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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 equa- tions 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 re- tarded 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 deri- vation 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 .
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