# Electrostatic T-matrix for a torus on bases of toroidal and spherical harmonics

## TL;DR

This paper derives semi-analytic static T-matrix expressions for a circular torus using toroidal and spherical harmonics, enabling efficient electromagnetic scattering analysis in static conditions.

## Contribution

It introduces a novel semi-analytic method for calculating the static T-matrix of a torus in both harmonic bases, including basis transformations and explicit expressions for thin rings.

## Key findings

- Derived static T-matrix expressions for a torus in toroidal and spherical harmonics
- Computed static optical cross-sections and limits for thin rings
- Provided analytic formulas for auxiliary matrices and basis transformations

## Abstract

Semi-analytic expressions for the static limit of the T-matrix for electromagnetic scattering are derived for a circular torus, expressed in bases of both toroidal and spherical harmonics. The scattering problem for an arbitrary static excitation is solved using toroidal harmonics and the extended boundary condition method to obtain analytic expressions for auxiliary Q and P-matrices, from which the T-matrix is given by their division. By applying the basis transformations between toroidal and spherical harmonics, the quasi-static limit of the T-matrix block for electric multipole coupling is obtained. For the toroidal geometry there are two similar T-matrices on a spherical basis, for computing the scattered field both near the origin and in the far field. Static limits of the optical cross-sections are computed, and analytic expressions for the limit of a thin ring are derived.