Short range electrostatic interactions in dielectric media

Abstract

The electrostatic theory for a dielectric sphere with arbitrary charge distribution within it is presented. The sphere is located in a dielectric medium which either contains another similar sphere or is separated from another dielectric medium by a planar interface. It is shown that the case of two dielectric spheres is simply related to that of a sphere near a planar interface. The cases of a point charge or a point dipole at the center of the sphere are analyzed in detail. For the point charge case, the numerical results for both the electrostatic potential energy and forces for a wide range of values of the ratio of dielectric constant of sphere to that of medium are given. In addition, approximate analytical expressions are given and constrasted with the exact numerical results. The asymptotic expression occasionally used is shown to provide a good approximation for equal and oppositely charged cavities embedded in a medium of very high dielectric constant. An analytical expression, similar in structure to the exact one for metallic spheres, is derived and shown to provide an excellent approximation over the whole range of dielectric constants and arbitrary charge combinations. Approximate expressions using Padé approximants and simple variational approaches are also given. For the point dipole case, extensive numerical results are presented. It is shown that, for the case of a dielectric cavity, embedded in a medium of high dielectric constant and in contact with another similar cavity with the dipoles oriented along the lines of centers, the potential energy can be expressed in terms of tabulated analytical functions. This also applies for a dielectric cavity in contact with a planar dielectric interface with dipoles oriented perpendicular to the interface.