# Vacuum energy between a sphere and a plane at finite temperature

## TL;DR

This paper investigates the Casimir effect between a sphere and a plane at finite temperature, analyzing scalar and electromagnetic fields, and compares exact results with proximity force approximations across temperature regimes.

## Contribution

It provides detailed calculations of the Casimir effect at finite temperature for a sphere-plane setup, including limiting cases and temperature corrections for scalar and electromagnetic fields.

## Key findings

- Low temperature correction for scalar field is of order T^2.
- Low temperature correction for electromagnetic field is of order T^4.
- At high temperature, the zeroth Matsubara frequency dominates.

## Abstract

We consider the Casimir effect for a sphere in front of a plane at finite temperature for scalar and electromagnetic fields and calculate the limiting cases. For small separation we compare the exact results with the corresponding ones obtained in proximity force approximation. For the scalar field with Dirichlet boundary conditions, the low temperature correction is of order $T^2$ like for parallel planes. For the electromagnetic field it is of order $T^4$. For high temperature we observe the usual picture that the leading order is given by the zeroth Matsubara frequency. The non-zero frequencies are exponentially suppressed except for the case of close separation.