Complete electromagnetic multipole expansion including toroidal moments

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1 ENSEÑANZA REVISTA MEXICANA DE FÍSICA E 52 (2) DICIEMBRE 26 Compete eectomgnetic mutipoe expnsion incuing tooi moments A. Góngo T. Cente fo Inteiscipiny Resech on Compex Systems, Physics Deptment, Nothesten Univesity Boston M.A. E. Ley-Koo Instituto e Físic, Univesi Ncion Autónom e México, Apto Post 2-364, 1 México D.F., México. Recibio e 3 e mzo e 26; cepto e 22 e myo e 26 The eectomgnetic mutipoe expnsion pesente in this ppe is compete on two ccounts: i) It is vi fo points in spce, n ii) it ecognizes the existence of tooi moments. The eectomgnetic fie ue to tenting pooi cuents in tooi soenoi is evute excty vi the soution of the inhomogeneous vecto Hemhotz equtions, using the outgoing wve Geen function technique n the Debye potentis fo souces n fies. The physic mening of the tooi moments cn be ppecite when they e compe with the fmii eectic n mgnetic moments; the nysis of the ong-wveength imit of the exct esuts so expins the pevious negect of the tooi moments. The mgnetosttic imit n the point souce imit e so physicy n icticy inteesting. Keywos: Eectomgnetic ition; mutipoe expnsion; eectic, mgnetic n tooi moments. E esoo mutipo eectomgnético que se pesent en este tícuo es competo po os zones: i) es váio p toos os puntos e espcio, y ii) econoce existenci e momentos tooies. E cmpo eectomgnético poucio po coientes pooies tens en un soenoie tooi se evú exctmente tvés e soución e ecuciones e Hemhotz inhomogénes, usno técnic e función e Geen e on siente y os potencies e Debye p fuentes y cmpos. E significo físico e os momentos tooies se estc compos con os momentos eécticos y mgnéticos fmiies; e náisis e ímite e ongitu e on gne e os esutos exctos tmbién expic ignonci pevi e os momentos tooies. E ímite mgnetostático y e ímite e fuente puntu son tmbién inteesntes físicmente y iácticmente. Desciptoes: Rición eectomgnétic; esoo mutipo; momentos eécticos, mgnéticos y tooies. PACS: 41.2 Jb 1. Intouction The stn pesenttions of the mutipoe expnsions of eectosttic, mgnetosttic n eectomgnetic fies in the textbooks e usuy incompete since they e esticte to the egions outsie ocize souces 1-6. Two tices my be cite fom the ictic itetue mking up fo this incompeteness; one of them es with the compete vecto spheic hmonic expnsion fo Mxwe s equtions 7, n the othe gives the mutipoe expnsions outsie n insie the souces fo the eectosttic n mgnetosttic cses 8. Anothe gp in the textbooks is the bsence of ny mention bout tooi moments. It wi soon be fifty yes since the viotion of pity ws estbishe in the wek intections, n the intouction of the npoe moment by Ze ovich in his note on Eectomgnetic intection with pity viotion 9. The Russin uthos hve been ctive investigting the mutipoe expnsion in cssic n quntum fie theoy, the eectomgnetic fies of tooi soenois n coesponingy the existence n impotnce of tooi moments 1-13, in the ensuing peio. It is time tht the stuy of such topics shou be incopote into the vnce couses in eectoynmics. This ppe pesents compete eectomgnetic mutipoe expnsion vi fo points in spce, with emphsis on the pesence of tooi moments on the sme footing s the fmii eectic n mgnetic moments. Section 2 povies the theoetic fmewok fo the stuy, incuing s the stting point Mxwe s equtions fo hmonic time vying souces, n thei tnsfomtion into inhomogeneous Hemhotz equtions. The coesponing soutions e constucte by using the outgoing wve Geen function n its spheic mutipoe expnsion, s we s the Debye potentis in oe to exhibit the ongituin n tnsvese components of the espective vecto souces n fies 14. Section 3 is evote to the constuction of the mgnetic fie ue to pooi cuents in tooi soenoi with cicu ing secto coss section in ech meiin pne, incuing the mutipoe expnsion, the Debye potentis, the ynmic tooi mutipoe moments, the ong wveength imit, the mgnetosttic imit, n the point souce imit. Section 4 contins iscussion bout the min new esuts n some of thei consequences. 2. Mxwe s equtions n Debye potentis The chge ensity ρ, the cuent ensity J, the eectic intensity fie E n the mgnetic inuction fie B e ete by Mxwe s equtions 1-6: E = 4πρ (1) E = iω c B (2)

2 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 189 B = (3) B = 4π c J iω c E, (4) coesponing to Guss w, Fy s w, nonexistence of mgnetic monopoes, n Ampée-Mxwe s w, fo hmonic time vitions e iωt with fequency ω fo the souces n fies. Hee ω/c = k is ientifie s the wve numbe. Equtions (1-4) e set of ine coupe equtions fo the fies. They cn be ecoupe, by tking the cu of Eqs. (2) n (4) n using the emining Eqs. (1) n (3), to obtin the espective Hemhotz inhomogeneous equtions fo the eectic n mgnetic fie: ( 2 + k 2 ) E( ) = 4π ρ 4πiω J( ) (5) c 2 ( 2 + k 2 ) B( ) = 4π c J( ). (6) Eqution (6) shows tht the mgnetic inuction fie is etemine by the tnsvese component of the cuent ensity. In contst, Eq. (5) shows tht the eectic intensity fie is etemine by the gient of the chge ensity n both ongituin n tnsvese components of the cuent ensity. The soutions to Eqs. (5), (6) cn be constucte with the hep of the Geen function of the Hemhotz eqution ( 2 + k 2 )G + ( ; ) = 4πδ( ). (7) The outgoing wve Geen function n its mutipoe expnsion e given by G + ( ; ) = eik = 4πik = j (k < ) (k > ) ( ) m NmP 2 m (cos θ )P m (cos θ)(2 δ m ) cos m(ϕ ϕ ). (8) m= The pticu soutions to Eqs. (5) n (6) invove the integtions of the espective souces n the Geen function E( ) = v ρ( )G + ( ; ) + iω c 2 v J( )G + ( ; ) (9) B( ) = v J( )G + ( ; ). (1) Use of the mutipoe expnsion of the Geen function of Eq. (8) in Eqs. (9) n (1) es to the coesponing gene n exct expnsion of the eectomgnetic fie. The pticu ppiction to the tooi soenoi is me in the foowing section. The Debye potentis, s hs been pointe out by Gy 14, e usefu fo exhibiting the ecomposition of the souce n foce fies into thei ongituin n tnsvese (tooi n pooi) components, incuing the etionships mong them. The gient of the chge ensity is ongituin fie, since its cu is ienticy zeo. The cuent ensity fie my be witten s { J( ) = i L ( ) + i T ( ) + i P ( ) = i L ( ) i i T ( ) i i P ( ), (11) whee the secon ine mkes use of the ngu momentum opeto = i. The continuity eqution stisfie by the souce ensities, J( ) iωρ( ) =, (12) invoves ony the ongituin component of the cuent ensity; in tems of the coesponing Debye potenti, it becomes 2 i L = iωρ, (13) showing tht this potenti n the chge ensity e ete though Poisson s eqution. On the othe hn, the cu of the cuent ensity, J( ) = i i T ( ) i ( )i P ( ) = i i T ( ) i 2 i P ( ) (14) shows tht its tooi n pooi Debye potentis e 2 i P ( ) n i T ( ), espectivey. The secon ine of Eq. (14) is obtine by pefoming the tipe vecto pouct n using the othogonity of the ivegence n ngu momentum opetos, s we s the commutbiity of the Lpce n ngu momentum opetos. The use of Eqs. (11) n (14) togethe with the symmety popeties of the Geen function n the hemiticity popeties of the gient n ngu momentum opetos ow us to ewite Eqs. (9) n (1) to exhibit the ongituin n tnsvese components of the espective foce fies,

3 19 A. GÓNGORA T. AND E. LEY-KOO E( ) = B( ) = i v ρ( ) iω c 2 il ( )G + ( ; ) i v 2 i P ( )G + ( ; ) i v iω c 2 it ( )G + ( ; ) i v iω c 2 ip ( )G + ( ; ) (15) v i T ( )G + ( ; ). (16) The coesponing Debye potentis cn be immeitey e off fom these equtions in nogy with Eq. (11): e L ( ) = v ρ( ) iω c 2 il ( )G + ( ; ) (17) e T ( ) = v iω c 2 it ( )G + ( ; ) (18) e P ( ) = v iω c 2 ip ( )G + ( ; ) (19) b L ( ) = (2) b T ( ) = v 2 i P ( )G + ( ; ) (21) b P ( ) = v i T ( )G + ( ; ). (22) Equtions (18) n (22) show tht tooi cuents pouce tooi eectic n pooi mgnetic fies, whie Eqs. (19) n (21) show tht pooi cuents pouce pooi eectic n tooi mgnetic fies. Aso Eq. (17) shows tht the chge ensity n the ongituin cuents e the souces of the ongituin eectic fie. 3. Eectomgnetic fie of tooi soenois This section stts by efining the toois with cicu ing secto coss section in ech meiin pne, n the tenting pooi cuents in the coesponing soenois. The cu of the cuent ensity is evute n use in Eq. (1) togethe with the mutipoe expnsion of the Geen function, Eq. (8), in oe to obtin the mutipoe expnsion of the mgnetic inuction fie. The eectic intensity fie cn be evute by the integtion of Eq. (9), but hee it is pefebe to obtin it though Eq. (4). The tooi chcte of the mgnetic fie n the pooi chcte of the eectic fie e expicity exhibite, obtining the mutipoe expnsions of the espective Debye potentis ong the wy. The nysis focuses on the competeness of the mutipoe expnsion 14, incuing the so-ce tooi moments 5. Then the imiting situtions of ong wveengths, incuing the mgnetosttic cse n the point souce cse, e stuie in pticu The mutipoe expnsion The tooi with cicu ing secto coss section is efine by its inne spheic ing ( =, θ 2 < θ < θ 1, ϕ), its uppe conic ing ( < < b, ϑ = θ 1, ϕ), its oute spheic ing ( = b, ϑ 1 < θ < ϑ 2, ϕ), n its owe conic ing (b > >, θ = ϑ 2, ϕ). The tenting pooi Ie iωt cuent in the tooi soenoi with N tuns hs the ensity J(, t) = NIe iωt 2π sin θ { ˆ δ(θ θ 1 ) δ(θ θ 2 ) Θ( ) Θ( b) +ˆθ δ( b) δ( ) Θ(θ θ 1 ) Θ(θ θ 2 ), (23) whee the Dic et functions efine the coi eements ong which the cuent fows n the Hevisie step functions efine the extent of such eements. The cu of the cuent ensity, J( ) = NI { Θ(θ 2π ˆϕ θ1 ) Θ(θ θ 2 ) δ( b) δ( ) sin θ 1 2 θ δ(θ θ1 ) δ(θ θ 2 ) is tooi, i.e., in the zimuth iection, n invint une ottions oun the xis of the tooi. sin θ Θ(θ θ 1 ) Θ(θ θ 1 ) (24)

4 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 191 As ey nticipte, the compete mutipoe expnsion of the mgnetic fie foows fom the combintions of Eqs. (1), (24) n (8), B( ) = NI 2πc π 2π 4πik = 2 sin θ ϑ ϕ ˆϕ j (k < ) (k > ) 1 2 θ { Θ(θ δ( b) δ( θ 1 ) Θ(θ θ 2 ) ) sin θ δ(θ θ 1 ) δ(θ θ 2 ) sin θ Θ( ) Θ( b) ( ) m NmP 2 m (cos θ )P m (cos θ)(2 δ m ) cos m(ϕ ϕ) (25) m= The zimuth nge integtion cn be one by using ˆϕ = ˆϕ cos(ϕ ϕ) ˆR sin(ϕ ϕ) n the othogonity of the cosine n sine functions, 2π ϕ ˆϕ cos m(ϕ ϕ) = ˆϕπδ m1. (26) It foows tht the mgnetic inuction fie is so zimuth n invint une xi ottions, i.e., tooi fie. The po nge integtions e so stightfow, n ccoing to the seection ue of Eq. (26) e imite to the tems with m=1 in the sum of Eq. (25). The fist integ, θ 2 θ 1 θ P 1 (cos θ ) = P (cos θ 1 ) P (cos θ 2 ), (27) foows immeitey fom the etion between the ssocite n oiny Legene poynomis, (cos ϑ) = sin θ (cos θ) P (cos θ). (28) P 1 The secon integ, invoving the Dic et functions, cn be one by pts: π θ θ sin θ P 1 (cos θ ) sin θ P 1 (cos ϑ ) = 1 sin θ θ sin θ P 1 (cos θ ) θ =θ sin θ θ sin θ P 1 (cos θ ) θ =θ 2 = ( + 1) P (cos θ 1 ) P (cos θ 2 ). (29) The st ine is obtine by gin using Eq. (28) n so the iffeenti eqution fo the oiny Legene poynomis: 1 sin θ θ sin ϑ θ P (cos θ) = ( + 1)P (cos θ). The common fcto in the po nge integs, Eqs. (27) n (29), shou be note. The integtions ove the i coointe e so iect, but the istinction mong the iffeent octions of the fie point must be me. The fist integ ove the Dic et functions is one by pts: δ( b) δ( ) j (k < ) (k > ) = { b (k ) (k ) { b j (k ) =b j (k), > j (k) + j (k ) (k), b <. = (k), < < b (3)

5 192 A. GÓNGORA T. AND E. LEY-KOO The secon integ cn be expicity witten s b j (k < ) (k > ) = b b (k )j (k), < j (k ) b (k) + j (k ) (k), b <. (k )j (k), < < b (31) The substitution of the integs of Eqs. (26)-(31) in Eq. (25) gives B( ) = NI c 4πik ˆϕ =1 ( )N 2 1P 1 (cos θ) P (cos θ 1 ) P (cos θ 2 ) { b (k ) + ( + 1) b (k ) =b { +( + 1) { j (k ) (k ) j (k), < j (k) + j (k ) j (k ) (k) + b b (k ) + ( + 1) b = (k ) (k )j (k), < < b j (k ) (k), b <. (32) The i fctos in Eq. (32) cn be simpifie by integting the iffeenti eqution fo the spheic Besse functions. The esut is B( ) = NI c 4πik ˆϕ =1 ( )N 2 1P 1 (cos θ) P (cos θ 1 ) P (cos θ 2 ) k 2 b { (k )j (k), < j (k) = +k 2 j (k ) k 2 b j (k ) (k), b <. (k) (k ) = (k) + k 2 b j (k) (k )j (k), < < b (33) It is ppopite t this point to ecognize tht the stn mutipoe expnsion of the eectomgnetic fie 5, 6, 14 is usuy imite to the egion outsie the souces, coesponing to b < in Eqs. (3-33). Foowing Lmbet 7, hee we hve so obtine the fie in the inne egion <, whee thee e no souces, n in the intemeite egion < < b, whee the souces e octe. A of this hs been one within one n the sme ccution by simpy istinguishing mong the iffeent octions of the fie point. It is so petinent to point out tht, s is to be expecte, the soutions in the souce fee egions, < n b <, e supepositions of soutions of the homogeneous Hemhotz equtions, whie the soutions in the egion whee the souces e octe, < < b, invove non-ine combintions of the spheic Besse functions n thei eivtives o integs The Debye potentis Eqution (33) my be witten in n expicity tooi fom by using the epesenttion of the pouct of the zimuth unit vecto n the ssocite Legene poynomis, so tht ˆϕP 1 (cos θ) = 4π ( i )Y (θ, ϕ), (34)

6 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 193 B(, θ, ϕ) = i =1 NI c 4πik( ) 4π N 2 1 P 1 (cos θ) P (cos θ 2 ) Y (θ, ϕ) k 2 b { (k )j (k), < (j (k)) + k 2 j (k ) + (h(1) (k)) + k 2 b (k ) k 2 b j (k ) (k), b <. (k) j (k), < < b (35) Compison with Eqs. (16) n (21) gives the ientifiction of the summtion in Eq. (35) s the mutipoe expnsion of the Debye potenti b T. Notice tht the Lpcin in Eq. (21), becuse of its hemiticity, my be me to opete on the Geen function, giving, ccoing to Hemhotz eqution, Eq. (7), k 2 times the Geen function pus the point souce ensity tem; such etionship is ppent in the i fctos of Eq. (35). As ey mentione t the beginning of this section, the eectic intensity fie my be obtine by using Eq. (4), when the mgnetic inuction, Eq. (35), n the cuent ensity, Eq. (23), e known. The eectic intensity is obviousy pooi s esut of the ppiction of the cu to the tooi mgnetic inuction n the pooi chcte of the cuent ensity itsef. The cuent ensity my be witten in its mutipoe expnsion fom by using the coesponing epesenttions of the Dic et n the Hevisie step functions in the po nge, espectivey, in Eq. (23). Fo the ske of spce, the compete expessions E fo n J e not witten out expicity The ynmic tooi mutipoe moments At this point it is moe instuctive to go on to wite the mgnetic n eectic fies in the egion outsie the exten sphee, b <, in the fom tht ows the chcteiztion of the mutipoe expnsions of the eectomgnetic fie of tooi soenois, B(b <, θ, ϕ) = E(b <, θ, ϕ) = whee ψ = 4πNIk3 c =1 =1 ψ h (1) (k)y (θ, ϕ) (36) ψ i k (k)y (θ, ϕ), (37) π 1 ( + 1) b P (cos θ 1 ) P (cos θ 2 ) j (k ). (38) Hee the expicit vue of the nomiztion constnt N 1 hs been incopote. Fo puposes of compison, we tnscibe next the coesponing gene equtions fo the mutipoe expnsions of the eectomgnetic fie 6, 14, ψm E (k)y m (θ, ϕ) B( ) = E( ) = m + ψm M ( i) m k ( i k ψ E m (k)y m (θ, ϕ) ) (k)y m (θ, ϕ) (39) + ψm M (k)y m (θ, ϕ), (4) whee the ynmic mutipoe moments, in the teminoogy of 14, e given by ψm= M 4πik2 v j (k )Y c(+1) m(θ, ϕ ) J( ) (41) ψm E = 4πik2 v j (k )Ym(θ, ϕ ) c( + 1) ik J( ) c(2 + )ρ( ) (42) Notice tht Eq. (5b) of Ref. 14, equivent to ou Eq. (42), is missing fcto of k on its hs, s cn be veifie by compison with Eq. (16-91) in Ref. 6. The compison of Eqs. (36-38) n (39-42) is iect. The pooi component of the mgnetic inuction fie is bsent in Eq. (36) n the tooi component of the eectic intensity fie is bsent in Eq. (37), becuse the ynmic mgnetic mutipoe moments in Eq. (41) vnish t the souce eve ue to the pooi chcte of the cuent, Eq. (23). The vnishing of the integn in Eq. (41) foows immeitey fom the othogonity of the i vecto n the cu of the cuent ensity, Eq.(24). On the othe hn, the ynmic eectic mutipoe moments, Eq. (42), e etemine by the i components of the cuent ensity n of the gient of the chge ensity, s we s by the chge ensity itsef. In the cse une stuy, the chge ensity is bsent, n the i component of the cuent ensity is the

7 194 A. GÓNGORA T. AND E. LEY-KOO fist tem insie the cuy bckets on the hs of Eq. (23). The evution of the coesponing integ in Eq. (42) es to the sme vue of Eq. (38). In concusion, the mutipoe expnsions of Eqs. (36,37,38) coespon to the tems in Eq. (42) invoving the ynmic tnsvese eectic mutipoe moments ssocite with the i component of the pooi cuent ensity, i.e. Eq. (42) with ρ =. One might be tempte to the supescipt E in Eqs. (36-38) to compete the chcteiztion of the mutipoe expnsion of the eectomgnetic fie of the tooi soenoi. Howeve fing into such tempttion is tntmount to missing the existence of the tooi moments. Foowing Ref. 1, the coect supescipt to be e in Eqs. (36-38) is fo tooi, n Eq. (42) gives n exct etionship mong the ynmic mutipoe moments of eectic, tooi n chge types. ψm(k) E = ψm(k) T + ψ Q m (k). (43) The istinction mong these types of ynmic mutipoe moments n the connection mong them cn be tce bck to the iffeent wys of septing the souce tems in the inhomogeneous Hemhotz equtions, Eqs. (5,6). In Eqs. (41), (42) it is ecognize tht the souce fctos e the i components of the espective souces in Eqs. (6) n (5). Let us consie fist the unmbiguous cse of the mgnetic moments. The coesponing souce fcto in the integn of Eq. (41) my be witten in tems of the ecomposition of the cuent ensity of Eq. (11) in the tentive foms J( ) = J( ) = i i L i i T i i P = ˆ 2 i T, (44) whee in the fist ine the ot n coss e exchnge in the tipe sc pouct, n in the secon ine the othogonity of the opetos is tken into ccount. Eqution (44) shows tht the ynmic mgnetic mutipoe moments epen ony on the tooi component of the cuent. On the othe hn, thee is oom fo mbiguity in the cse of the eectic moments. In fct, the coesponing souce fcto tken fom Eq. (5), ρ( ) + ik J( c ) = ρ( ) ik c il ( ) + iˆ 2 i P ( ) (45) invoves both ongituin n pooi components of the souce. Tking into ccount tht the bsis functions fo the mutipoe expnsions e eigenfunctions of the 2 n ˆ 2 opetos, it cn be ecognize tht Eq.(41) is in iect coesponence with Eq. (22), whie Eq.(42) is ete to both Eqs. (17) n (21). The stn teminoogy of Eqs. (39)-(42) of tnsvese eectic n mgnetic moments cn be me moe pecise by mking the istinction of Eq. (43), n ecognizing tht the mgnetic moments ising fom tooi cuents cou be ppopitey ce pooi moments, just s the moments ising fom the pooi cuents e ce tooi moments The ong wveength imit When the souces e confine in egion with i extent tht is sm compe with the wveength λ = 2π/k, it is usu to ppoximte the spheic Besse functions in Eqs. (34-42) by the ominnt tems of thei powe seies expnsions cose to the oigin. The esut of such ppoximtion is tht the ynmic mutipoe moments, which e efine in tems of spheic Besse functions s weight functions in the integs of Eqs. (41) n (42), e connecte to the sttic mutipoe moments efine in tems of the powes of the i coointe s weight functions, ψ M m(k ) = 4πik +2 c( + 1)(2 + 1)!! ψm(k E 4πik +2 ) = c( + 1)(2 + 1)!! = 4πik+2 1 c(2 + 1)!! + 1 v Y m(θ, ϕ ) J( ) = v Ym(θ, ϕ ) ik J( ) c(2 + )ρ( ) v Ym(θ, ϕ )ik J( ) + c 4πik+2 (2 + 1)!! M m (46) v Ym(θ, ϕ )ρ( ) M m. (47) Eqution (46) is the sme s Eqs. (58) n (17) of Ref 14. The chge pt of Eq. (47) is the sme s Eqs. (59), (33) n (32) of Ref. 14, whee the st integ is the fmii eectosttic mutipoe moment Q m 6, 8. The pt ssocite with the i component of the cuent ensity in Eq. (47) is usuy oppe in the ong wveength ppoximtion, but this is not justifie s iustte in this ppe n Refs. 1 to 13. In the nottion of Ref. 1, the ynmic mutipoe moments in Eqs. (41), (42) n (43) e nomize to give the time epenent fom fctos M m ( k 2, t)= (2+1)!! 4πik +2 ψm m(k)e iωt M m e iωt (48)

8 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 195 E m ( k 2, t) = c(2 + 1)!! 4πk +1 ψ E m(k)e iωt (49) Q m ( k 2, t)= (2+1)!! 4πik +2 ψq m (k)e iωt Q m e iωt (5) T m ( k 2, t) = (2 + 1)!! 4πk +3 ψt m(k)e iωt. (51) In tems of these fom fctos, Eq.(43) becomes E m ( k 2, t) = k 2 T m ( k 2, t) + Q m ( k 2, t) (52) which is the cent esut of Ref. 1. The stn systems of eectomgnetic souces n fies 5 7, 14 e escibe in tems of the chge fom fctos, Eq.(5), the mgnetic fom fctos, Eqs.(48) n the tnsvese eectic fom fctos, Eq. (49). In the ong wveength ppoximtion the tte becomes E m ( k 2, t) = Q m ( k 2, t) (53) s foows fom Eq. (52). Howeve, in systems with pooi cuents ike the tooi soenois, Eq. (53) oes not ho. Inste of the tnsvese eectic fom fctos, it is moe gene to use the tooi fom fctos of Eq. (51), which e inepenent of the mgnetic n the chge fom fctos. In Secs. 3.3 n 3.4, we hve consiee the eectomgnetic fies in the egion b < in oe to compe with the stn vibe esuts of Refs. 6 n 14. The tetment of Sec. 3.1 incues the fies fo the emining egions < b, since we hve the compete fies, s Lmbet 7 pointe out. In Sec. 3.5, we stuy the mgnetic inuction fie in the sttic imit ω, k, o infinite wveength imit. In Sec. 3.6, the point souce imit coesponing so to n infinitey ong wveength imit, but fo finite fequency, is nyze The mgnetosttic imit The imit of sttiony cuents with fequency ω in the tooi soenois coespons to the mgnetosttic cse. It is ce tht Eq. (4) becomes Ampee s w, Eqs. (6) n (7) become Poisson s eqution, the Geen function of Eq. (8) becomes the Couomb potenti, n the spheic Besse functions e epce by thei powe ppoximtions. We tke up the pobem t the eve of Eq. (33) epcing, j (k < ) (k) k i < k(2 + 1) +1 <. (54) Notice tht the common fcto ik befoe the sum in Eq. (33) when mutipie by the fcto ( i/k) in Eq. (54) gives one. Fo the souceess egions, < n b <, the i integs e finite, but they e to be mutipie by the fcto k 2 which vnishes in the sttic imit. In the nottion of Secs. 3.3 n 3.4, the tooi fom fctos e finite, but the tooi moments vnish in the inne n oute egions bouning the tooi. The mgnetic inuction coesponingy vnishes in both egions. Fo the egion whee the souces e octe, the i fcto becomes (+1 ) 1 +1 ( 1 ) = (55) The coefficient in the numeto of Eq (55) wi cnce the coefficient in the enominto of Eq. (54), n Eq. (33) tkes the fom B( < < b,θ, ϕ) = NI c 4π ˆϕ ( ) π P 1 (cos θ) t=1 P (cos θ 1 ) P (cos θ 2 ) = 2NI Θ(θ θ 1 ) Θ(θ θ 2 ) c sin θ (56) In the fist ine, the expicit vue of the nomiztion constnt ws substitute. In the st ine, the sum is ientifie with the iffeence of the Hevisie step functions in the po nge, which foow fom the competeness of the othonom Legene bsis = n its integtion Θ(θ θ i ) = = = P (cos θ)p (cos θ i ) = δ(cos θ cos θ i ) 2 = =1 θ θ δ(θ θ i ) θ = δ(θ θ i) sin θ θ sin θ P (cos θ )P (cos θ i ) 1 ( + 1) (57) sin θp 1 (cos θ)p (cos θ i ). (58) In concusion, the mgnetic inuction vnishes outsie the soenoi n is zimuth n invesey popotion to the i istnce fom the xis in the inteio. Notice tht the tooi moments between the two sphees e iffeent fom zeo, n the summtion of the mutipoe components of the fie ws cie out in Eq. (56) The point souce imit Equtions (36), (37) fo the eectomgnetic fie of tooi soenois coespon to the tnsvese eectic fies of Eqs. (39), (4), with the ientifiction ψ E = ψ T foowing fom Eq. (43), since the chge moments vnish in this cse. Consequenty, the poiztion n ngu istibution of the ition of ech of the mutipoe components of the fie ue

9 196 A. GÓNGORA T. AND E. LEY-KOO to the tooi soenoi hve the sme chcteistics s the ition of the coesponing tnsvese eectic mutipoe fies 6, 14. The iffeence is in the mpitues given by Eq. (38), in contst with the usu cse ominte by the chge moments, st tem in Eqs. (47). In the cse of soenoi with sm imensions compe to the wveength, i.e. kb 1, the tooi moments in Eq. (38) become ψ(kb T 1) = NI 4π(2 + 1) c P (cos θ 1 ) P (cos θ 2 ) ( + 1) k(kb) +2 (k) +2 (59) + 2 Since the tio of the moments of two consecutive mutipoes is of the oe kb 1, the component with the owest mutipoity is the ominnt one. The most ominnt of is the tooi ipoe with = 1. Compison of Eq. (54) with the coesponing chge mutipoe moment of Eq. (47) shows one ext fcto of k fo the tooi moments. This is tnste into n ition fcto of ω 2 in the powe ite by tooi mutipoe etive to tht of the coesponing eectic mutipoe. Thus the powe ite by tooi soenoi, ppoximte s tooi ipoe, goes s ω 6, in contst with the we known ω 4 epenence of eectic n mgnetic ipoes. Since the tooi n eectic mutipoes of given oe hve the sme ngu momentum n pity popeties, the simutneous pesence of both types of mutipoes gives fequency epenence of the ite powe tht is moe compicte thn the coesponing epenence fo ech iniviu type. 4. Discussion The gene eements fo constucting compete mutipoe expnsion fo the eectomgnetic fie pouce by ny ocize istibution of chges n cuents hve been ientifie in Sec. 2. They incue: 1) Mxwe s equtions (1)-(4) n the coesponing inhomogeneous Hemhotz equtions (5), (6), connecting the eectic n mgnetic fies n thei souces; 2) the outgoing spheic wve Geen function n its spheic mutipoe expnsion, Eq. (8); n 3) the Debye potentis fo the souces, Eqs. (11) n (13), n the fies, Eqs. (15), (16), owing the immeite ientifictions of thei espective ongituin n tnsvese - tooi n pooi - components, Eqs. (17)-(19) n (2)-(22). Specificy, the eectic intensity fie my hve ongituin, tooi o pooi components ising fom chge ensity istibution o ongituin cuent istibution, tooi cuent o pooi cuent, espectivey; whie the mgnetic inuction fie is tnsvese, n its pooi n tooi components ise fom tooi n pooi cuents, espectivey. The eectic n mgnetic moments stuie in the stn textbooks ise fom ongituin n tooi souces; the constuction of the fies ising fom pooi cuents in soenoi pesente in Section 3 of this wok is wy to compete the stuy of the mutipoe expnsion intoucing the tooi moments. The compete mutipoe expnsion of the eectomgnetic fie ising fom tenting pooi cuents in tooi soenois hs been expicity constucte, Eq. (33). The expnsion is compete in the sense of Ref. 7 tht the fie is escibe t points in spce < <, n so in the sense tht it exhibits the existence of the tooi mutipoe moments 1. In fct, the pooi cuents in the tooi soenois possess vnishing mgnetic n chge mutipoe moments; the tooi moments e the moments of the pooi cuents, just s the mgnetic (pooi) moments e the moments of tooi cuents. The expicit, exct etionships mong the tnsvese eectic, tooi n chge mutipoe moments n fom fctos e given though Eqs. (43) n (48-52). The stn mutipoe expnsion of the eectomgnetic fie is fomute in tems of the set of fom fctos Q m ( k 2, t), M m ( k 2, t) n E m ( k 2, t), o the coesponing ynmic mutipoe moments, in the ong wveength ppoximtion, E m ( k 2, t) Q m ( k 2, t). It is in this ppoximtion tht the pesence of the tooi fom fctos is ost. Refeence 1 poposes the use of the tentive set of fom fctos Q m ( k 2, t), M m ( k 2, t), n T m ( k 2, t), which ccoing to the iscussion t the en of ou Secs. 3.3 n 3.4 e espectivey connecte to the chge ensity n ongituin component of the cuent, the tooi component of the cuent, n the pooi component of the cuent, Eqs. (44) n (45). The esciption of the eectomgnetic fie of the tooi soenois with pooi cuents equies these tooi moments; fo such systems the exct vues of the fom fctos e Q m ( k 2, t) =, M m ( k 2, t) = n E m ( k 2, t) = k 2 T m ( k 2, t). The ong wveength ppoximtion ws ppie in two pticu cses. In the mgnetosttic cse of Sec. 3.5, the mutipoe expnsion of the mgnetic inuction fie invoves vnishing components of ech mutipoity outsie the soenoi, n the components in the inteio of the soenoi up to the fie being invesey popotion to the i istnce fom the xis. In pticu it cn be pointe out tht the Ze ovich npoe in subsection 3.6 coespons to point tooi soenoi with sttiony pooi cuent 9. The tooi mutipoe components of the ition fie wee nyze in the point souce imit n compe with the stn tnsvese eectic mutipoe components, ecognizing the pesence of n ext fcto of ω 2 in the powe ite by the tooi moments etive to tht of the coesponing eectic moments. Specificy, the tooi ipoe moment is the ominnt ynmic ppoximtion to

10 COMPLETE ELECTROMAGNETIC MULTIPOLE EXPANSION INCLUDING TOROIDAL MOMENTS 197 the npoe 15, 16. The inteeste ee my fin in the st efeences some iusttive woks on the evutions of tooi ipoe moments inuce in nuceons, eptons, nucei, toms n moecues, by pity non-conseving wek intections, n exten eectic fies. 1. E.M. Puce, Eecticity n Mgnetism (Mc Gw-Hi, New Yok, 1965) p P. Loin n D. Coson, Eectomgnetic Fies n Wves, Secon Eition (Feemn, Sn Fncisco, 197) p J.R.Reitz n F.J. Mifo, Fountions of Eectomgnetic Theoy, Secon Eition (Aison-Wesey, Reing, MA 1979) p R.P. Feynmn, R.B. Leighton, n M. Sns, The Feynmn Lectues on Physics (Aison - Wesey, Reing, MA, 1964) p W.K.H. Pnofsky n M. Phiips, Cssic Eecticity n Mgnetism, Secon Eition (Aison -Wesey, Reing, MA 1962) Chp J.D. Jckson, Cssic Eectoynmics, Secon Eition (Wiey, New Yok, 1975) Chp R.H. Lmbet, Am. J. Phys. 46 (1978) E. Ley-Koo n A. Góngo-T, Rev. Mex. Fis. 34 (1988) Y.B. Ze ovich, Sov. Phys. JETP. 6 (1958) V.M. Dubovik n A.A. Cheskov, Sov. J. Ptices Nuc. 5 (1975) V.M. Dubovik n V.V. Tugushev, Phys. Repots 187 (199) G.N. Afnsiev, J. Phys. A: Mth. Gen. 23 (199) G.N.Afnsiev n V.M.Dubovik, J. Phys. A: Mth. Gen. 25 (1992) C.G.Gy, Am. J. Phys. 46 (1978) R.R. Lewis n S.M. Bine, Phys. Rev. A 52 (1995) Lewis R.R. n A. Góngo-T, J.Phys. B: At. Mo. Opt. Phys. 31 (1998) 3565.