# Going beyond PFA: a precise formula for the sphere-plate Casimir force

## TL;DR

This paper introduces a highly accurate, easy-to-use semi-analytic formula for calculating the sphere-plate Casimir force, improving computational efficiency while maintaining precision across all distances.

## Contribution

The authors derive a new semi-analytic formula for the sphere-plate Casimir force that surpasses previous approximations in accuracy and computational simplicity.

## Key findings

- The formula is accurate at all distances tested.
- Error is smaller than experimental uncertainties.
- Comparison with exact scattering confirms high precision.

## Abstract

Quantum fluctuations of the electromagnetic field in the medium surrounding two discharged macroscopic polarizable bodies induce a force between the two bodies, the so called Casimir force. In the last two decades many experiments have accurately measured this force, and significant efforts are made to harness it in the actuation of micro and nano machines. The inherent many body character of the Casimir force makes its computation very difficult in non-planar geometries, like the standard experimental sphere-plate configuration. Here we derive an approximate semi-analytic formula for the sphere-plate Casimir force, which is both easy to compute numerically and very accurate at all distances. By a comparison with the fully converged exact scattering formula, we show that the error made by the approximate formula is indeed much smaller than the uncertainty of present and foreseeable Casimir experiments.