  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.
