On the $T$-matrix in the electrostatic problem for the spheroidal particle with a spherical core

A solution to the electrostatic problem for the spheroidal particle with a spherical core is constructed. To involve the problem geometry in a full manner, in a vicinity of the particle surface the potentials of the fields are represented by their expansions in terms of spheroidal harmonics of Laplace's equation, while in a vicinity of the core surface by the expansions in terms of spherical harmonics. Matching of the fields inside the particle shell is made by using the relations between the spheroidal and spherical harmonics. The $T$-matrix relates the coefficients of expansions of the incident and “scattered” fields. In the paper, both the particle polarizability related to the main matrix element $T_{11}$, and the whole $T$-matrix are considered. The symmetry of the matrix as well as its dependence on the size of the layered particle are shown. A relation between the $T$-matrixes in the spherical and spheroidal systems was also found. Numerical calculations made for particles of small and large aspect ratios ($a/b = 1.5 - 5.0$) confirmed high efficiency of the suggested solution in contrast to the methods that use a single spherical basis.