Traceless cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media

Abstract

A solid spherical harmonic of degree n at a point r takes the tensor contraction form A(n).n.rn, where A(n) is a totally symmetric and traceless nth-rank cartesian tensor. The utility of this form rests on the properties of a detracer operator Fn which transforms any totally symmetric nth-rank tensor to a totally traceless form. In particular, the components of the tensor Fnrn, which is related to the nth gradient of r-1, are spherical harmonics with properties analogous to those of the tesseral harmonics Ynm( theta , phi ), including adherence to an addition theorem and an 'Unsold' theorem. These properties lead to new formulae for the Legendre polynomials and their derivatives in terms of cartesian tensors. The traceless cartesian tensor forms are used to treat problems in electrostatics of a dielectric medium requiring spherical harmonic expansions of the potential. These include the potentials arising from an arbitrary charge distribution in a spherical dielectric cavity, the reaction field gradients in the cavity, the response of a dielectric sphere embedded in a dielectric medium to an arbitrary external field, and the gradients of the Lorentz internal field in a homogeneous dielectric. Expressions are obtained for nth-order field gradients and induced multipole moments in cartesian tensor form.