Electrostatic images and T-matrices for simple geometries

Abstract

This thesis is concerned with electrostatic boundary problems and how their solutions behave depending on the chosen basis of harmonic functions and the location of the fundamental singularities of the potential. The first part deals with the method of images for simple geometries where the exact nature of the image/fundamental singularity is unknown; essentially a study of analytic continuation for Laplace's equation in 3 dimensions. For the sphere, spheroid and cylinder, new deductions are made on the location of the images of point charges and their linear or surface charge densities, by using different harmonic series solutions that reveal the image. The second part looks for analytic expressions for the T-matrix for electromagnetic scattering of simple objects in the low frequency limit. In this formalism the incident and scattered fields are expanded on an orthogonal basis such as spherical harmonics, and the T-matrix is the transformation between the coefficients of these series, providing the general solution of any electromagnetic scattering problem by a given particle at a given wavelength. For the spheroid, bispherical system and torus, the natural basis of harmonic functions for the geometry of the scatterer are used to determine T-matrix expressed in that basis, which is then transformed onto a basis of canonical spherical harmonics via the linear relationships between different bases of harmonic functions.