and (14) together with the symmetry = ∇iL(~r) − i~liT (~r) − i∇ × ~liP (~r); (11) properties of the Green function and the hermiticity properties of the gradient and angular momentum operators allow where the second line makes use of the angular momentum us to rewrite Eqs. (9) and (10) to exhibit the longitudinal and operator ~l = −i~r × ∇. The continuity equation satisfied by transverse components of the respective force fields,

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197 190 A. GONGORA´ T. AND E. LEY-KOO

Z Z i! i! E~ (~r) = −∇ dv0[‰(~r 0) − iL(~r 0)]G (~r; ~r 0) − i~l dv0[− iT (~r 0)]G (~r; ~r 0) c2 + c2 + Z i! − i∇ × ~l dv0[− iP (~r 0)]G (~r; ~r 0) (15) c2 + Z Z ~ 0 02 P 0 0 ~ 0 T 0 0 B~ (~r) = −il dv [−∇ i (~r )]G+(~r; ~r ) − i∇ × l dv i (~r )]G+(~r; ~r ): (16)

The corresponding Debye potentials can be immediately read off from these equations in analogy with Eq. (11): Z i! ing poloidal currents in the corresponding solenoids. The curl eL(~r) = dv0[‰(~r 0) − iL(~r 0)]G (~r; ~r 0) (17) of the current density is evaluated and used in Eq. (10) to- c2 + Z gether with the multipole expansion of the Green function, i! Eq. (8), in order to obtain the multipole expansion of the eT (~r) = dv0[− iT (~r 0)]G (~r; ~r 0) (18) c2 + magnetic induction field. The electric intensity field can be Z evaluated by the integration of Eq. (9), but here it is prefer- P 0 i! P 0 0 e (~r) = dv [− i (~r )]G+(~r; ~r ) (19) able to obtain it through Eq. (4). The toroidal character of c2 the magnetic field and the poloidal character of the electric bL(~r) = 0 (20) field are explicitly exhibited, obtaining the multipole expan- Z sions of the respective Debye potentials along the way. The T 0 02 P 0 0 b (~r) = dv [−∇ i (~r )]G+(~r; ~r ) (21) analysis focuses on the completeness of the multipole ex- Z pansion [14], including the so-called toroidal moments [5]. P 0 T 0 0 Then the limiting situations of long wavelengths, including b (~r) = dv i (~r )]G+(~r; ~r ): (22) the magnetostatic case and the point source case, are studied Equations (18) and (22) show that toroidal currents pro- in particular. duce toroidal electric and poloidal magnetic fields, while Eqs. (19) and (21) show that poloidal currents produce poloidal electric and toroidal magnetic fields. Also Eq. (17) shows that the charge density and the longitudinal currents 3.1. The multipole expansion are the sources of the longitudinal electric field. The toroid with a circular ring sector cross section is defined 3. Electromagnetic field of toroidal solenoids by its inner spherical ring (r = a; 2 < < 1;’), its up- per conical ring (a < r < b; # = 1;’), its outer spherical This section starts by defining the toroids with a circular ring ring (r = b; #1 < < #2;’), and its lower conical ring −iωt sector cross section in each meridian plane, and the alternat- (b > r > a; = #2;’). The alternating poloidal Ie current in the toroidal solenoid with N turns has the density

½ · ¸· ¸ NIe−iωt rˆ J~(~r; t) = –( − ) − –( − ) Θ(r − a) − Θ(r − b) 2…r sin r 1 2 · ¸· ¸¾ ˆ + –(r − b) − –(r − a) Θ( − 1) − Θ( − 2) ; (23) where the Dirac delta functions define the coil elements along which the current flows and the Heaviside step functions define the extent of such elements. The curl of the current density, ½ · ¸ NI d Θ( − ) − Θ( − ) ∇ × J~(~r) = ’ˆ –(r − b) − –(r − a) 1 2 2…r dr sin · ¸· ¸¾ 1 d –( − ) − –( − ) − 1 2 Θ( − ) − Θ( − ) (24) r2 d sin 1 1 is toroidal, i.e., in the azimuthal direction, and invariant under rotations around the axis of the toroid.

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 191

As already anticipated, the complete multipole expansion of the magnetic ﬁeld follows from the combinations of Eqs. (10), (24) and (8), Z∞Zπ Z2π ½ · ¸ NI ’ˆ0 d Θ(0 − ) − Θ(0 − ) B~ (~r) = r02dr0 sin 0d#0d’0 –(r0 − b) − –(r0 − a) 1 2 2…c r0 dr0 sin 0 0 0 0 · ¸· ¸¾ 1 d –(0 − ) − –(0 − ) − 1 2 Θ(r0 − a) − Θ(r0 − b) r02 d0 sin 0 X∞ Xl (1) m 2 m 0 m 0 4…ik jl(kr<)hl (kr>) (−) NlmPl (cos )Pl (cos )(2 − –m0) cos m(’ − ’) (25) l=0 m=0

The azimuthal angle integration can be done by using ’ˆ0 =’ ˆ cos(’0 − ’) − Rˆ sin(’0 − ’) and the orthogonal- terms with m=1 in the sum of Eq. (25). The first integral, ity of the cosine and sine functions, Zθ2 Z2π 0 1 0 d P (cos ) = Pl(cos 1) − Pl(cos 2); (27) 0 0 0 l d’ ’ˆ cos m(’ − ’) =’…– ˆ m1: (26) θ1 0 follows immediately from the relation between the associated It follows that the magnetic induction field is also az- and ordinary Legendre polynomials, imuthal and invariant under axial rotations, i.e., a toroidal d field. P 1(cos #) = sin P (cos ): (28) l d(cos ) l The polar angle integrations are also straightforward, and according to the selection rule of Eq. (26) are limited to the The second integral, involving the Dirac delta functions, can be done by parts:

Zπ · ¸ · ¸¯ · ¸¯ ¯ ¯ 0 d 0 1 0 0 1 0 1 d 0 1 0 ¯ 1 d 0 1 0 ¯ d 0 sin Pl (cos ) sin Pl (cos # ) = − 0 0 sin Pl (cos ) ¯ + 0 0 sin Pl (cos ) ¯ d sin d θ0=θ sin d θ0=θ 0 1 2 · ¸

= −l(l + 1) Pl(cos 1) − Pl(cos 2) : (29)

The last line is obtained by again using Eq. (28) and also the differential equation for the ordinary Legendre polynomi- The common factor in the polar angle integrals, Eqs. (27) als: and (29), should be noted. 1 d d The integrations over the radial coordinate are also direct, sin # P (cos ) = −l(l + 1)P (cos ): but the distinction among the different locations of the ﬁeld sin d d l l point must be made. The ﬁrst integral over the Dirac delta functions is done by parts:

Z∞ · ¸ d r0dr0 –(r0 − b) − –(r0 − a) j (kr )h(1)(kr ) dr0 l < l > 0  ½ · ¸¯ ¾  ¯b  − d r0h(1)(kr0) ¯ j (kr) ; r > a  dr0 l ¯ l  · ¸¯ a · ¸¯  ¯ ¯ = − d r0h(1)(kr0) ¯ j (kr) + d r0j (kr0) ¯ h(1)(kr) ; r < a < b (30)  dr0 l ¯ l dr0 l ¯ l  ½ · ¸¯r0=¾b r0=a  ¯b  d 0 0 ¯ (1)  − dr0 r jl(kr ) ¯ hl (kr) ; b < r: a

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197 192 A. GONGORA´ T. AND E. LEY-KOO

The second integral can be explicitly written as

 b  R 0  dr h(1)(kr0)j (kr) ; r < a  r0 l l Zb  a  Rr Rb d (1) dr0 0 (1) dr (1) 0 0 jl(kr<)hl (kr>) = r0 jl(kr )hl (kr) + r0 hl (kr )jl(kr) ; a < r < b (31) dr  a r a  b  R 0  dr 0 (1)  r0 jl(kr )hl (kr) ; b < r: a

The substitution of the integrals of Eqs. (26)-(31) in Eq. (25) gives

· ¸ NI X∞ B~ (~r) = 4…ik’ˆ (−)N 2 P 1(cos ) P (cos ) − P (cos ) c l1 l l 1 l 2 l=1  ½ · ¸¯ ¾  ¯b R  − d r0h(1)(kr0) ¯ + l(l + 1) b d h(1)(kr0) j (kr) ; r < a  dr0 l ¯ a dr0 l l  ½ · ¸¯a · ¸¯  ¯ ¯  d 0 (1) 0 ¯ d 0 0 ¯ (1) 0  − dr0 r hl (kr ) ¯ jl(kr) + dr0 r jl(kr ) ¯ hl (kr ) × · r0=b r0=a ¸¾ (32)  R r 0 (1) R b 0 (1)  +l(l + 1) dr j (kr0)h (kr) + dr h (kr0)j (kr) ; a < r < b  a r0 l l a r0 l l  ½ · ¸ ¸¯ ¾  ¯b R  d 0 0 (1) 0 ¯ b d 0 (1)  − dr0 r jl(kr ) hl (kr ) ¯ + l(l + 1) a dr0 jl(kr ) hl (kr) ; b < r: a

The radial factors in Eq. (32) can be simpliﬁed by integrating the differential equation for the spherical Bessel functions. The result is

· ¸ NI X∞ B~ (~r) = 4…ik’ˆ (−)N 2 P 1(cos ) P (cos ) − P (cos ) c l1 l l 1 l 2 l=1  R 2 b 0 0 (1) 0  k½ a·dr r hl ¸(¯kr )jl(kr) ; r < a · ¸¯  ¯ ¯  d 0 ¯ (1) d 0 (1) 0 ¯  dr0 r jl(kr) ¯ hl (kr) − dr0 r hl (kr ) ¯ jl(kr) × r0=r r0=r ¾ (33)  R r (1) R b (1)  +k2 dr0r0j (kr0)h (kr) + k2 dr0r0h (kr0)j (kr) ; a < r < b  a l l r l l  R  2 b 0 0 0 (1) k a dr r jl(kr )hl (kr) ; b < r:

It is appropriate at this point to recognize that the standard multipole expansion of the electromagnetic field [5, 6, 14] is are located, a < r < b, involve non-linear combinations of usually limited to the region outside the sources, correspond- the spherical Bessel functions and their derivatives or inte- ing to b < r in Eqs. (30-33). Following Lambert [7], here grals. we have also obtained the field in the inner region r < a, where there are no sources, and in the intermediate region 3.2. The Debye potentials a < r < b, where the sources are located. All of this has been done within one and the same calculation by simply dis- Equation (33) may be written in an explicitly toroidal form tinguishing among the different locations of the field point. It by using the representation of the product of the azimuthal is also pertinent to point out that, as is to be expected, the unit vector and the associated Legendre polynomials, solutions in the source free regions, r < a and b < r, are r superpositions of solutions of the homogeneous Helmholtz 4… ’Pˆ 1(cos ) = (−i~l)Y (; ’); (34) equations, while the solutions in the region where the sources l 2l + 1 l0 so that

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 193

r · ¸ X∞ NI 4… B~ (r; ; ’) = −i~l 4…ik(−) N 2 P 1(cos ) − P (cos ) Y (; ’) c 2l + 1 l1 l l 2 l0 l=1  R 2 b 0 0 (1) 0  k½· a dr r hl (kr )jl(kr) ; r < a ¸  R  d 2 r 0 0 0 (1)  dr (rjl(kr)) + k a dr r jl(kr ) hl (kr) × · ¸ ¾ (35)  (1) R b (1)  + − d (rh (kr)) + k2 dr0r0h (kr0) j (kr) ; a < r < b  dr l r l l  R  2 b 0 0 0 (1) k a dr r jl(kr )hl (kr) ; b < r:

Comparison with Eqs. (16) and (21) gives the identification of the summation in Eq. (35) as the multipole expan- Here the explicit value of the normalization constant Nl1 sion of the Debye potential bT . Notice that the Laplacian in has been incorporated. Eq. (21), because of its hermiticity, may be made to operate For purposes of comparison, we transcribe next the cor- on the Green function, giving, according to Helmholtz equa- responding general equations for the multipole expansions of tion, Eq. (7), −k2 times the Green function plus the point the electromagnetic field [6, 14], source density term; such a relationship is apparent in the ra- X X · ~ E ~ (1) dial factors of Eq. (35). B(~r) = ˆlmlhl (kr)Ylm(; ’) As already mentioned at the beginning of this section, l m ¸ the electric intensity field may be obtained by using Eq. (4), (−i) + ˆM ∇ × ~lh(1)(kr)Y (; ’) (39) when the magnetic induction, Eq. (35), and the current den- lm k l lm sity, Eq. (23), are known. The electric intensity is obviously · µ ¶ X X i poloidal as a result of the application of the curl to the toroidal E~ (~r) = Ê ∇ × lh(1)(kr)Y (; ’) lm k l lm magnetic induction and the poloidal character of the current l m density itself. The current density may be written in its mul- ¸ tipole expansion form by using the corresponding represen- M ~ (1) + ˆlmlhl (kr)Ylm(; ’) ; (40) tations of the Dirac delta and the Heaviside step functions in the polar angle, respectively, in Eq. (23). For the sake of where the dynamic multipole moments, in the terminology space, the complete expressions E~ for and J~ are not written of [14], are given by out explicitly. Z 4…ik2 ˆM = dv0j (kr0)Y ∗ (0;’0)~r · ∇0×J~(~r 0) (41) lm cl(l+1) l lm 3.3. The dynamic toroidal multipole moments Z 4…ik2 Ê = − dv0j (kr0)Y ∗ (0;’0) At this point it is more instructive to go on to write the mag- lm cl(l + 1) l lm netic and electric fields in the region outside the external · ¸ sphere, b < r, in the form that allows the characterization × ik~r 0 · J~(~r 0) − c(2 + ~r 0 · ∇0)‰(~r 0) (42) of the multipole expansions of the electromagnetic field of toroidal solenoids, Notice that Eq. (5b) of Ref. 14, equivalent to our Eq. (42), is missing a factor of k on its rhs, as can be verified by com- ∞ X parison with Eq. (16-91) in Ref. 6. B~ (b < r; ; ’) = ˆ ~lh(1)(kr)Y (; ’) (36) l0 l l0 The comparison of Eqs. (36-38) and (39-42) is direct. l=1 The poloidal component of the magnetic induction field is X∞ i absent in Eq. (36) and the toroidal component of the elec- E~ (b < r; ; ’) = ˆ ∇ × ~lh(1)(kr)Y (; ’); (37) l0 k l l0 tric intensity field is absent in Eq. (37), because the dynamic l=1 magnetic multipole moments in Eq. (41) vanish at the source where level due to the poloidal character of the current, Eq. (23). r The vanishing of the integrand in Eq. (41) follows immedi- 4…NIk3 2l + 1 1 ately from the orthogonality of the radial vector and the curl ˆ = − l0 c 4… l(l + 1) of the current density, Eq.(24). On the other hand, the dynamic electric multipole moments, Eq. (42), are determined · ¸ b Z by the radial components of the current density and of the 0 0 0 × Pl(cos 1) − Pl(cos 2) dr r jl(kr ): (38) gradient of the charge density, as well as by the charge den- a sity itself. In the case under study, the charge density is absent, and the radial component of the current density is the

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197 194 A. GONGORA´ T. AND E. LEY-KOO

first term inside the curly brackets on the rhs of Eq. (23). The of the operators is taken into account. Equation (44) shows evaluation of the corresponding integral in Eq. (42) leads that the dynamic magnetic multipole moments depend only to the same value of Eq. (38). In conclusion, the multi- on the toroidal component of the current. On the other hand, pole expansions of Eqs. (36,37,38) correspond to the terms there is room for ambiguity in the case of the electric mo- in Eq. (42) involving the dynamic transverse electric multi- ments. In fact, the corresponding source factor taken from pole moments associated with the radial component of the Eq. (5), poloidal current density, i.e. Eq. (42) with ‰ = 0. · ¸ ik One might be tempted to add the superscript E in ~r 0 · − ∇0‰(~r 0) + J~(~r 0) Eqs. (36-38) to complete the characterization of the multi- c · ¸ pole expansion of the electromagnetic field of the toroidal ik ˆ = −~r 0 · ∇0 ‰(~r 0) − iL(~r 0) + i~l 2iP (~r 0) (45) solenoid. However falling into such a temptation is tan- c tamount to missing the existence of the toroidal moments. Following Ref. 10, the correct superscript to be added in involves both longitudinal and poloidal components of the Eqs. (36-38) is for toroidal, and Eq. (42) gives an exact rela- source. Taking into account that the basis functions for tionship among the dynamic multipole moments of electric, the multipole expansions are eigenfunctions of the ∇2 and ˆ toroidal and charge types. ~l 2 operators, it can be recognized that Eq.(41) is in di- Q rect correspondence with Eq. (22), while Eq.(42) is re- Ê (k) = ˆT (k) + ˆ (k): (43) lm lm lm lated to both Eqs. (17) and (21). The standard terminology The distinction among these types of dynamic multipole of Eqs. (39)-(42) of transverse electric and magnetic mo- moments and the connection among them can be traced back ments can be made more precise by making the distinction of to the different ways of separating the source terms in the in- Eq. (43), and recognizing that the magnetic moments arising homogeneous Helmholtz equations, Eqs. (5,6). In Eqs. (41), from toroidal currents could be appropriately called poloidal (42) it is recognized that the source factors are the radial com- moments, just as the moments arising from the poloidal cur- ponents of the respective sources in Eqs. (6) and (5). Let rents are called toroidal moments. us consider first the unambiguous case of the magnetic moments. The corresponding source factor in the integrand of 3.4. The long wavelength limit Eq. (41) may be written in terms of the decomposition of the current density of Eq. (11) in the alternative forms When the sources are confined in a region with a radial ex- · ¸ · ¸ tent that is small compared with the wavelength ‚ = 2…=k, ~r 0 · ∇0 × J~(~r 0) = ~r 0 × ∇0 · J~(~r 0) it is usual to approximate the spherical Bessel functions in Eqs. (34-42) by the dominant terms of their power series ex- · ¸¸ pansions close to the origin. The result of such approximation ˆ = i~l · ∇iL − i~liT − i∇ × ~liP = ~l 2iT ; (44) is that the dynamic multipole moments, which are defined in terms of spherical Bessel functions as weight functions in the where in the first line the dot and cross are exchanged in the integrals of Eqs. (41) and (42), are connected to the static triple scalar product, and in the second line the orthogonality multipole moments defined in terms of the powers of the radial coordinate as weight functions,

Z 4…ikl+2 4…ikl+2 ˆM (k → 0) = dv0r0lY ∗ (0;’0)~r 0 · ∇0 × J~(~r 0) = M (46) lm cl(l + 1)(2l + 1)!! lm l(2l + 1)!! lm Z · ¸ 4…ikl+2 ˆE (k → 0) = − dv0r0lY ∗ (0;’0) ik~r 0 · J~(~r 0) − c(2 + ~r · ∇0)‰(~r 0) lm cl(l + 1)(2l + 1)!! lm · Z Z ¸ 4…ikl+2 1 = − dv0r0lY ∗ (0;’0)ik~r 0 · J~(~r 0) + c dv0r0lY ∗ (0;’0)‰(~r 0) M : (47) cl(2l + 1)!! l + 1 lm lm lm

Equation (46) is the same as Eqs. (58) and (17) of Ref 14. The charge part of Eq. (47) is the same as Eqs. (59), (33) and (32) of Ref. 14, where the last integral is the familiar and Refs. 10 to 13. In the notation of Ref. 10, the dynamic electrostatic multipole moment Qlm [6, 8]. The part asso- multipole moments in Eqs. (41), (42) and (43) are normalized ciated with the radial component of the current density in to give the time dependent form factors Eq. (47) is usually dropped in the long wavelength approximation, but this is not justiﬁed as illustrated in this paper l(2l+1)!! M (−k2; t)= ˆM (k)e−iωt → M e−iωt (48) lm 4…ikl+2 lm lm

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 195

regions bounding the toroid. The magnetic induction correspondingly vanishes in both regions. For the region where 2 cl(2l + 1)!! E −iωt Elm(−k ; t) = l+1 ˆlm(k)e (49) the sources are located, the radial factor becomes 4…k · ¸ · µ ¶¸ l(2l+1)!! d l+1 1 d 1 l 2l + 1 Q (−k2; t)=− ˆQ (k)e−iωt → Q e−iωt (50) (r ) − r = : (55) lm 4…ikl+2 lm lm dr rl+1 dr rl r

2 l(2l + 1)!! T −iωt The coefficient in the numerator of Eq (55) will cancel Tlm(−k ; t) = ˆ (k)e : (51) 4…kl+3 lm the coefficient in the denominator of Eq. (54), and Eq. (33) In terms of these form factors, Eq.(43) becomes takes the form NI X∞ 2l + 1 E (−k2; t) = k2T (−k2; t) + Q˙ (−k2; t) (52) B~ (a < r < b;; ’) = 4…’ˆ (−) P 1(cos ) lm lm lm cr 4… l t=1 which is the central result of Ref. 10. · ¸ The standard systems of electromagnetic sources and × Pl(cos 1) − Pl(cos 2) fields [5–7, 14] are described in terms of the charge form · ¸ factors, Eq.(50), the magnetic form factors, Eqs.(48) and the 2NI = Θ( − ) − Θ( − ) (56) transverse electric form factors, Eq. (49). In the long wave- cr sin 1 2 length approximation the latter becomes In the first line, the explicit value of the normalization 2 2 Elm(−k ; t) = Q˙ lm(−k ; t) (53) constant was substituted. In the last line, the sum is identified with the difference of the Heaviside step functions in as follows from Eq. (52). the polar angle, which follow from the completeness of the However, in systems with poloidal currents like the orthonormal Legendre basis toroidal solenoids, Eq. (53) does not hold. Instead of the ∞ transverse electric form factors, it is more general to use the X 2l + 1 P (cos )P (cos ) = –(cos − cos ) toroidal form factors of Eq. (51), which are independent of 2 l l i i the magnetic and the charge form factors. l=0 –( − ) In Secs. 3.3 and 3.4, we have considered the electromag- = − i (57) netic fields in the region b < r in order to compare with the sin standard available results of Refs. 6 and 14. The treatment of and its integration Sec. 3.1 includes the fields for the remaining regions r < b, since we have the complete fields, as Lambert [7] pointed out. Zθ 0 0 In Sec. 3.5, we study the magnetic induction field in the static Θ( − i) = d –( − i) limit ! → 0; k → 0, or infinite wavelength limit. In Sec. 3.6, 0 the point source limit corresponding also to an infinitely long Zθ wavelength limit, but for a finite frequency, is analyzed. X∞ 2l + 1 = − d0 sin 0P (cos 0)P (cos ) 2 l l i 3.5. The magnetostatic limit l=0 0 ∞ The limit of stationary currents with frequency ! → 0 in the X 2l + 1 1 = − sin P 1(cos )P (cos ): (58) toroidal solenoids corresponds to the magnetostatic case. It 2 l(l + 1) l l i l=1 is clear that Eq. (4) becomes Ampere’s law, Eqs. (6) and (7) become Poisson’s equation, the Green function of Eq. (8) be- In conclusion, the magnetic induction vanishes outside comes the Coulomb potential, and the spherical Bessel func- the solenoid and is azimuthal and inversely proportional to tions are replaced by their power approximations. the radial distance from the axis in the interior. Notice that We take up the problem at the level of Eq. (33) replacing, the toroidal moments between the two spheres are all different from zero, and the summation of all the multipole com- l (1) ir< ponents of the field was carried out in Eq. (56). jl(kr<)hl (kr)|k→0 − l+1 : (54) k(2l + 1)r< 3.6. The point source limit Notice that the common factor ik before the sum in Eq. (33) when multiplied by the factor (−i=k) in Eq. (54) Equations (36), (37) for the electromagnetic field of toroidal gives one. For the sourceless regions, r < a and b < r, solenoids correspond to the transverse electric fields of the radial integrals are finite, but they are to be multiplied Eqs. (39), (40), with the identification Ê = ˆT following by the factor k2 which vanishes in the static limit. In the from Eq. (43), since the charge moments vanish in this case. notation of Secs. 3.3 and 3.4, the toroidal form factors are Consequently, the polarization and angular distribution of the finite, but the toroidal moments vanish in the inner and outer radiation of each of the multipole components of the field due

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197 196 A. GONGORA´ T. AND E. LEY-KOO to the toroidal solenoid have the same characteristics as the induction field is transverse, and its poloidal and toroidal radiation of the corresponding transverse electric multipole components arise from toroidal and poloidal currents, re- fields [6, 14]. The difference is in the amplitudes given by spectively. The electric and magnetic moments studied in Eq. (38), in contrast with the usual case dominated by the the standard textbooks arise from longitudinal and toroidal charge moments, last term in Eqs. (47). sources; the construction of the fields arising from poloidal In the case of a solenoid with small dimensions compared currents in a solenoid presented in Section 3 of this work is a to the wavelength, i.e. kb ¿ 1, the toroidal moments in way to complete the study of the multipole expansion intro- Eq. (38) become ducing the toroidal moments. p The complete multipole expansion of the electromagnetic NI 4…(2l + 1) ˆT (kb ¿ 1) = − field arising from alternating poloidal currents in toroidal l0 c l(l + 1) solenoids has been explicitly constructed, Eq. (33). The ex- · ¸ k[(kb)l+2 − (ka)l+2] pansion is complete in the sense of Ref. 7 that the field is × P (cos ) − P (cos ) (59) described at all points in space 0 < r < ∞, and also in l 1 l 2 l + 2 the sense that it exhibits the existence of the toroidal mul- Since the ratio of the moments of two consecutive multi- tipole moments [10]. In fact, the poloidal currents in the poles is of the order kb ¿ 1, the component with the lowest toroidal solenoids possess vanishing magnetic and charge multipolarity is the dominant one. The most dominant of all multipole moments; the toroidal moments are the moments is the toroidal dipole with l = 1. of the poloidal currents, just as the magnetic (poloidal) mo- Comparison of Eq. (54) with the corresponding charge ments are the moments of toroidal currents. The explicit, ex- multipole moment of Eq. (47) shows one extra factor of k act relationships among the transverse electric, toroidal and for the toroidal moments. This is translated into an addi- charge multipole moments and form factors are given through tional factor of !2 in the power radiated by a toroidal mul- Eqs. (43) and (48-52). tipole relative to that of the corresponding electric multipole. The standard multipole expansion of the elec- Thus the power radiated by a toroidal solenoid, approximated tromagnetic field is formulated in terms of the 6 2 2 as a toroidal dipole, goes as ! , in contrast with the well set of form factors Qlm(−k ; t);Mlm(−k ; t) and 4 2 known ! dependence of electric and magnetic dipoles. Elm(−k ; t), or the corresponding dynamic multi- Since the toroidal and electric multipoles of a given or- pole moments, in the long wavelength approximation, 2 2 der have the same angular momentum and parity properties, Elm(−k → 0; t) → Qlm(−k → 0; t). It is in this approx- the simultaneous presence of both types of multipoles gives imation that the presence of the toroidal form factors is lost. a frequency dependence of the radiated power that is more Reference 10 proposes the use of the alternative set of form 2 2 2 complicated than the corresponding dependence for each in- factors Qlm(−k ; t);Mlm(−k ; t), and Tlm(−k ; t), which dividual type. according to the discussion at the end of our Secs. 3.3 and 3.4 are respectively connected to the charge density and longitudinal component of the current, the toroidal component 4. Discussion of the current, and the poloidal component of the current, The general elements for constructing a complete multipole Eqs. (44) and (45). The description of the electromagnetic expansion for the electromagnetic field produced by any lo- field of the toroidal solenoids with poloidal currents requires these toroidal moments; for such systems the exact values of calized distribution of charges and currents have been identi- 2 2 the form factors are Qlm(−k ; t) = 0;Mlm(−k ; t) = 0 and fied in Sec. 2. They include: 2 2 2 Elm(−k ; t) = k Tlm(−k ; t). 1) Maxwell’s equations (1)-(4) and the corresponding in- The long wavelength approximation was applied in two homogeneous Helmholtz equations (5), (6), connect- particular cases. In the magnetostatic case of Sec. 3.5, ing the electric and magnetic fields and their sources; the multipole expansion of the magnetic induction field involves vanishing components of each multipolarity outside 2) the outgoing spherical wave Green function and its the solenoid, and the components in the interior of the spherical multipole expansion, Eq. (8); and solenoid add up to the field being inversely proportional to the 3) the Debye potentials for the sources, Eqs. (11) radial distance from the axis. In particular it can be pointed and (13), and the fields, Eqs. (15), (16), allowing the out that the Zel’dovich anapole in subsection 3.6 corresponds immediate identifications of their respective longitu- to a point toroidal solenoid with a stationary poloidal cur- dinal and transverse - toroidal and poloidal - compo- rent [9]. The toroidal multipole components of the radia- nents, Eqs. (17)-(19) and (20)-(22). tion field were analyzed in the point source limit and compared with the standard transverse electric multipole compo- Specifically, the electric intensity field may have longitudinal, nents, recognizing the presence of an extra factor of !2 in toroidal or poloidal components arising from a charge density the power radiated by the toroidal moments relative to that of distribution or a longitudinal current distribution, a toroidal the corresponding electric moments. Specifically, the toroidal current or a poloidal current, respectively; while the magnetic dipole moment is the dominant dynamical approximation to

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 197 the anapole [15, 16]. The interested reader may ﬁnd in the atoms and molecules, by parity non-conserving weak interac- last references some illustrative works on the evaluations of tions, and external electric ﬁelds. toroidal dipole moments induced in nucleons, leptons, nuclei,

1. E.M. Purcell, Electricity and Magnetism (Mc Graw-Hill, New 8. E. Ley-Koo and A. Gongora-T,´ Rev. Mex. Fis. 34 (1988) 645. York, 1965) p. 250. 9. Y.B. Zel’dovich, Sov. Phys. JETP. 6 (1958) 1184. 2. P. Lorain and D. Corson, Electromagnetic Fields and Waves, Second Edition (Freeman, San Francisco, 1970) p. 347. 10. V.M. Dubovik and A.A. Cheskov, Sov. J. Particles Nucl. 5 (1975) 318. 3. J.R.Reitz and F.J. Milford, Foundations of Electromagnetic Theory, Second Edition (Addison-Wesley, Reading, MA 1979) 11. V.M. Dubovik and V.V. Tugushev, Phys. Reports 187 (1990) p. 172. 145. 4. R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lec- 12. G.N. Afanasiev, J. Phys. A: Math. Gen. 23 (1990) 5755. tures on Physics (Addison - Wesley, Reading, MA, 1964) p. 16. 13. G.N.Afanasiev and V.M.Dubovik, J. Phys. A: Math. Gen. 25 5. W.K.H. Panofsky and M. Phillips, Classical Electricity and (1992) 4869. Magnetism, Second Edition (Addison -Wesley, Reading, MA 1962) Chap. 14. 14. C.G.Gray, Am. J. Phys. 46 (1978) 169. 6. J.D. Jackson, Classical Electrodynamics, Second Edition (Wi- 15. R.R. Lewis and S.M. Blinder, Phys. Rev. A 52 (1995) 4439. ley, New York, 1975) Chap. 16. 16. Lewis R.R. and A. Gongora-T,´ J.Phys. B: At. Mol. Opt. Phys. 7. R.H. Lambert, Am. J. Phys. 46 (1978) 849. 31 (1998) 3565.

Rev. Mex. F´ıs. E 52 (2) (2006) 188–197