Exact solutions for some spherical electrostatic scattering problems

Abstract

The purpose of this thesis is to analyze some basic spherical structures in electrostatics using separable coordinate systems. The main emphasis is on a dielectric body immersed in a constant electric field. This setting gives rise to the concept of polarizability, which encapsulates the scattering properties of the dielectric body in a single matrix called a polarizability tensor. For simple structures, such as a sphere and ellipsoid, the polarizability tensor can be found in a closed-form. For more complicated geometries, where there is no separable coordinate system available, one usually must resort to numerical methods. This thesis focuses on intersecting dielectric double spheres (both two- and three-dimensional). The coordinate system considered in three dimensions is a toroidal coordinate system (R-separable), which leads to an elegant numerical solution scheme (Neumann series) that can be implemented efficiently, for example, in a Java Applet. A special case of the toroidal coordinate system is the tangent sphere frame, representing spheres that intersect each other at a single point, in which the solution of the scattering problem is reduced to a second order linear ordinary differential equation with elementary coefficients. The two-dimensional double hemisphere (or double half-disk) is considered in the bipolar coordinate system, which leads to a closed form solution for the polarizability.