# Electrostatic limit of the $T$-matrix for electromagnetic scattering: Exact results for spheroidal particles

## TL;DR

This paper derives exact electrostatic expressions for the T-matrix of spheroidal particles, clarifying the long-wavelength limit of electromagnetic scattering and providing tools for analyzing convergence and the Rayleigh hypothesis.

## Contribution

It introduces a fully electrostatic version of the T-matrix method (ES-EBCM), deriving analytic expressions for spheroids and connecting them to the electromagnetic T-matrix in the long-wavelength limit.

## Key findings

- Explicit electrostatic T-matrix expressions for spheroids.
- Symmetry properties of the ES-EBCM matrices.
- Insights into the convergence of the T-matrix formalism.

## Abstract

The $T$-matrix, often obtained with Waterman's extended boundary condition method (EBCM), is a widely-used tool for fast calculations of electromagnetic scattering by particles. Here we investigate the quasistatic or long-wavelength limit of this approach, where it reduces to an electrostatics problem. We first present a fully electrostatic version of the EBCM/T-matrix method (dubbed ES-EBCM). Explicit expressions are then given to quantitatively express the long-wavelength limit of the EBCM matrix elements in terms of those of the ES-EBCM formalism. From this connection we deduce a number of symmetry properties of the ES-EBCM matrices. We then investigate the matrix elements of the ES-EBCM formalism in the special case of prolate spheroids. Using the general electrostatic solution in spheroidal coordinates, we derive fully analytic expressions (in the form of finite sums) for all matrix elements. Those can be used for example for studies of the convergence of the $T$-matrix formalism. We illustrate this by discussing the validity of the Rayleigh hypothesis, where analytical expressions highlight clearly the link with analytical continuation of series.