An improved proximity force approximation for electrostatics

TL;DR

This paper introduces an improved approximation method for calculating electrostatic forces between conducting surfaces, enhancing the traditional proximity force approximation by incorporating a derivative expansion to better account for geometric effects.

Contribution

The paper presents a novel enhancement to the proximity force approximation using a derivative expansion, providing more accurate electrostatic force estimates for complex geometries.

Findings

The improved approximation better captures geometric dependencies of electrostatic forces.

The method offers a useful benchmark for numerical simulations in atomic force microscopy.

Results demonstrate increased accuracy over traditional proximity force approximation.

Abstract

A quite straightforward approximation for the electrostatic interaction between two perfectly conducting surfaces suggests itself when the distance between them is much smaller than the characteristic lengths associated to their shapes. Indeed, in the so called "proximity force approximation" the electrostatic force is evaluated by first dividing each surface into a set of small flat patches, and then adding up the forces due two opposite pairs, the contribution of which are approximated as due to pairs of parallel planes. This approximation has been widely and successfully applied to different contexts, ranging from nuclear physics to Casimir effect calculations. We present here an improvement on this approximation, based on a derivative expansion for the electrostatic energy contained between the surfaces. The results obtained could be useful to discuss the geometric dependence of the electrostatic force, and also as a convenient benchmark for numerical analyses of the tip-sample electrostatic interaction in atomic force microscopes.