Exact results for Casimir interactions between dielectric bodies: The weak-coupling or van der Waals Limit

TL;DR

This paper derives exact closed-form formulas for Casimir interactions between various dielectric bodies in the weak-coupling limit, connecting Casimir forces with van der Waals interactions.

Contribution

It simplifies the weak-coupling technique and applies it to finite dielectric bodies, providing explicit formulas for spheres, cylinders, and plates.

Findings

Exact formulas for sphere-sphere interactions

Exact formulas for cylinder-cylinder interactions

Analysis of torque and equilibrium points on slabs

Abstract

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 and between cylinders, and between an infinite plate and a retangular slab of finite size. For such a slab, we consider the torque acting on it, and find non-trivial equilibrium points can occur.