Construction of the Rayleigh approximation for axisymmetric multilayered particles using the eigenfunctions of the Laplace operator

The Rayleigh approximation is constructed from solution of the electrostatic problem for axisymmetric multilayered particles. The approach used is based on consideration of the surface integral equations analogous to those used in the extended boundary condition method (EBCM) applied to solve the electromagnetic problems. The electric fields are related with the scalar potentials that are represented by their expansions in terms of the eigenfunctions of the Laplace operator written in spheroidal and spherical coordinates. The unknown expansion coefficients are derived from infinite linear algebraic equations. The explicit solution found in spheroidal coordinates for multilayered spheroids coincides with the known solutions for the homogeneous and core-mantle particles.