On the corrections beyond proximity force approximation (PFA)

TL;DR

This paper recalculates the first correction beyond the proximity force approximation for a sphere-plane setup, revealing boundary condition-dependent differences and confirming some results with a derivative expansion method.

Contribution

It provides corrected analytic expressions for the first correction beyond PFA for various boundary conditions, clarifying previous discrepancies.

Findings

Confirmed previous results for Dirichlet conditions.

Identified sign errors affecting Robin, Neumann, and conductor conditions.

Validated corrections using a derivative expansion approach.

Abstract

We recalculate the first analytic correction beyond PFA for a sphere in front of a plane for a scalar field and for the electromagnetic field. We use the method of Bordag and Nikolaev [J.Phys.A, {\bf 41} (2008) p.164002]. We confirm their result for Dirichlet boundary conditions whereas we find a different one for Robin, Neumann and conductor boundary conditions. The difference can be traced back to a sign error. As a result, the corrections depend on the Robin parameter. Agreement is found with a very recent method of derivative expansion.