Casimir potential of a compact object enclosed by a spherical cavity

TL;DR

This paper analyzes the electromagnetic Casimir interaction between a compact object and a spherical cavity, deriving formulas for the force based on scattering matrices and exploring special cases like metallic spheres.

Contribution

It introduces a general scattering matrix approach to compute Casimir forces in spherical geometries and provides exact and approximate results for specific configurations.

Findings

Casimir force expressed via scattering and translation matrices.

Force depends on the object's displacement from the sphere's center.

Exact force computed for two metallic spheres at arbitrary separations.

Abstract

We study the electromagnetic Casimir interaction of a compact object contained inside a closed cavity of another compact object. We express the interaction energy in terms of the objects' scattering matrices and translation matrices that relate the coordinate systems appropriate to each object. When the enclosing object is an otherwise empty metallic spherical shell, much larger than the internal object, and the two are sufficiently separated, the Casimir force can be expressed in terms of the static electric and magnetic multipole polarizabilities of the internal object, which is analogous to the Casimir-Polder result. Although it is not a simple power law, the dependence of the force on the separation of the object from the containing sphere is a universal function of its displacement from the center of the sphere, independent of other details of the object's electromagnetic response. Furthermore, we compute the exact Casimir force between two metallic spheres contained one inside the other at arbitrary separations. Finally, we combine our results with earlier work on the Casimir force between two spheres to obtain data on the leading order correction to the Proximity Force Approximation for two metallic spheres both outside and within one another.