Multipole Radiation Fields from the Jefimenko Equation For

Multipole radiation fields from the Jefimenko equation for the magnetic field and the Panofsky-Phillips equation for the electric field ͒ R. de Melo e Souza,a M. V. Cougo-Pinto, and C. Farina Universidade Federal do Rio de Janeiro, Instituto de Fisica, Rio de Janeiro, RJ 21945-970, Brazil M. Moriconi Universidade Federal Fluminense, Instituto de Fisica, Niteroi, RJ 24210-340, Brazil ͑Received 12 February 2008; accepted 8 September 2008͒ We show how to obtain the first multipole contributions to the electromagnetic radiation emitted by an arbitrary localized source directly from the Jefimenko equation for the magnetic field and the Panofsky-Phillips equation for the electric field. This procedure avoids the unnecessary calculation of the electromagnetic potentials. © 2009 American Association of Physics Teachers. ͓DOI: 10.1119/1.2990666͔

I. INTRODUCTION tion only in the temporal coordinate. We shall see that this method, although longer than the previous one, avoids any The derivation found in most textbooks of the electromag- possibility of misleading manipulations with retarded quan- netic fields generated by arbitrary sources in vacuum starts tities. The subtleties in calculations involving retarded quan- by calculating the corresponding electromagnetic potentials tities have been discussed in Refs. 13–15. We then present an ͑ ͒ see, for instance, Ref. 1 and 2,or3 . After the retarded alternative method that is a variation of the first one; the ͑ ͒ potentials are obtained assuming the Lorenz gauge the main difference is the order of integration. electromagnetic fields are calculated with the aid of the rela- -١ϫA ͑we use Gauss=t͒ and Bץ/Aץ١⌽−͑1/c͒͑−=tions E ian units͒. The resulting expressions for the fields are usually called Jefimenko’s equations because they appeared for the A. Direct calculation using the retarded Green function first time in the textbook by Jefimenko.4 Jefimenko’s equations are obtained in Ref. 1 from the retarded potentials and The following approach is similar to that in Refs. 5, 6, and are obtained directly from Maxwell’s equations in Ref. 5. 13. Maxwell’s equations with sources in vacuum are given ͑Heras6 had already derived Jefimenko’s equations directly by ͒ E =4␲␳, ͑1͒ · ١ from Maxwell’s equations. References 8 and 7 obtain a less common form of Jefimenko’s equations for the electric field, but this form is more convenient for studying radiation. B =0, ͑2͒ · ١ Griffiths and Heald9 illustrate Jefimenko’s equations by obtaining the standard Liénard-Wiechert fields for a point Bץ 1 10 ϫ E =− , ͑3͒ ١ -charge. Ton provides an alternative derivation of Jefimen tץ ko’s equations and three applications, including that of a c point charge in arbitrary motion. Heras has generalized Jefi- Eץ menko’s equations to include magnetic monopoles and ob- 4␲ 1 ϫ B = J + . ͑4͒ ١ tץ tained the electric and magnetic fields of a particle with both c c electric and magnetic charge in arbitrary motion.11 He has also discussed Jefimenko’s equations in material media to If we take the curl of Eq. ͑3͒ and the time derivative of Eq. obtain the electric and magnetic fields of a dipole in arbitrary ͑4͒, we obtain motion12 and has derived Jefimenko’s equations from Max- Jץ 1 2ץ 1 ١2 − ͪE =4␲ͩ١␳ + ͪ, ͑5͒ͩ well’s equations using the retarded Green function of the tץ t2 c2ץ wave equation.6 c2 The main purpose of this paper is to enlarge the list of ١2 ١2 ͒ ١͑١ ͒ ١ϫ͑١ϫ problems that are solved directly from Jefimenko’s equations where we have used F = ·F − F, with F ͒ ١2͑ ͒ ١2͑ ͒ ١2͑ ͑or the equivalent͒. Our procedure avoids completely the use = Fx xˆ + Fy yˆ + Fz zˆ with F an arbitrary function. of electromagnetic potentials. Specifically, we shall obtain Similarly, we obtain for the magnetic field 4␲ 2ץ the electric dipole, the magnetic dipole, and the electric 1 ͒ ͑ ͪ ١2ͩ -quadrupole terms of the multipole expansion due to the ra − 2 2 B =− J. 6 t cץ diation fields of an arbitrary localized source. c The solutions of Eqs. ͑5͒ and ͑6͒ can be obtained with the aid ͑ ͒ II. JEFIMENKO’S EQUATIONS FROM MAXWELL’S of the retarded Green function Gret x,t;xЈ,tЈ , which satis- EQUATIONS fies the inhomogeneous differential equation 2ץ In this section we present three methods of calculating 1 G ͑x,t;xЈ,tЈ͒ = ␦͑x − xЈ͒␦͑t − tЈ͒, ͑7͒ͪ − ١2ͩ t2 retץ Jefimenko’s equations directly from Maxwell’s equations. c2 The first method closely follows Ref. 5. The second method Ͻ Ͼ makes use of a Fourier transformation as discussed by Ref. and is zero for t−tЈ 0. The solution for Gret for t−tЈ 0is 16. For our purposes it suffices to do a Fourier transforma- given by3

67 Am. J. Phys. 77 ͑1͒, January 2009 http://aapt.org/ajp © 2009 American Association of Physics Teachers 67 1 ␦͑tЈ − ͑t − ͉x − xЈ͉/c͒͒ 1 ͓J͔ 1 1 ͪ ١Јͩ G ͑x,t;xЈ,tЈ͒ =− B͑x,t͒ = ͵ dxЈ١Ј ϫ ͩ ͪ + ͵ dxЈ͓J͔ ϫ ret 4␲ ͉x − xЈ͉ c R c R 1 ␦͑tЈ − ͑t − R/c͒͒ ˆ ϫ ͓˙͔ =− , ͑8͒ 1 R J ␲ − ͵ dxЈ . ͑15͒ 4 R c2 R where Rϵ͉R͉=͉x−xЈ͉. With the help of Eq. ͑8͒ the solution The first term on the right-hand side of Eq. ͑15͒ vanishes of Eq. ͑5͒ can be written as because the current distribution is localized in space. We use ١Ј͑1/R͒=−Rˆ /R2 to obtain dxЈ the relation E͑x,t͒ =−͵ ͵ dtЈ␦͑tЈ − ͑t − R/c͒͒ R ͓J͔ ϫ Rˆ ͓J˙͔ ϫ Rˆ B͑x,t͒ = ͵ dxЈͫ + ͬ. ͑16͒ J͑xЈ,tЈ͒ 2 2ץ 1 ϫͩ١Ј␳͑xЈ,tЈ͒ + ͪ ͑9͒ cR c R tЈץ c2 The first term on the right-hand side of Eq. ͑15͒ is the retarded Biot-Savart term; the transverse radiation field is con- -١Ј␳͔ ͓J˙͔ tained in the last term. Equations ͑12͒ and ͑16͒ are the Jefi͓ =− ͵ dxЈ − ͵ dxЈ , ͑10͒ R c2R menko equations. ͓ ͔ where the notation ¯ means that the quantity inside the brackets is a function of xЈ and is evaluated at the retarded B. Fourier method time tЈ=t−͉x−xЈ͉/c. -١Ј͓␳͔ In Sec. II A we showed how to obtain Jefimenko’s equa At this point much care must be taken, because ١Ј␳͔. Due to a bad choice of notation, the author in Ref. tions by a careful treatment of the derivatives of retarded͓ ١Ј͓␳͔ quantities. This point is crucial—spatial derivatives cannot ١Ј␳͔ as if it were͓ incorrectly used the quantity 13 and, after an integration by parts, an incomplete result for the be commuted with retarding the functions, because the re- electric field was found, as pointed out in Ref. 14. The cor- tarded function depends on the coordinates in its time argu- ment. A simple way to circumvent this difficulty is to use ͑ /␳͔ ˆ ͓␳͔ ١͓ ␳͔͓ ١ rect relation is given by Ј = Ј +R ˙ c see, for in- Fourier transforms and factor out the time dependence of the stance, Ref. 5͒. If we use the correct relation, Eq. ͑10͒ be- functions so these subtleties are not encountered. We will comes show how to obtain the electric field and will leave the derivation of the magnetic field as an exercise for the interested ͒ ͑ ͔˙١Ј͓␳͔ ͓␳˙͔Rˆ ͓J E͑x,t͒ =−͵ dxЈ + ͵ dxЈ − ͵ dxЈ ͑11͒ reader see, for example, Ref. 16 . R cR c2R We start with the electric field given by Eq. ͑9͒. The term involving the current density takes the form after integration ͓␳͔Rˆ ͓␳˙͔Rˆ ͓J˙͔ over time, =͵ dxЈ + ͵ dxЈ − ͵ dxЈ . ͑12͒ Jץ R2 cR c2R dxЈ 1 − ͵ ͵ dtЈ␦͑tЈ − ͑t − R/c͒͒ ͑xЈ,tЈ͒ tЈץ R c2 In the last step we integrated by parts and discarded surface terms, because the charge distribution is localized. Equation ͓J˙͔ ͑ ͒ 4,5 ͵ Ј ͑ ͒ 12 is one of the Jefimenko’s equations. Note the retarded =− dx 2 . 17 character of the electric field. The first term on the right-hand c R ͑ ͒ side of Eq. 12 is the retarded Coulomb term. We introduce the Fourier transformation ˜␳͑xЈ,␻͒ of ␳͑xЈ,tЈ͒ As shown by Panofsky and Phillips,7 there is an equivalent ␳͑ Ј Ј͒ ͐˜␳͑ Ј ␻͒ −i␻tЈ ␻ way of deriving Eq. ͑12͒ for the electric field which mani- as x ,t = x , e d and express the remaining ͑ ͒ festly shows the transverse character of the radiation field. A contribution to the electric field in Eq. 10 as discussion of this point can be found in Ref. 8; we will dxЈ discuss this point in Sec. III. − ͵ ͵ dtЈ␦͑tЈ − ͑t − R/c͒͒١Ј␳͑xЈ,tЈ͒ To obtain the desired expression for the magnetic field, we R use the retarded Green function in Eq. ͑8͒ to rewrite Eq. ͑6͒ dxЈ as =−͵ ͵ dtЈ␦͑tЈ − ͑t − R/c͒͒ R 1 dxЈ ١Ј ϫ ͑ Ј Ј͒ ␻͒͒ / ͑ Ј␦͑ Ј ͵ ͵ ͒ ͑ B x,t = dt t − t − R c J x ,t −i tЈ c R ϫ ͵ d␻١Ј˜␳͑xЈ,␻͒e ͑18a͒ 1 dxЈ ͒ ͑ ͔ ١Ј ϫ͓ = ͵ J . 13 eikR ١Ј˜␳͑xЈ,␻͒͑18b͒ c R =− ͵ d␻e−i␻t ͵ dxЈ R ١ЈϫJ͔+͑Rˆ /c͒ϫ͓J˙͔͑see͓=١Јϫ͓J͔ If we use the relation ͒ ͑ ͒ Ref. 5 , Eq. 13 takes the form eikR ␻ −i␻t Јͭ١Ј ␳͑ Ј ␻͒ =− ͵ d e ͵ dx ͩ ˜ x , ͪ 1 dxЈ 1 Rˆ ϫ ͓J˙͔ R ͒ ͑ ١Ј ϫ ͓ ͔ ͵ Ј ͵ ͒ ͑ B x,t = J − 2 dx . 14 c R c R Rˆ Rˆ ␳͑ Ј ␻͒ͩ ͪ ikRͮ ͑ ͒ − ˜ x , − ik e 18c We integrate by parts and obtain R2 R

68 Am. J. Phys., Vol. 77, No. 1, January 2009 de Melo e Souza et al. 68 Rˆ III. MULTIPOLE RADIATION =͵ dxЈ ͵ d␻␳˜͑xЈ,␻͒e−i␻͑t−R/c͒ VIA JEFIMENKO’S EQUATIONS R2 The main purpose of this section is to add to the list of Rˆ + ͵ dxЈ ͵ d␻␳˜͑xЈ,␻͒͑− i␻͒e−i␻͑t−R/c͒ ͑18d͒ problems that can be handled directly with Jefimenko’s equa- cR tions by calculating the first multipole contributions to the radiation fields of an arbitrary localized source. We shall obtain the first three contributions, namely, the electric dipole, Rˆ ͓␳͔ Rˆ ͓␳˙͔ =͵ dxЈ + ͵ dxЈ . ͑18e͒ the magnetic dipole, and the electric quadrupole terms. Most R2 cR textbooks treat this problem by calculating first the electromagnetic potentials. ͑ ͒ To obtain Eq. 18c we integrated by parts; to obtain Eq. Although Eq. ͑12͒ for the electric field gives the correct ͑ ͒ 18d we discarded the surface term. expression for the electric field of moving charges, it is pref- ͑ ͒ ͑ ͒ ͑ ͒ We combine Eq. 9 with Eqs. 17 and 18 and obtain the erable for our purposes to write it in an equivalent form as ͑ ͒ electric field generated by arbitrary but localized sources, given in Refs. 7 and 8: namely, Eq. ͑12͒. On the right-hand side of Eq. ͑18a͒ it is ˆ١Ј acts only on ˜␳͑xЈ,␻͒. Hence, in Eq. ͑18b͒ ͓␳͔R obvious that E͑x,t͒ = ͵ dxЈ ١Ј also acts on the R2 there is no possibility of thinking that exponential eikR. The subtleties of dealing with retarded quantities are circumvented by the Fourier method. ͓͑J͔ · Rˆ ͒Rˆ + ͓͑J͔ ϫ Rˆ ͒ ϫ Rˆ + ͵ dxЈ cR2 ͓͑J˙͔ ϫ Rˆ ͒ ϫ Rˆ C. Postponing the delta-function integration + ͵ dxЈ . ͑22͒ c2R When we are faced with integrals involving delta func- We obtain Eq. ͑22͒ starting with Jefimenko’s equation for the tions, we are usually tempted to do them first, because they ͑ ͒ are so easy. To derive Jefimenko’s equation for the electric electric field in Eq. 12 . Our derivation will follow the one ͑ ͒ in Ref. 8. Note that ͑the Einstein convention of implicit sum- field given by Eq. 12 , this procedure is not optimum. Let us ͒ start with Eq. ͑9͒: mation over repeated indices is assumed ץ Ј ͑ Ј Ј͉͒ ͉ ͑ Ј Ј͉͒ץ͉ ͔ ͓ ١Ј dxЈ · J = i Ji x ,t tЈ=t−R/c + Ji x ,t tЈ=t−R/c tЈץ E͑x,t͒ =−͵ ͵ dtЈ␦͑tЈ − ͑t − R/c͒͒ R 1 ͒ ͑ ͪ Јץ ϫͩ J − i R 23aץ 1 ϫͩ١Ј␳͑xЈ,tЈ͒ + ͑xЈ,tЈ͒ͪ. ͑19͒ c tЈץ c2 X Rˆ ١Ј · J͔ + ͓J˙ ͔ͩ i ͪ =−͓␳˙͔ + ͓J˙͔ · , ͑23b͓͒= Instead of first performing the time integration using the Dirac delta function, we integrate by parts on both terms on i cR c the right-hand side of Eq. ͑19͒ so that it takes the form ϵ Ј where Xi xi −xi and in the last step we used the continuity ,١Ј·͓J͔+͓J˙͔·Rˆ /c−=͔˙equation. From Eq. ͑23b͒ we have ͓␳ E͑x,t͒ = ͵ dxЈ ͵ dtЈͭ␦Ј͑tЈ − ͑t − R/c͒͒ which can be substituted into the second term on the right- hand side of Eq. ͑12͒ to yield 1 1 ˆ ˆ ϫͩ ␳͑xЈ,tЈ͒١ЈR + J͑xЈ,tЈ͒ͪ ١Ј · ͓J͔͒R͑ Rc Rc2 1 ͓␳˙͔R 1 ͵ dxЈ = ͵ dxЈ 1 c R c R tЈ − ͑t − R/c͒͒ͩ١Ј ͪ␳͑xЈ,tЈ͒ͮ. ͑20͒͑␦ + R 1 ͓͑J˙͔ · Rˆ ͒Rˆ + ͵ dxЈ. ͑24͒ c2 R -١Ј1/R=Rˆ /R2,weob ١ЈR=−Rˆ and If we use the relations tain We show that the first term on the right-hand side of Eq. ͑24͒ is proportional to 1/R2, so that it does not contribute to the Rˆ radiation field. Observe that ͑ ͒ Ј Јͭ␦Ј͑ Ј ͑ / ͒͒ͩ ␳͑ Ј Ј͒ E x,t = ͵ dx ͵ dt t − t − R c − x ,t ˆ١Ј · ͓J͔͒R͑ Rc 1 − ͵ dxЈ 1 Rˆ c R ͑ Ј Ј͒ͪ ␦͑ Ј ͑ / ͒͒␳͑ Ј Ј͒ ͮ + J x ,t + t − t − R c x ,t . Rc2 R2 eˆ X Ј͓J ͔͒ k dxЈ ͑25a͒ץ͑ ͵ k −= ͑21͒ c i i R2

By using the properties of the Dirac delta function, we can eˆ X eˆ X Јͩ k ͪ Јץ͔ ͓ ͵ Ј͓ͩ ͔ k ͪ Ј kץ ͵ Ј ͑ ͒ k easily perform the integration over t to obtain Eq. 12 .An =− i Ji 2 dx + Ji i 2 dx analogous calculation provides an expression for the mag- c R c R netic ﬁeld. ͑25b͒

69 Am. J. Phys., Vol. 77, No. 1, January 2009 de Melo e Souza et al. 69 eˆ 2 X ␦ 1 xˆ · xЈ =0 + k ͵ ͓J ͔ͭX i − kiͮdxЈ ͑25c͒ E ͑x,t͒Ӎ ͵ dxЈͫJ˙ͩxЈ,t + ͪ ϫ xˆͬ ϫ xˆ , c i k R3 R R2 rad c2r 0 c ͑29͒ / 1 2͓͑J͔ · Rˆ ͒Rˆ − ͓J͔ where t0 =t−r c is the retarded time of the origin. A com- = ͵ dxЈ ͑25d͒ parison of Eqs. ͑28͒ and ͑37͒ leads to the relation c R2 ͑ ͒ ͑ ͒ ϫ ͑ ͒ Erad x,t = Brad x,t xˆ . 30 Now we are ready to calculate the first multipole contri- 1 ͓͑J͔ · Rˆ ͒Rˆ + ͓͑J͔ ϫ Rˆ ͒ ϫ Rˆ ͵ Ј ͑ ͒ butions for the radiation fields. We need to calculate only one = 2 dx , 25e c R of the radiation fields because the other is readily obtained by Eq. ͑30͒, which also shows that the fields in the radiation where we have used the identity ͑aϫb͒ϫc=͑a·c͒b zone are mutually orthogonal. We start by calculating the −͑b·c͒a, and the fact that the surface term appearing in Eq. electric dipole term and then consider the next order contri- ͑25b͒ vanishes because the sources are localized. We substi- bution given by both the magnetic dipole and electric quad- tute Eq. ͑25͒ into Eq. ͑24͒ and insert the result into Eq. ͑12͒ rupole terms. to obtain

͓␳͔Rˆ A. The electric dipole contribution E͑x,t͒ = ͵ dxЈ R2 The lowest order contribution to the radiation ﬁelds comes from the electric dipole term. For simplicity, we calculate the 1 ͓͑J͔ · Rˆ ͒Rˆ + ͓͑J͔ ϫ Rˆ ͒ ϫ Rˆ + ͵ dxЈ lowest order contribution to the radiation magnetic ﬁeld, c R2 ͑1͒ which we denote by Brad,

1 ͓͑J˙͔ · Rˆ ͒Rˆ 1 ͓J˙͔ ͑ ͒ 1 ͵ Ј ͵ Ј ͑ ͒ B 1 ͑x,t͒ = ͭ͵ dxЈJ˙͑xЈ,t ͒ͮ ϫ xˆ . ͑31͒ + dx − dx 26a rad 0 c2 R c2 R c2r ١Ј Ј We write the unit vectors of the Cartesian basis as eˆi = xi , ͑i=1,2,3͒, and write any vector v as v=eˆ v =eˆ ͑v·eˆ ͒. The ͓␳͔Rˆ i i i i =͵ dxЈ integral in Eq. ͑31͒ can be expressed as R2 ͵ Ј˙͑ Ј ͒ ͵ Ј˙͑ Ј ͒ ͓͑J͔ · Rˆ ͒Rˆ + ͓͑J͔ ϫ Rˆ ͒ ϫ Rˆ dx J x ,t0 = ei dx J x ,t0 · ei + ͵ dxЈ cR2 e ͵ dxЈJ˙͑xЈ,t ͒ · ٌЈxЈ ͑32a͒ = ͓͑J˙͔ ϫ Rˆ ͒ ϫ Rˆ i 0 i + ͵ dxЈ , ͑26b͒ c2R ͒͒ e ͵ dxЈ١Ј · ͑xЈJ˙͑xЈ,t= which is Eq. ͑22͒. If we consider arbitrary time-varying i i 0 sources at rest and use Eqs. ͑16͒ and ͑22͒, the ͑transverse͒ ͒ ͑ ͒ magnetic and electric radiation ﬁelds are given by ͵ Ј ЈٌЈ ˙͑ Ј − ei dx xi · J x ,t0 , 32b

1 ͓J˙͔ ϫ Rˆ where in the last step we integrated by parts. Because we are B ͑x,t͒ = ͵ dxЈ , rad c Rc considering localized sources, the first integral on the right- hand side of Eq. ͑32a͒ vanishes ͑after the use of Gauss’ ͑27͒ theorem this integral is converted to a zero surface term͒. ͓͑J˙͔ ϫ Rˆ ͒ ϫ Rˆ E ͑x,t͒ = ͵ dxЈ . The remaining integral may be cast into a convenient form if rad c2R we use the relation ͒ 2␳͑xЈ,tץ ١Ј · J˙͑xЈ,t ͒ =− 0 , ͑33͒ 17 2 ץ It can be shown that for time-varying sources in motion, 0 Eq. ͑27͒ gives the radiation fields plus additional nonradia- t tive terms of order O͑1/R2͒. For instance, if a Hertz dipole is which is a direct consequence of the continuity equation. To ץ/͒ ␳͑ץ ͒ ͑ ͒ ͑ accelerated, integration of Eq. 27 yields radiation fields obtain Eq. 33 we used the relation xЈ,t0 t ͒ ͑ ͒ ͑ ץ/͒ ␳͑ץ O͑ / 2͒ plus nonradiative terms of order 1 R which are induced = xЈ,t0 t0 see Ref. 18, note 5 . We substitute Eqs. 33 by the dipole motion ͑see Ref. 17 for details͒. and ͑32a͒ into Eq. ͑31͒ and obtain In the radiation zone we can write Rˆ Ӎxˆ,1/RӍ1/r, and RӍr−xˆ ·xЈ, where we defined r=͉x͉. If we substitute these approximations into Eq. ͑27͒, we obtain

1 xˆ · xЈ B ͑x,t͒Ӎ ͵ dxЈJ˙ͩxЈ,t + ͪ ϫ xˆ , ͑28͒ ͑34͒ rad c2r 0 c

70 Am. J. Phys., Vol. 77, No. 1, January 2009 de Melo e Souza et al. 70 ͑ ͒ where p t0 is the electric dipole moment of the distribution where we have identified the second time derivative of the at the retarded time t0. Now it is clear why this first term is magnetic dipole moment of the charge distribution at the ͑ ͒ called the electric dipole term. The radiation electric field is retarded time as m¨ t0 . Hence, the antisymmetric contribu- readily obtained from Eq. ͑30͒ tion for the radiation magnetic field is given by ͓ ͑ ͒ ϫ ͔ ϫ ͑1͒ p¨ t0 xˆ xˆ E ͑x,t͒ = . ͑35͒ ͑ ͒ ͓m¨ ͑t ͒ ϫ xˆ͔ ϫ xˆ rad c2r B 2a ͑x,t͒ = 0 . ͑40͒ rad c2r Equations ͑31͒ and ͑35͒ are the radiation fields of the electric dipole term, that is, the first-order contribution to the The appearance of the magnetic dipole moment of the distri- multipole expansion. These expressions are valid for arbi- bution justifies the name given to this contribution. The cor- trary but localized, sources in vacuum such as an oscillating responding radiation electric field is readily given by electric dipole. ͑ ͒ xˆ ϫ m¨ ͑t ͒ E 2a ͑x,t͒ = 0 . ͑41͒ B. Next order contribution rad c2r To calculate the next order term we need to take into ac- count the second term of the expansion Expressions ͑40͒ and ͑41͒ are valid for an arbitrary, but localized, time-varying source at rest in vacuum such as an xˆ · xЈ xˆ · xЈ J˙ͩxЈ,t + ͪ Ϸ J˙͑xЈ,t ͒ + J¨͑xЈ,t ͒. ͑36͒ oscillating magnetic dipole. 0 0 0 ͑ ͒ ͑2s͒ c c We denote the symmetric term of Eq. 38 as Brad .We have We substitute Eq. ͑36͒ into Eq. ͑28͒ and identify the next ͑ ͒ order contribution to the radiation magnetic field, B 2 , given rad ͑ ͒ 1 1 by B 2s ͑x,t͒ = ͵ dxЈ͕J¨͑xЈ,t ͒͑xˆ · xЈ͒ rad c2r 2c 0 1 ͑2͒͑ ͒ ͩ͵ Ј¨͑ Ј ͒͑ Ј͒ͪ ϫ ͑ ͒ Brad x,t = dx J x ,t0 xˆ · x xˆ . 37 ͑¨͑ Ј ͒ ͒ Ј͖ ϫ ͑ ͒ c3r + J x ,t0 · xˆ x xˆ . 42 For reasons that will become clear we will split the integral We manipulate the first integral on the right-hand side of Eq. in Eq. ͑37͒ into antisymmetric and symmetric contributions ͑42͒ in the same way as before. We have under the exchange of J and xЈ. This rearrangement will give rise to the magnetic dipole and electric quadrupole terms of ͵ Ј¨͑ Ј ͒͑ Ј͒ ͵ Ј͑ Ј͒¨͑ Ј ͒ ͑ ͒ the multipole expansion for the radiation fields. dx J x ,t0 xˆ · x = ei dx xˆ · x J x ,t0 · ei 43a ¨͑ Ј ͒͑ Ј͒ 1 ¨͑ Ј ͒͑ Ј͒ 1 ¨͑ Ј ͒ If we write J x ,t0 xˆ ·x as 2 J x ,t0 xˆ ·x + 2 J x ,t0 ϫ͑ Ј͒ 1 ͓¨͑ Ј ͒ ͔ Ј xˆ ·x and sum and subtract 2 J x ,t0 ·xˆ x in the inte- ͒ ͑ grand of the right-hand side of Eq. ͑37͒, we obtain ͵ Ј͑ Ј͒¨͑ Ј ͒ ٌЈ Ј =ei dx xˆ · x J x ,t0 · xi . 43b ͑ ͒ 1 1 B 2 ͑x,t͒ = ͭ ͵ dxЈ͓J¨͑xЈ,t ͒͑xˆ · xЈ͒ rad c2r 2c 0 If we integrate by parts and remember that the surface term vanishes, we obtain ͑¨͑ Ј ͒ ͒ Ј͔ͮ ϫ − J x ,t0 · xˆ x xˆ ͵ dxЈJ¨͑xЈ,t ͒͑xˆ · xЈ͒ 1 1 0 + ͭ ͵ dxЈ͓J¨͑xЈ,t ͒͑xˆ · xЈ͒ c2r 2c 0 ͒ ͑ ͔͒ Ј Ј١Ј ͓͑ Ј͒¨͑ Ј ͵ =−ei dx xi · xˆ · x J x ,t0 44a ͑¨͑ Ј ͒ ͒ Ј͔ͮ ϫ ͑ ͒ + J x ,t0 · xˆ x xˆ . 38

͒ For pedagogical reasons we shall treat the magnetic dipole =e ͵ dxЈxЈٌ͓͑Ј͑xˆ · xЈ͒͒ · J¨͑xЈ,t and electric quadrupole cases separately. i i 0 Consider the antisymmetric term of Eq. ͑38͒. We denote ͒ ͑ ͔͒ ͑¨ ٌ͒ ͑ 2a͒͑ ͑ ϫ ͒ϫ ͑ ͒ ͑ ͒ + xˆ · xЈ Ј · J xЈ,t0 . 44b this contribution by Brad , use a b c= c·a b− c·b a, and write it in the following suggestive way: t3, the relation18ץ/͒ 3␳͑xЈ,tץ−=͒ ١Ј·J¨͑xЈ,t We use 1 1 0 0 ١Ј͑xˆ ·xЈ͒=xˆ to obtain t ͒, andץ/͒ ␳͑xЈ,tץ=tץ/͒ ␳͑xЈ,tץ 2a͒͑ ͒ ͵ Ј͓¨͑ Ј ͒ Ј͑ B x,t = dx J x ,t xˆ · x 0 0 0 rad c2r 2c 0 ͑¨͑ Ј ͒ ͒ Ј͔ ϫ ͑ ͒ − J x ,t0 · xˆ x xˆ 39a ͵ Ј¨͑ Ј ͒͑ Ј͒ ͵ Ј ¨͑ Ј ͒ Ј dx J x ,t0 xˆ · x =−ei dx xˆ · J x ,t0 xi

3 ץ ͒ xЈ ϫ J¨͑xЈ t 1 ͩ , 0 ͪ + e ͵ dxЈ͑xˆ · xЈ͒xЈ␳͑xЈ,t ͒, ͑45͒ ͵ dxЈ ϫ xˆ ϫ xˆ i 3 i 0 ץ , ͮ ͭ 2 = c r 2c t

m͑t ͒ ¨ 0 ͑39b͒ which implies

71 Am. J. Phys., Vol. 77, No. 1, January 2009 de Melo e Souza et al. 71 given by Eqs. ͑40͒ and ͑41͒, are the first corrections to the ͵ Ј͓¨͑ Ј ͒͑ Ј͒ ͑¨͑ Ј ͒ ͒ Ј͔ dx J x ,t0 xˆ · x + J x ,t0 · xˆ x leading order term given by Eqs. ͑34͒ and ͑35͒. In this sense, we can write the first multipole contributions to the multipole ,expansion for the radiation fields of a completely arbitrary 3ץ = ͵ dxЈ͑xˆ · xЈ͒xЈ␳͑xЈ,t ͒. ͑46͒ t3 0 but localized, time-varying source at rest in vacuum asץ 1 The left-hand side of Eq. ͑46͒ is the integral in Eq. ͑42͒.We ͑ ͒ ͭ ͑ ͒ ϫ ͓ ͑ ͒ ϫ ͔ ϫ Brad x,t = 2 p¨ t0 xˆ + m¨ t0 xˆ xˆ write the factor 1/2 in Eq. ͑42͒ as 3/6 and obtain c r ᠮ 1 3ץ ͑2s͒ 1 ͑ ͒ ϫ ͮ ͑ ͒ B ͑x,t͒ = ͭ ͵ dxЈ͑3xˆ · xЈ͒xЈ␳͑xЈ,t ͒ͮ ϫ xˆ + Q xˆ,t0 xˆ + ¯ , 52 t3 0 3cץ rad 6c3r ͑ ͒ 47a ͑ ͒ ͑ ͒ ϫ ͑ ͒ Erad x,t = Brad x,t xˆ . 53 -With these radiation fields, we can calculate the correspond 3ץ 1 = ͭ ͵ dxЈ͑3͑xˆ · xЈ͒xЈ + rЈ2xˆ͒ .t3 ing Poynting vector and the power radiated by the sourceץ 6c3r ACKNOWLEDGMENTS ϫ␳͑ Ј ͒ͮ ϫ ͑ ͒ x ,t0 xˆ , 47b The authors are indebted to the referees for their valuable comments and suggestions. R.S.M. and C.F. would like to where in the last step we included in the integrand the term thank CNPq for partial financial support and M.M. would 2 ␳͑ ͒ ͉ ͉ rЈ xˆ xЈ,t0 , where rЈ= rЈ , which gives a vanishing contri- like to thank FAPERJ for financial support. bution to the result because xˆ ϫxˆ =0. Now, we define the transformation Q such that ͒ a Electronic mail: [email protected] 1 David J. Griffiths, Introduction to Electrodynamics, 3rd ed. ͑Prentice Q͑␰,t͒ = ͵ dxЈ͓3͑␰ · xЈ͒xЈ + rЈ2␰͔␳͑xЈ,t͒. ͑48͒ Hall, Englewood Cliffs, NJ, 1999͒, pp. 427–428. 2 Mark A. Heald and Jerry B. Marion, Classical Electromagnetic Radia- tion, 3rd ed. ͑Saunders College Publishing, New York, 1995͒. This transformation takes a vector and an instant of time 3 J. D. Jackson, Classical Electrodynamics, 2nd ed. ͑Wiley, New York, ͑␰,t͒ and transforms it into the vector Q͑␰,t͒, given by Eq. 1975͒, Chap. 9. ͑48͒. If we use the definition ͑48͒, Eq. ͑47a͒ for the antisym- 4 Oleg D. Jefimenko, Electricity and Magnetism ͑Appleton-Century-Crofts, ͑ ͒ ͒ metric contribution for B 2 becomes New York, 1996 , Sec. 15.7. rad 5 J. D. Jackson, Classical Electrodynamics, 3rd ed. ͑Wiley, New York, 1 1999͒, Chap. 9. ͑2s͒͑ ͒ ត ͑ ͒ ϫ ͑ ͒ 6 Brad x,t = 3 Q xˆ,t0 xˆ . 49 J. A. Heras, “Time-dependent generalizations of the Biot-Savart and Cou- 6c r lomb laws: A formal derivation,” Am. J. Phys. 63, 928–932 ͑1995͒. 7 W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism The corresponding symmetric contribution for the radiating ͑Addison-Wesley, Reading, MA, 1962͒, Sec. 14.3. electric field is obtained from Eq. ͑30͒: 8 Kirk T. McDonald, “The relation between expressions for time-dependent 1 electromagnetic fields given by Jefimenko and by Panofsky and Phillips,” ͑2s͒͑ ͒ ͓ ត ͑ ͒ ϫ ͔ ϫ ͑ ͒ Am. J. Phys. 65, 1074–1076 ͑1997͒. E x,t = Q xˆ,t0 xˆ xˆ . 50 rad 6c3r 9 David J. Griffiths and Mark A. Heald, “Time-dependent generalizations of the Biot-Savart and Coulomb laws,” Am. J. Phys. 59, 111–117 ͑1991͒. In order to interpret the above term, let us define a linear 10 Tran-Cong Ton, “On the time-dependent, generalized Coulomb and Biot- operator Qt, for a fixed instant t,byQt͑␰͒ϵQ͑␰,t͒. Note Savart laws,” Am. J. Phys. 59, 520–528 ͑1991͒. t t͑␣ ␰ ␣ ␰ ͒ 11 J. A. Heras, “Jefimenko’s formulas with magnetic monopoles and the that Q is a linear operator so that Q 1 1 + 2 2 ␣ t͑␰ ͒ ␣ t͑␰ ͒ ␣ ␣ ෈R t Lienard-Wiechert fields of a dual-charged particle,” Am. J. Phys. 62, = 1Q 1 + 2Q 2 , for 1, 2 . The linear operator Q 525–531 ͑1994͒. t͑ ͒ 12 is called the electric quadrupole operator. Because Q eî is a J. A. Heras, “Radiation fields of a dipole in arbitrary motion,” Am. J. R3 Phys. 62, 1109–1115 ͑1994͒. vector in , we can write it as a linear combination of the 13 basis vectors, namely F. Rohrlich, “Causality, the Coulomb field, and Newton’s law of gravitation,” Am. J. Phys. 70, 411–414 ͑2002͒. 3 14 Oleg D. Jefimenko, “Comment on ‘Causality, the Coulomb field, and t͑ ͒ t ͑ ͒ ͑ ͒ Newton’s law of gravitation,’” Am. J. Phys. 70, 964–964 ͑2002͒. Q eî = ͚ Qjieˆ j i = 1,2,3 . 51 15 j=1 F. Rohrlich, “Reply to Comment on ‘Causality, the Coulomb field, and Newton’s law of gravitation,’” Am. J. Phys. 70, 964–964 ͑2002͒. t 16 M. Bornatici and U. Bellotti, “Time-dependent, generalized Coulomb and The coefficients Qji are the Cartesian elements of the electric quadrupole tensor ͑a second rank tensor͒ of the distribution Biot-Savart laws: A derivation based on Fourier transforms,” Am. J. Phys. 64, 568–570 ͑1996͒. at instant t. That is why we interpret Eqs. ͑49͒ and ͑50͒ as the 17 J. A. Heras, “New approach to the classical radiation fields of moving electric quadrupole contribution for the radiation fields. dipoles,” Phys. Lett. A 237, 343–348 ͑1998͒. The electric quadrupole radiation fields given by Eqs. ͑49͒ 18 J. A. Heras, “Can Maxwell’s equations be obtained from the continuity and ͑50͒, together with the magnetic dipole radiation fields equation?,” Am. J. Phys. 75, 652–657 ͑2007͒.

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