Magnetic field corrections to the repulsive Casimir effect at finite temperature

I investigate the finite temperature Casimir effect for a charged and massless scalar field satisfying mixed (Dirichlet-Neumann) boundary conditions on a pair of plane parallel plates of infinite size. The effect of a uniform magnetic field, perpendicular to the plates, on the Helmholtz free energy and Casimir pressure is studied. The zeta-function regularization technique is used to obtain finite results. Simple analytic expressions are obtained for the zeta function and the free energy, in the limits of small plate distance, high temperature and strong magnetic field. The Casimir pressure is obtained in each of the three limits and the situation of a magnetic field present between and outside the plates, as well as that of a magnetic field present only between the plates is examined. It is discovered that, in the small plate distance and high temperature limits, the repulsive pressure is less when the magnetic field is present between the plates but not outside, than it is when the magnetic field is present between and outside the plates.