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Radiation of the Electromagnetic Field Beyond the Dipole Approximation

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Radiation of the Electromagnetic Field Beyond the Dipole Approximation Radiation of the electromagnetic field beyond the dipole approximation Andrij Rovenchak and Yuri Krynytskyi Department for Theoretical Physics, Ivan Franko National University of Lviv, Ukraine July 9, 2018 Abstract An expression for the intensity of the electromagnetic field radia- tion is derived up to the order next to the dipole approximation. Our approach is based on the fundamental equations from the introduc- tory course of classical electrodynamics and the derivation is carried out using straightforward mathematical transformations. Key words: multipole expansion; radiation; higher multipole mo- ments; anapole; toroidicity. 1 Introduction Multipole expansion is a well-known technique applied in calculations of electromagnetic [1, 2] and gravitational fields [3, 4] at large distances from sources. In stationary cases, such as electrostatic problems, this approach is mainly linked to simple series expansions and certain sym- metrization procedures [5, 6]. Treatment of magnetostatic problems requires more elaborate techniques beyond the lowest order [7]. How- ever, it becomes more tricky in the case of the electromagnetic field radiation. Moreover, in textbooks on classical electrodynamics one arXiv:1803.07889v2 [physics.class-ph] 20 Sep 2018 of the terms beyond the dipole approximation is typically omitted in expressions for the radiated power even if the detailed derivations are given (see, e.g., [8, 9, 10]; for an exception see [11]). The goal of this article is to derive an expression for the radiated power in the approximation next to the dipole one using the basic 1 equations of electrodynamics and the techniques from vector and ten- sor calculus. From the methodological point of view, our approach is advantageous comparing to the derivations based on gauge symmetries [12] or solutions to the scattering problem [13]. Indeed, our deriva- tion only requires knowledge of the fundamental relations from the introductory course of classical electrodynamics and involves straight- forward mathematical transformations. In addition, the simplicity of our approach allows one to obtaining a correction to the dipole ra- diation sufficient for any practical purposes; more general derivations might be found in [7, 14, 15, 16, 17]. The rest of the paper is organized as follows. An expression for the radiated power via the Poynting vector and magnetic field is obtained in Section 2. The multipole expansion of the radiative part of the vector potential is given in Section 3. The detailed derivations of each contribution into the radiated power are presented in Section 4. Brief discussion of the results is given in Section 5. 2 Poynting vector and radiated power In a region of the space without charges and currents, the energy conservation law for the electromagnetic field in the differential form is @w + div S = 0; (1) @t where w is the energy density (we use the Gaussian units) E2 + B2 w = ; (2) 8π and S the energy flux density (the Poynting vector) c S = E × B: (3) 4π Applying Gauss's theorem to Eq. (1), the radiated power dW d Z I = − ≡ w dV; dt dt can be reduced to the surface integral of the Poynting vector: I Z I = S · dΣ = jSjr2 dΩ; (4) Σ Ω=4π 2 where on the r.h.s the integration is performed over the complete solid angle. From this expression, one can conclude that only those fields contribute to the radiation, which ensure the dependence jSj / 1=r2 as r ! 1. In the far zone, the magnetic and electric fields can be written as the sums E = E0 + E1; B = B0 + B1; (5) where the second terms (E1 and B1) are proportional to 1=r and thus correspond to the radiation part of the field. Since the electric (E1) and magnetic (B1) field vectors are orthogonal and equal by magnitude, it is sufficient to consider the norm of the Poynting vector for the radiation fields c c 2 jS1j = E1 × B1 = jB1j : (6) 4π 4π Then the radiated power is given by c Z I = jB j2r2 dΩ: (7) 4π 1 Ω=4π The magnetic field is B(r; t) = rot A(r; t); (8) where A(r; t) is vector potential, which we calculate in the next Sec- tion. 3 Vector potential We consider a system of electric charges in a volume V near the origin and assure that the observation point r is far enough: jrj ≡ r V 1=3, see Fig. 1. The vector potential of such a system in the Lorenz gauge is given by (cf. [8, p. 408]): 1 Z 1 1 A(r; t) = dV 0 j r0; t − jr − r0j ; (9) c jr − r0j c V 3 Figure 1: The system of charges is located near the origin and a distant observer sits at point r. where dV 0 ≡ dx0 dy0 dz0, and we have taken into account the retarda- tion effects. We write for r0 r the following approximations over r0=r: 1 1 ' ; (10) jr − r0j r r0 · r jr − r0j ' r − r0 · rr = r − ; (11) r or, using the unit vector n = r=r, jr − r0j ' r − n · r0: (12) Further terms bringing higher powers of r in the denominators can be neglected when considering the radiation part of the field. Within this approximation, the vector potential yields 1 Z r 1 A(r; t) = dV 0 j r0; t − + n · r0 : (13) cr c c V 1 0 It can be expanded in c n · r using the Taylor series as follows: 1 Z r d 1 Z r A(r; t) = dV 0 j r0; t − + dV 0 (n · r0) j r0; t − + cr c dt c2r c V V d2 1 Z r + dV 0 (n · r0)2 j r0; t − + :::: (14) dt2 2c3r c V The current density for a system of point changes ei at ri moving with velocities vi = r_ i can be written using the Dirac delta-function 4 as follows: X j(r; τ) = eivi(τ)δ r − ri(τ) ; (15) i where the retarded time τ = t − r=c was introduced for convenience. The first term in expansion (14) easily simplifies to the time deriva- tive of the dipole moment: Z r X d X d dV 0 j r0; t − = e v (τ) = e r (τ) = d(τ) = d_ (τ): c i i dτ i i dτ V i i (16) Thus, we have in the leading order d_ t − r A(r; t) = c + :::: (17) cr To obtain the next-order correction, we make the following trans- formations: Z r X Z dV 0 (n · r0) j r0; t − = e v (τ) dV 0 n · r0 δr0 − r (τ) = c i i i V i V X X dri = e v (n · r ) = e (n · r ) = i i i i dτ i i i 1 d X 1 X n o = e r (n · r ) + e v (n · r ) − r (n · v ) = 2 dτ i i i 2 i i i i i i i 1 d X 1 X = e r (n · r ) + e n × (v × r ): (18) 2 dτ i i i 2 i i i i i As we will see further, in the radiation parts of the field, the vec- tor potential appears only as cross product with the unit vector n. Thus, one can add to A an arbitrary vector proportional to n without changing the results. It is convenient to add to the first term of (18): 1 d X 1 d X r2 e r (n · r ) ! e r (n · r ) − n i : 2 dτ i i i 2 dτ i i i 3 i i We will further work with the vector potential A shifted as described above. The sum over i is a contraction of the electric quadrupole moment tensor 2 X (i) (i) r Q = e x x − i δ jk i j k 3 jk i 5 with the unit vector n, yielding some vector D with components X Dj = Qjknk: (19) k The remaining part of this correction involves the cross product of the magnetic dipole moment 1 X m = e r × v (20) 2c i i i i with the unit vector n. The last term, which is written explicitly in (14), can be trans- formed as follows: d2 1 Z r dV 0 (n · r0)2 j r0; t − = dt2 2c3r c V 1 d2 Z r = dV 0 e j r0; t − n x0 n x0 = 2c3r dt2 i i c k k l l V 1 r = M¨ r0; t − n n e ; (21) 2c2r ikl c k l i where ei are the unit vectors of the Cartesian coordinate system and the summation over repeating indices is implied. The third-rank cur- rent quadrupole tensor is defined as: 1 Z M = dV 0 j x0 x0: (22) ikl c i k l V Note that we do not attempt to make this tensor traceless and just re- 0 0 tain the main xkxl term, which is sufficient for the purposes of further derivations. Collecting all contributions in Eq. (14), we arrive at the vector potential in the following form: d_ m_ × n 1 1 A(r; t) = + + D¨ + M¨ n n e ; (23) cr cr 2c2r 2c2r ikl k l i where all the quantities are evaluated at the retarded time τ = t−r=c. The obtained expression might be compared, e.g., to the radiation part of the vector potential in [18, p. 17]. 6 4 Bringing it all together The next step is to calculate the magnetic field B using expression (23) for the vector potential. We have: 1 r 1 r B = rot A = r × d_ t − + m_ t − × n + cr c cr c 1 r 1 r + D¨ t − + M¨ t − n n e : 2c2r c 2c2r ikl c k l i where we have written the time argument explicitly for convenience.
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