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23 November 1998 PHYSICS LETTERS A Physics Letters A 249 ( 1998) 1-9 ELXWER Electric and magnetic fields of a toroidal dipole in arbitrary motion Jo& A. Heras lnstituto de Fkica, Universidad National Autdnoma de MPxico, Apartado Postal 20-364. OlooO M.kxico, D.F:. Mexico Received I5 August 1998; accepted for publication 10 September Communicated by V.M. Agranovich 1998 Abstract Electric and magnetic fields of an arbitrarily moving particle possessing a constant toroidal moment -r are derived from the general solution of Maxwell’s equations for electric and magnetic fields of a toroidal moment density. The fields divide themselves naturally into five parts: the very near fields which vary as l/p and depend on I,@; the near fields which vary as i/R” and depend on T, /3 and 8; the intermediate fields which vary as l/R2 and depend on T, 0, fi and & the far ... fields which vary as 1/R and depend on 7, fi, p, p and p; and the delta-terms which represent the fields inside the source. The total power radiated by this dipole is then calculated for the special case in which the velocity and its derivatives are parallel. @ 1998 Elsevier Science B.V. PACS: 03.5O.De: 41.20.Bt; 4l.lO.H~ 1. Introduction In 1967 Dubovik and Cheshkov [ l] discovered a third family of moments in the classical electrodynamics, the so-called toroidal moments which are independent of the electric and magnetic moments. Since then the interest in this new class of moments has increased considerably (an early review on toroidal moments was presented by Dubovik and Tosunyan [2] and a more recent review by Dubovik and Tugushev [ 31). Ginzburg and Tsytovich 141 were apparently the first in investigating the fields of a point toroid dipole in uniform motion. They discussed the Cherenkov radiation emitted by a toroidal dipole. The fields of a dipole at rest possessing a time-varying toroidal moment were discussed by Dubovik and Shabanov [ 5 J . Afanasiev and Stepanovsky [ 61 have investigated the radiation fields of toroidal-like time-dependent sources. Explicit expressions for the radiation fields of a toroidal dipole in arbitrary motion and formulas for the radiated power and the radiation reaction force of a nonrelativistic toroidal dipole have recently been derived by the present author [7]. The angular momentum loss by a radiating toroidal dipole has recently been discussed by Radescu and Vlad [ 81. In this paper a more general treatment of the theory of moving toroidal dipoles is presented. Specifically, the general solution of Maxwell’s equations for electric and magnetic fields of a toroidal moment density in vacuum is obtained by means of a time-dependent extension of Helmholtz’s theorem [9,10]. The solution 037%9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PIISO375-9601(98)00712-9 2 J.A. Heras/Physics Letters A 249 (1998) 1-9 shows that both the electric field and the magnetic field are composed of two terms: a retarded-integral term representing the corresponding field outside the toroidal source plus a contact term (that evaluated at the field point and the present time) describing the field inside the source. It is shown that the magnetostatic field of a time-independent toroidal source is entirely confined. The general solution is used first to derive the fields of a point dipole at rest possessing a time-varying toroidal moment. In a second application, general expressions for the electric and magnetic fields of a moving particle possessing a time-varying toroidal moment are derived. These fields are written in terms of conventional parameters (n, B, B, . ..) when the toroidal moment is constant. These fields divide themselves naturally into five parts: the expected l/R, l/R*, 1/R3 parts; the novel l/R” part and the delta-function part. The total power radiated by this toroidal dipole is then calculated when the velocity and its derivatives are parallel. 2. Time-dependent extension of Helmholtz’s theorem The classical Helmholtz theorem of vector analysis can be formulated for a time-dependent vector field F(x, t) which propagates hyperbolically in vacuum and vanishes at infinity [9,10]. This extension of the theorem states that F is determined by specifying its divergence, curl and time derivative. An expression for F is given by [ lo] F(x,r) = & SS( {v’.F(x’,t’))V’G(x,t;x’,t’) + $G(x,t;x’,i’)-&{ + (V’ x F’(x’J’)} x V’G(x,t;x’,t’) dF(a:;‘r’)})d3x’dr’, (1) where the time integration is from -KJ to +oc and the spatial integration is over all space. G( x, t; x’, t’) = S( t’ + R/c - t) /R is the free-space retarded Green function of the wave equation, 6 being the delta function, R =) x - x’ 1, x the field point, x’ the source point and c the speed of light. 3. The electric and magnetic fields of a toroidal source As pointed out by Dubovik and Shabanov [ 51, in classical electrodynamics there exist toroidal sources apart from the magnetization and polarization sources. The effective current associated with the toroidal moment density T(x, t) (which might be called “toroidization”) is given by Jeff = cV x (V x T). Thus, Maxwell’s equations for electric and magnetic fields of a toroidal sources in vacuum and Gaussian units take the form V.E=O, (2a) V.B=O, (2b) VxE+;y=O, (2c) VXB-;~ 1 37E =&rVx(VxT). These equations can be integrated by applying Eq. ( 1). In fact, if the quantities specified by means of Eqs. (2a), (2~) and (2d), then IQ. ( 1) yields the result (2d) V . E, V x E and dE/dt are J.A. Heras/Physics Similarly, B=- the use of Eqs. (2b)-(2d) Letters A 249 (1998) l-9 into Eq. ( 1) yields the expression J.I V’G x [V’ x (V’ x T)] d3xrdt’. (3b) Eqs. (3) constitute the solution of Eqs. (2) subject to the conditions that the fields E and B, and their derivatives, vanish at infinity and the source T is confined. The spatial derivatives in Eqs. (3) can be transformed into time derivatives. To do this the following results are required, d as(u) - cR -g- a/G = $6(u) 3nsni a”af’G = ( _ pi (da) f$ 6”‘s (x - - R3 , x’) ) S( u ) (3”“5,@) - (u = t’ + R/c - t, (n = R/R)’ = n’ and S’j is the Kronecker can be transformed into the convenient form E=- ss( 3n(n + R3c IZXT -+R=c= B= * F) - F nxT 3n(n * T) - i; R=c= aTt4 I az,“,jd, (g) (4b) delta). With the aid of these expressions, Eqs. (3) ... + ’ ’ ‘,“,: S(u) d3x’dt’ - g T, $ , (5a) > ... RC3 > S(u) d3x’ dt’ + 47rV x T , (5b) where a overdot means differentiation with respect to t’. In obtaining Eqs. (5), the boundary conditions have been used. It is interesting to note that the terms -( 8~/3c) (aT/at) and 497V x T in Eqs. (5) are vector functions evaluated at the field point and the present time - without the presence of these terms, Eqs. (5) do not strictly satisfy Maxwell’s equations (2). These “contact” terms represent the values of the fields inside the toroidal source. The values of the fields outside the toroidal source are given by the corresponding retarded integrals appearing in Eqs. (5). Evidently, Eqs. (5) reduce to B=4rVxT, (6) for a time-independent toroidal source. Since T is confined, the field B in Eq. (6) is entirely confined. This is a remarkable property of the magnetostatic field of a toroidal source. The magnetostatic field of a magnetized source does not exhibit this property [ 111. 4. The oscillating toroidal dipole As a first application of Eqs. (5) consider a particle at rest (at the point xc) with a toroidal moment oscillating in time r = r(t) e, where r(t) is the magnitude of the toroidal moment which is a periodic function of time and e is its direction. The associated toroidization vector is given by T(x, t) = eT( t) S(x - x0). With this specific source, Eqs. (5) are integrated. The result is E= B= _ T(t’){3(e*n)n R2c2 _+(t’){3(e.n)n-e} R3c fqt’> e x II R2C2 + ‘i’(t’)e Rc3 x n - 47n-(t)e -e} - ?(t’)n x VS{x -x0}, x (n x e) Rc3 --- 81r dr e&x 3c dt - ~0) , (74 (7b) 4 J.A. Hews/Physics Letters A 249 (1998) 1-9 where now n = (x - xo)/{x - x0\ and the overdot means differentiation with respect to t’ = t - R/c with R = Ix - x01. Eqs. (7) are similar to those given by Dubovik and Shabanov in Ref. [ 51. Eq. (7a) (without the delta term) was also considered in Ref. [ 81. 5. A toroidal dipole in arbitrary motion The problem of finding the fields of a toroidal dipole in motion is considerably more complicated than that of computing the fields of a toroidal dipole at rest. The relativistic transformation of the toroidization demands that, in the same sense that a moving magnetization has an associated polarization, a moving toroidization has necessarily an associated axial vector density which should appear on the right-hand side of Eq. (2a). However, this problem may be avoided to a certain extent by assuming that the dipole is observed in a frame where there is only toroidization and it is given by T(x, t) = 7(t) S{x - r(t)}. After an integration by parts, Eqs. (5) can be written in the equivalent form (W (8b) It is should be noted that the 1/R parts of these equations, (94 (9b) have been derived previously by the author [ 71. For point toroidal sources, Eqs. (9) yield the associated radiation fields: if the time-dependent toroidal source is at rest, as the oscillating toroidal dipole, then Eqs. (9) give only radiation fields, that is, E1iR = E,d and Bl/R = Brad. But if the source is moving in arbitrary manner then Eqs. (9) yield radiation fields plus nonradiative terms, that is, in general E~IR = Erad+ nonradiative terms and Bl JR = B,,j + nonradiative terms. With the source T(x, t) = 7(t) S{x - r(t)}, Eqs. (8) yield the electric and magnetic fields of an arbitrarily moving dipole possessing a time-varying toroidal moment, 3n(n R2(1-n.@)c2 -=j-f{x 87r dr r(t)} - ~$S{X ltX7 R2(1 -n *7)-7 1 [ 1 [ ret d3 dt3 I1x (nx7) R(l-n./?)c3 1 ret (104 -r(r)}, nxr ./3)c2 Etf dt3 R(1 - n~fi)c3 d3 _- +47rVS{x -r(t)} x 7, (lob) where now n = (X - r (t’) )/IX - r( t’) ( and the square brackets with the subscript “ret” means that the bracketed with R = Ix - r( t’) I. Although Eqs. (10) quantity is to be evaluated at the retarded time t’ = t - R(t’)/c have a relatively simple form, they do not exhibit explicitly the useful separation of the fields into their l/R, /.A. Hems/Physics Letters A 249 (1998) 1-9 5 1/R2, . . . parts. Such a separation of the fields, however, can be accomplished by performing all the specified time derivatives in Eqs. (10). This task, although straightforward, is extremely laborious. It involves long and complicated vector manipulations and the full expressions obtained for the fields turn out to be very lengthy. By performing the specified derivatives in Eqs. (10) and making dT/dt = 0 and K = 1 - n - /?, one obtains E=- {3n(n.7) -T} [ {:s(X&)+~~(&)} 3 dn +cdix fnx (lla) B= 1 d3n 1 >gRK+Fp 1 d2n (1 lb) are given by [ 111 where the explicit time-derivatives 1 dn --_= c dt 1 d2n --= c= dt2 n x (n x p) RK ’ nx{(n-P)XP}+nx(nx8)(1-P2) RK3c ld 1 4 -c dt ( RK ) =z+ n-P =jgg+ Id 1 _ n*P -_ c dt ( R3K ) R3 K3c ld= 1 -= 3(n.P)= c= dt* ( RK > RK5c2 Pm (n-P> R=K3 P.(n-P> R3K3 I P*(n-P> pK3 + n-i) - n(nx/?)2_nx(nxp) R=K= R2K3 ’ ’ n-/3 +m’ +2n.p R4K2 ’ + 2{2P* RK4c2 (n -P) - (n x f02}(n.j?) R2KSc +{nX(nX8)-2P+n}.B+P.(n-P){28.(n-8)-(nxP)*} R* K4c R=K R3K5 _ (n x p>= R3K4 (120 J.A. Heras/Physics Letters A 249 (1998) l-9 3(?24)2 R2 K5c2 + {n x (n xp) + r2.S -2P+n}.p+3(n.p)(n.b) R3 K4c + P*(n-P>{W*(n-PI -(n R4K5 The third time derivatives 1 d3n --_=__ c3 dt3 1 d2 212P. R2K4c2 + (n - P) - (n X B2}(n R3 K=c + .i?) n-b R3 K3c x P12} _ (n x/3>2+3n./3{/3.(n-/3)} R4K4 (1%) in Eqs. ( 11) are much lengthier. They can be written as i, nx c2 dt2 ( RK > (nxP>l ,f~(~)[(n.P)t~+n(~~.B)+n(n.~~)+~~] ) ( 13a) . ( 13b) The explicit form of Eqs. ( 13) follows from using Eqs. ( 12a)-( 12g) and 1dP b c dr Kc’ --=a 1 d2/3 -----_c2 dt2 ldp --=c dt (14a) p K2c2 jj Kc’ + (14b) ( 14c) (14d) J.A. Heras/Physics 3n.p 7 Letters A 249 (1998) 1-9 2(nxP)2-&(n-j3) t R2K5c (nxB2-2P.(n-P) R3K5 t 14f) 2(n x p)’ =-- RK4c 8(n.B)2 K6c3 + 14e) (14g) ’ 16(n x b)2(n. + 2(n - S) K5C’- 2nx cnx P) *P-d(n X p> RK5c2 p) RK6c’ * (n X p) + 8(n x /3)4 _ 6(n x P)2(n R2K6c R2K5c .p) ( 14h) In Eqs. ( 11) and ( 13) the notation [ dF/dt],, means dF( t’) /dt and not dF( t’)/dt’, that is, the “ret” outside the square brackets applies to the arguments of the functions inside and not to the variable of differentiation. Eqs. ( 11) are the electric and magnetic fields of an arbitrarily moving particle possessing a constant toroidal moment. A detailed interpretation of these fields is complicated. However, there are some points that are relevant for an interpretation of these equation: (i) Very neal; near; intermediate and fur jields. When Eqs. (12)-( 14) are used in Eqs. ( 1 1 ), the fields separate naturally into five parts: the very near fields Every near and B,, nex, which vary as 1/R4 and depend on p and r; the near fields E,,, and B,,,, which vary as l/R3 and depend on j3, b and r; the intermediate fields Eint and Bi,t, which vary as I/R2 and depend on p, fi, $ and r; the far fields Ef, and Bf,, which vary as 1/R and depend on /?, /% j), b and r; and the delta-fields source. Thus, the complete fields read E = Eve, near + Enem+ Einr+ Far + EM B = &ry mill + E de1and Bdel which represent the fields within the 3 t &em+ Bint+ Bfx + Bdel. 15a) (1%) It should be emphasized that the delta fields E &I = -(8rrr/3c)d[6{x - r(t)}]/dt and &jet = 4rrVS{x r(t)} x T are evaluated at the field point and the present time; they are essential for achieving the consistency of Eqs. ( 11) with the Maxwell equations (2). The novelty in Eqs. (15) is the presence of the very near fields, which are not present in the fields of electric and magnetic dipoles. (ii) Static limit. When the velocity and its derivatives are zero, that is, when the toroidal dipole is at absolute rest, Eq. ( 1 la) becomes E = 0 and Eq. (1 lb) reduces to the static form: B = 47rV6{x - x0} x 7, where x0 is the point where the dipole is at rest. As may be seen, the field B is completely confined to a point. (iii) Uniform motion. It follows from Eqs. ( 11) that the fields of a toroidal dipole in uniform motion (/I = B = 2 = 0) exhibit an exceedingly complicated form. The electric field is given by the l/R” part of Eq. ( 1 la) plus the associated contact term and the magnetic field by the l/R” part of Eq. ( 1 lb) plus the delta term, that is, E = Every nem + E&l and B = Bvery near+ Bdel. (iv) RudiutionJields. The far fields derived from Eqs. ( I 1), ( 16b) are radiation fields. These fields depend on j?,&& to assume p = 0 and B = 0 at least instantaneously. and p, which are in general independent. In this case Eqs. ( 16) reduce to This allows one J.A. Heras/Physics Letters A 249 (1998) 1-9 ... = [nlret x &ad. &ad 1 ret ’ (174 Therefore, even when both the velocity and the acceleration of a toroidal dipole are instantaneously equal to zero (at the retarded time), the dipole can still produce a radiation field on account of the second derivative of its acceleration. 6. Radiated power by a toroidal dipole The radiated power P( t’) expressed dP(t’) dLI IREI [ 1 - = ; in terms of the dipole’s own time is given by n * j?],, (18) . By using Eq. ( 16a), one obtains .. dP( t’) -= da A@ [g (:2y$j3 - (n. T)2] + 3oqy)4;;,;B) [ ... + 2O(n-jQ(n.i))(n./Lq + 30(?24)3(n4) (1 -n.P)‘O (I -n.j3)” In order to find the total radiated power P(t’) ... (n*P12 + (1 --n*P)9 + 1oo~;wm&~)2 II (19) *et’ at a fixed time t’, it is necessary to specify the direction of the vectors 7, p,p, p and j. The simplest example is one in which the vectors /3,& fi and p are parallel. Thus, consider a dipole with toroidal moment in direction of the z-axis and moving along this axis. Therefore, p = tp, jj = e+, fl = ijj, n = f(sin8cos~$) P(t’) jj = ij + j(sinesin+) 225r2 = y& i = p” s an d r = ir. With these specific values and with dR = sin 8 dr9d4 + tcos0, Eq. (19) is first integrated over 4, sin3 8 co@ 8 delret+ y )J delret 1 sin3 ec0s4e (I -pcose)l* K ...2 ,;",";;deo + de,ret+ 0 77 J ret 0 j j (;$;;;;;,2 0 0 5opD2+15& lo+ [$bJ (1 -pcos8)'" 7r + g and 6 sin3 ecos2e (1 -pcosep 0 de 1 (20) ret The integrals over 0 are computed directly and the resulting expressions can be written in terms of powers of y= (1 -p2>- ‘/2 . After some laborious calculation one obtains an expression for the total power radiated by a moving toroidal dipole when the velocity and its derivatives are parallel, r2 lop6 PC+>= -$ 77 (26880~~~ - 64512~~’ + 52416~‘~ - 16016~‘~ + 1287~‘~) [ + 16O@fi 231 (4032~~’ - 6496~‘~ + 2926~‘~ - 297~‘~) ... (896~‘~ - 1624~‘~ + 836~‘~ - 99~‘~) > J.A. Heras/Physics . .. $ WPPP I-9 . ..2 ..’ (56~‘~ - 56~‘~ + 9~‘~) + $ 63 Letter.7 A 249 (1998) (120~‘~ - 140$* + 27~“) 9 . 1ret (21) Finally, consider the low velocity limit of Eq. (21). The approximation /3 < 1 implies y M 1 and thereby /3y = ,/m M 0. This approximation also implies that the effect of retardation becomes unimportant [ 121. ... Therefore, by writing b = a/c, B = h/c and P = a/c, Eq. (21) for low velocities reduces to the expression p=- 50r2a6 7c” which 40?a2b2 + 7c9 has been previously 1272a3ii + 7c9 272ii2 ~ + 15c7 ’ (22) derived in Ref. [7]. Acknowledgement The author thanks Dr. Octavia Novaro for his valuable support and Dr. Karo Michaelian of the manuscript. for a careful reading References ] I] V.M. Dubovik, A.A. Cheshkov, Sov. Phys. JETP 24 ( 1967) 924. (2 1 V.M. Dubovik, L.A. Tosunyan, Sov. Part. Nucl. 14 ( 1983) 504. [ 31 V.M. Dubovik. V.V. Tugushev, Phys. Rep. 187 (1990) 147. 14 I V.L. Ginzburg, V.N. Tsytovich, Sov. Phys. JETP 61 (1985) 48. ]S] V.M. Dubovik, S.V. Shabanov in: Essays on the Formal Aspects of Electromagnetic Theory, A. Lakhtakia, ed. (World Scientific, Singapore, 1993) pp. 399474. ]6] G.N. Afanassiev, Y.P. Stepanovsky, J. Phys. A 28 (1995) 4565. ]7] J.A. Hems, Phys. Lea. A 237 ( 1998) 343. [S] E.E. Radescu, D.H. Vlad, Phys. Rev. E 57 (1998) 6030. [ 91 J.A. Hems, Am. J. Phys. 62 ( 1994). 525. [IO] J.A. Hems, Am. J. Phys. 63 (1995), 928. [ I I I J.A. Hems, Phys. Rev. E, to be published. [ 121L. Eiges, The Classical Electromagnetic Field (Dover, New York. 1972) p. 287.