vector satisfies (VXE)r>0. Hence, Maxwell's equation R (×E)=-c可>0 Thus, the assumption that genuine spherically symmetric radiation exists yields a contradiction, because for any spherical radiation, the longitudinal field component Brto- gether with its derivative with respect to the time, vanishes identically. This result completes the proof that a genuine Fig. 1. A sphere at the wave zone. The thin line denotes a great circle called spherically symmetric electromagnetic radiation cannot he equator. The path Pe is tangent to the equator at point P. The path gRP ist. It can be easily seen that the proof holds not only for an outgoing radiation but for an incoming one as well As stated, there is an infinite number of different systems emitting electromagnetic radiation. Each of these systems is Thus, cos 0 of the scalar product takes the maximal value of a particular arrangement of electric and magnetic dipoles or higher multipoles and appropriate classical currents. Thus unity. Because E is uniform on S, the integral on C is there are infinitely many patterns of electromagnetic radia eater than the absolute value of that of Ch tion. However, in spite of the infinite degrees of freedom econd case Here C coincides with available, not every pattern of radiation can be realized the equator which is taken as the closed path. Hence, in this case C'reduces to a point, but Eq. (3)still holds. Using eLectronic mail: eli@tauphy tau acil vector analysis, we find from Eq(3)that L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields(Perga- mon, Oxford, 1975),. 76 Edl=(V×E)ds>0 D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), P. 237. Seee2,pp.744-74 where S is any surface whose boundary is the closed path D E. Rutherford, Vector Methods(Oliver and Boyd, Edinburgh, 1962),9th C+C. Let us choose the corresponding part of the spherical E. L. ince, Ordinary Differential Equation (Dover, New York, 1956) shell S as the surface S. It follows from Eq. (4)that there exists a region on S where the radial part of the following See Ref 1, p 162, Ref 2, p.657 Am. J. Phys., Vol. 70, No. 7, July 2002 E Comay 716Thus, cos u of the scalar product takes the maximal value of unity. Because uEu is uniform on S, the integral on C is greater than the absolute value of that of C8. Now, let us turn to the second case. Here C coincides with the equator which is taken as the closed path. Hence, in this case C8 reduces to a point, but Eq. ~3! still holds. Using vector analysis, we find from Eq. ~3! that R C1C8 E"dl5 E S8 ~“ÃE!"ds.0, ~4! where S8 is any surface whose boundary is the closed path C1C8. Let us choose the corresponding part of the spherical shell S as the surface S8. It follows from Eq. ~4! that there exists a region on S8 where the radial part of the following vector satisfies (“ÃE)r.0. Hence, Maxwell’s equation yields ~“ÃE!r52 1 c ]Br ]t .0. ~5! Thus, the assumption that genuine spherically symmetric radiation exists yields a contradiction, because for any spherical radiation, the longitudinal field component Br to￾gether with its derivative with respect to the time, vanishes identically.7 This result completes the proof that a genuine spherically symmetric electromagnetic radiation cannot ex￾ist. It can be easily seen that the proof holds not only for an outgoing radiation but for an incoming one as well. As stated, there is an infinite number of different systems emitting electromagnetic radiation. Each of these systems is a particular arrangement of electric and magnetic dipoles or higher multipoles and appropriate classical currents. Thus, there are infinitely many patterns of electromagnetic radia￾tion. However, in spite of the infinite degrees of freedom available, not every pattern of radiation can be realized. a! Electronic mail: eli@tauphy.tau.ac.il 1 L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields ~Perga￾mon, Oxford, 1975!, p. 76. 2 J. D. Jackson, Classical Electrodynamics ~Wiley, New York, 1975!, p. 237. 3 See Ref. 2, pp. 744–747. 4 See Ref. 1, p. 162, and Ref. 2, p. 657. 5 D. E. Rutherford, Vector Methods ~Oliver and Boyd, Edinburgh, 1962!, 9th ed., pp. 16–19. 6 E. L. Ince, Ordinary Differential Equation ~Dover, New York, 1956!, Chap. III. 7 See Ref. 1, p. 162, Ref. 2, p. 657. Fig. 1. A sphere at the wave zone. The thin line denotes a great circle called the equator. The path PQ is tangent to the equator at point P. The path QRP is a portion of a great circle ~see the text!. 716 Am. J. Phys., Vol. 70, No. 7, July 2002 E. Comay 716