Three-dimensional electric potential induced by a point singularity in a multilayered dielectric medium

Abstract

A simple and effective method is proposed to derive the three-dimensional electric potential induced by a point singularity of any type in an N-phase dielectric medium composed of N−2 intermediate dielectric layers of equal thickness encased in two semi-infinite dielectric media. The point singularity can include a point charge or a point electric dipole. The original boundary value problem for the N-phase medium is reduced to the determination of a single unknown three-dimensional harmonic function through satisfaction of the continuity conditions across all of the N−1 perfect planar interfaces. The single harmonic function can be completely determined after analytically solving the resulting linear recurrence relations, which are independent of the type and the specific location of the singularity. When the singularity is a point charge, we obtain the self-energy of the point charge expressed in terms of a single function and the Coulomb force on the point charge expressed in terms of the negative derivative of this function.