  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.
