The interior Casimir problem

Abstract

We study the electromagnetic Casimir interaction of a metallic compact object with a compact and bounded metallic surface in which it is contained. 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 external conductor is a sphere and much larger than the internal conductor, the Casimir force can be expressed in terms of the static electric and magnetic multi-pole polarizabilities of the inner object, which is the interior analog of 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 universal. Additionally, we compute the exact Casimir force between two metallic spheres contained one inside the other for arbitrary separations. Finally, we combine our results with earlier work on the Casimir force between two spheres to obtain data on the first order correction to the Proximity Force Approximation for two metallic spheres both outside and within one another.