Exact solutions for electromagnetic fields inside and outside a spherical surface with magnetic/electric dipole distributed sources

Abstract

Exact solutions of the Maxwell equations for the electromagnetic fields inside and outside a spherical surface, with time alternating magnetic or electric dipole source distributions, are constructed as alternatives to the respective familiar point-dipole solutions in undergraduate and graduate books. These solutions are valid for all positions, inside and outside the sphere, including the quasi-static, induction and radiation zones; the solutions inside make the difference from the point-dipole solutions; the definitions of the dynamic dipole moments must be based on the ordinary spherical Bessel functions for the solutions outside, and on the outgoing spherical Hankel functions for the solutions inside,
instead of the powers of the radial coordinate as solutions of the Laplace equation valid for the static case. The solutions for the resonating cavities are associated with the nodes of the spherical Bessel function for the TE modes of the magnetic dipole source, and with the extremes of the product of the radial coordinate times the same spherical function for the TM modes of the electric dipole source; both conditions also guarantee the vanishing of the fields outside.