Repulsive Casimir Force

TL;DR

This paper investigates the repulsive Casimir force between a dielectric and a magnetic plate, using a microscopic model and statistical mechanics to understand the conditions for attraction or repulsion.

Contribution

It extends the harmonic oscillator model to analyze the microscopic origin of the repulsive Casimir force and generalizes Boyer's case to arbitrary permittivities and permeabilities.

Findings

Repulsive Casimir force can be explained microscopically via duality in electromagnetic modes.

The nature of the force depends on the polarizabilities and electromagnetic properties of the objects.

Equal particles or plates always attract, regardless of their properties.

Abstract

The Casimir force between two parallel thick plates, one perfectly dielectric, the other purely magnetic, has been calculated long ago by Boyer [T. H. Boyer, Phys. Rev. A {\bf 9}, 2078 (1974)]. Its most characteristic property is that it is repulsive. The problem is actually delicate and counterintuitive, since it implies, for instance, that in the boundary layer of the electric plate the square $E^2$ of the electric field becomes a negative quantity. In the present paper we analyze the problem by first considering the simple harmonic oscillator model introduced by us earlier [J. S. H{\o}ye {\it et al.}, Phys. Rev. E {\bf 67}, 056116 (2003); Phys. Rev. A {\bf 94}, 032113 (2016)]. Extension of this model shows how the repulsive behavior can be understood on a microscopic basis, due to the duality between canonical and mechanical momenta in presence of the electromagnetic vector potential. This duality corresponds to the TM and TE modes in electrodynamics. We analyze the generalized Boyer case where the permittivities and permeabilities of the parallel plates are arbitrary. In this respect we first find the induced interaction between a pair of particles with given electric and magnetic polarizabilities and then find it for a pair of parallel plates. The method used for our evaluations is the statistical mechanical one that we have introduced and applied earlier. Whether the pair of particles or plates attract or repel each other depends on their polarizabilities or permittivities and permeabilities respectively. For equal particles or equal plates there is always attraction.