The Rayleigh hypothesis and the region of applicability of the extended boundary condition method in electrostatic problems for nonspherical particles

Abstract

The electrostatic problem of excitation of a homogeneous axisymmetric particle in a constant electric field is considered. The approach is based on the surface integral equations arising in the extended boundary condition method (EBCM). The electrostatic fields are related to the scalar potentials, which are represented as expansions in the eigenfunctions of the Laplace operator in the spherical coordinate system. Unknown expansion coefficients are determined from infinite systems of linear algebraic equations (ISLAE). Analytic analysis of the solvability of the ISLAE is performed, and the convergence radii of expansions are obtained. It is shown that the EBCM can be applied in the far zone of a particle, i.e., the T matrix, can be constructed even if the Rayleigh hypothesis (the expansions converge up to the boundary of the particle) is not satisfied. However, a weaker restriction appears, which can be reduced to the requirement that there exists a spherical shell inside which the expansions of excited and internal fields simultaneously converge. The case of spheroids, as well as pseudospheroids that arise from spheroidal particles by inversion, is studied in detail. It is shown that the EBCM is applicable to spheroids for any ratio a/b of semiaxes and to pseudospheroids for a/b < 1 + \(\sqrt 2 \). The external Rayleigh hypothesis, i.e., the convergence of the expansion of a “scattered” field up to the surface of a particle is valid for a spheroid if a/b < \(\sqrt 2 \). The internal hypothesis, i.e., the convergence of an internal field, is always valid because the field is uniform inside the spheroid. For a pseudospheroid, the hypotheses are valid for a/b < \(\sqrt 2 \) and a/b < \(\sqrt 2 \), respectively. The relation and similarity between the results obtained for the wave and electrostatic problems are discussed.