Casimir Force Between Dielectric Media with Free Charges

The statistical mechanical approach to Casimir problems for dielectrics separated by a vacuum gap turns out to be compact and effective. A central ingredient of this method is the effect of interacting fluctuating dipole moments of the polarizable particles. At arbitrary temperature the path integral formulation of quantized particles, developed by H{\o}ye-Stell and others, is needed. At high temperature - the limit considered in the present paper - the classical theory is however sufficient. Our present theory is related to an idea put forward earlier by Jancovici and {\v{S}}amaj (2004), namely to evaluate the Casimir force between parallel plates invoking an electronic plasma model and the Debye-H\"uckel theory for electrolytes. Their result was recently recovered by H{\o}ye (2008), using a related statistical mechanical method. In the present paper we generalize this by including a constant permittivity in the description. The present paper generalizes our earlier theory for parallel plates (1998), as well as for spherical dielectrics (2001). We also consider the Casimir force between a polarizable particle and a conductor with a small density of charges, finding agreement with the result recently derived by Pitaevskii (2008).