  The Casimir mutual free energy F for a system of two dielectric concentric
nonmagnetic spherical bodies is calculated, at arbitrary temperatures. The
present paper is a continuation of an earlier investigation [Phys. Rev. E {\bf
63}, 051101 (2001)], in which F was evaluated in full only for the case of
ideal metals (refractive index n=infinity). Here, analogous results are
presented for dielectrics, for some chosen values of n. Our basic calculational
method stems from quantum statistical mechanics. The Debye expansions for the
Riccati-Bessel functions when carried out to a high order are found to be very
useful in practice (thereby overflow/underflow problems are easily avoided),
and also to give accurate results even for the lowest values of l down to l=1.
Another virtue of the Debye expansions is that the limiting case of metals
becomes quite amenable to an analytical treatment in spherical geometry. We
first discuss the zero-frequency TE mode problem from a mathematical viewpoint
and then, as a physical input, invoke the actual dispersion relations. The
result of our analysis, based upon the adoption of the Drude dispersion
relation at low frequencies, is that the zero-frequency TE mode does not
contribute for a real metal. Accordingly, F turns out in this case to be only
one half of the conventional value at high temperatures. The applicability of
the Drude model in this context has however been questioned recently, and we do
not aim at a complete discussion of this issue here. Existing experiments are
low-temperature experiments, and are so far not accurate enough to distinguish
between the different predictions. We also calculate explicitly the
contribution from the zero-frequency mode for a dielectric. For a dielectric,
this zero-frequency problem is absent.
