# Classical Casimir interaction of perfectly conducting sphere and plate

## TL;DR

This paper analyzes the classical high-temperature Casimir interaction between a perfectly conducting sphere and plate, deriving a correction beyond the proximity force approximation and developing a fast numerical scheme for precise calculations.

## Contribution

It introduces a small-distance expansion correction involving a logarithmic term and a new numerical method using bispherical partial waves for high-precision computations.

## Key findings

- Correction to Casimir energy includes a $ ext{ln}^2(d/R)$ term.
- The short-distance formula remains accurate at larger distances.
- A fast numerical scheme with bispherical waves is developed.

## Abstract

We study the Casimir interaction between perfectly conducting sphere and plate in the classical limit of high temperatures. By taking the small-distance expansion of the exact scattering formula, we compute the leading correction to the Casimir energy beyond the commonly employed proximity force approximation. We find that for a sphere of radius $R$ at distance $d$ from the plate the correction is of the form $\ln^2 (d/R)$, in agreement with indications from recent large-scale numerical computations. We develop a fast-converging numerical scheme for computing the Casimir interaction to high precision, based on bispherical partial waves, and we verify that the short-distance formula provides precise values of the Casimir energy also for fairly large distances.