Repulsive Casimir Force

The Casimir force between two parallel thick plates, one perfectly dielectric, the other purely magnetic, was calculated long ago by Boyer [T. H. Boyer, Phys Rev A 9, 2078 (1974)]. Its most characteristic property is that it is repulsive. The problem of this repulsive behavior is actually delicate and counterintuitive. In the present paper we analyze the problem by first considering the simple harmonic oscillator model introduced by us earlier [J. S. Hoye et al., Phys. Rev. E 67, 056116 (2003); Phys. Rev A 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 the presence of the electromagnetic vector potential. This duality corresponds to the transverse magnetic and transverse electric 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 parr of parallel plates. The method used for our evaluations is the statistical mechanical one that we 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.