Maxwell’s Equations in Spherical Coordinates

Material systems in which homogeneous, linear, isotropic media exhibit spherical symmetry around a given point in space support electromagnetic waves that can be expressed as a superposition of certain eigen-modes of Maxwell’s equations. These eigen-modes, known as vector spherical harmonics, are expressed in terms of Bessel functions of various types and orders, associated Legendre functions, and ordinary sinusoidal functions. In contrast to the integer-order Bessel functions which describe the radial dependence of the eigen-modes in systems of cylindrical symmetry, the Bessel functions representing the radial dependence of vector spherical harmonics are of half-integer order. In this chapter, we derive the exact solutions of Maxwell’s equations for transverse electric (TE) as well as transverse magnetic (TM) modes of the electromagnetic field in systems of spherical symmetry. Several applications such as the excitation of whispering gallery modes within dielectric spheres and the scattering of plane-waves from spherical particles will then be discussed.