# Casimir Force at a Knife's Edge

## TL;DR

This paper calculates the Casimir force between a parabolic cylinder and an infinite plate, providing exact results for a new geometry and exploring edge and inclination effects beyond the proximity force approximation.

## Contribution

It introduces an exact computation of the Casimir force for a parabolic cylinder geometry, extending known solutions to include edge and inclination effects.

## Key findings

- Exact Casimir interaction energy as a function of separation and inclination.
- Proximity force approximation becomes exact as $H/R\to 0$.
- Edge and inclination effects are significant in the $R/H\to 0$ limit.

## Abstract

The Casimir force has been computed exactly for only a few simple geometries, such as infinite plates, cylinders, and spheres. We show that a parabolic cylinder, for which analytic solutions to the Helmholtz equation are available, is another case where such a calculation is possible. We compute the interaction energy of a parabolic cylinder and an infinite plate (both perfect mirrors), as a function of their separation and inclination, $H$ and $\theta$, and the cylinder's parabolic radius $R$. As $H/R\to 0$, the proximity force approximation becomes exact. The opposite limit of $R/H\to 0$ corresponds to a semi-infinite plate, where the effects of edge and inclination can be probed.