  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).
