Exact results for Casimir interactions between dielectric bodies: the weak-coupling or van der Waals limit.

In earlier papers, we have applied multiple-scattering techniques to calculate Casimir forces due to scalar fields between different bodies described by delta function potentials. When the coupling to the potentials became weak, closed-form results were obtained. We simplify this weak-coupling technique and apply it to the case of tenuous dielectric bodies, in which case the method involves the summation of van der Waals (Casimir-Polder) interactions. Once again, exact results for finite bodies can be obtained. We present closed formulas describing the interaction between spheres, between cylinders, and between an infinite plate and a rectangular slab of finite size. For such a slab, we consider the torque acting on it and find that nontrivial equilibrium points can occur.