Casimir Friction Between a Magnetic and a Dielectric Material, in the Nonretarded Limit

The repulsive nature of the static Casimir force between two half-spaces, one perfectly dielectric and the other purely magnetic, has been known since Boyer's work [T. H. Boyer, Phys. Rev. A {\bf 9}, 2078 (1974)]. We here analyze the corresponding friction force in the magnetodielectric case. Our main method is that of quantum mechanical statistical mechanics. The basic model we introduce is a harmonic oscillator model: an electric dipole oscillating in the $x$ direction and a magnetic one oscillating in the $y$ direction, while their separation is in the $z$ direction. This is then extended to particles with isotropic polarizabilities. We evaluate the friction force in a variety of cases: force between moving particles, between a moving particle and a half-plane, and between half-spaces sliding against each other. At the end explicit results are obtained both for finite and zero temperatures. We restrict ourselves to the nonretarded limit.