  The present paper - a continuation of our recent series of papers on Casimir
friction for a pair of particles at low relative particle velocity - extends
the analysis so as to include dense media. The situation becomes in this case
more complex due to induced dipolar correlations, both within planes, and
between planes. We show that the structure of the problem can be simplified by
regarding the two half-planes as a generalized version of a pair of particles.
It turns out that macroscopic parameters such as permittivity suffice to
describe the friction also in the finite density case. The expression for the
friction force per unit surface area becomes mathematically well-defined and
finite at finite temperature. We give numerical estimates, and compare them
with those obtained earlier by Pendry (1997) and by Volokitin and Persson
(2007). We also show in an appendix how the statistical methods that we are
using, correspond to the field theoretical methods more commonly in use.
