Theory of perturbation of electrostatic field by an anisotropic dielectric sphere

TL;DR

This paper develops a mathematical framework to analyze how an anisotropic dielectric sphere perturbs an electrostatic field, providing series solutions and boundary condition methods for potential calculation.

Contribution

It introduces a novel series representation and boundary condition approach for the electrostatic perturbation caused by an anisotropic dielectric sphere.

Findings

Perturbation potential decays as inverse square of distance from the sphere

Derived a transition matrix relating interior and exterior potential coefficients

Numerical confirmation of potential decay behavior

Abstract

The boundary-value problem for the perturbation of an electric potential by a homogeneous anisotropic dielectric sphere in vacuum was formulated. The total potential in the exterior region was expanded in series of radial polynomials and tesseral harmonics, as is standard for the Laplace equation. A bijective transformation of space was carried out to formulate a series representation of the potential in the interior region. Boundary conditions on the spherical surface were enforced to derive a transition matrix that relates the expansion coefficients of the perturbation potential in the exterior region to those of the source potential. Far from the sphere, the perturbation potential decays as the inverse of the distance squared from the center of the sphere, as confirmed numerically.