The Legendre Expansion of Potentials

Abstract

In electrostatics a basic problem is to find the potential function for a given charge distribution that can be solved with the Laplace-Poisson equations under specified boundary conditions. Although the cause for distributed charges is unspecified in the given system, here we are only concerned about the electrostatic problem. For many applications we are interested in the potential at a distant point from distributed charges in a small region. The corresponding field is often dominated by deviations from spherical symmetry, a deviation that can be conveniently viewed with polar coordinates with respect to the center of distribution. Such a charge distribution is often described with respect to a unique direction for deformation that arises from an internal origin.