Exact results for classical Casimir interactions: Dirichlet and Drude model in the sphere-sphere and sphere-plane geometry

TL;DR

This paper derives exact analytic expressions for classical Casimir interactions between two spheres and sphere-plane configurations, considering Dirichlet and Drude boundary conditions, revealing detailed deviations from approximations at short distances.

Contribution

It provides the first exact analytic formulas for classical Casimir interactions in sphere-sphere and sphere-plane geometries, including interior and exterior configurations, for Dirichlet and Drude models.

Findings

Exact formulas derived for all separations

Short-distance expansions show deviations from proximity force approximation

Results applicable to both Dirichlet and Drude boundary conditions

Abstract

Analytic expressions that describe Casimir interactions over the entire range of separations have been limited to planar surfaces. Here we derive analytic expressions for the classical or high-temperature limit of Casimir interactions between two spheres (interior and exterior configurations), including the sphere-plane geometry as a special case, using bispherical coordinates. We consider both Dirichlet boundary conditions and metallic boundary conditions described by the Drude model. At short distances, closed-form expansions are derived from the exact result, displaying an intricate structure of deviations from the commonly employed proximity force approximation.