  Casimir entropies due to quantum fluctuations in the interaction between
electrical bodies can often be negative, either caused by dissipation or by
geometry. Although generally such entropies vanish at zero temperature,
consistent with the third law of thermodynamics (the Nernst heat theorem),
there is a region in the space of temperature and separation between the bodies
where negative entropy occurs, while positive interaction entropies arise for
large distances or temperatures. Systematic studies on this phenomenon in the
Casimir-Polder interaction between a polarizable nanoparticle or atom and a
conducting plate in the dipole approximation have been given recently. Since
the total entropy should be positive according to the second law of
thermodynamics, we expect that the self-entropy of the bodies would be
sufficiently positive as to overwhelm the negative interaction entropy. This
expectation, however, has not been explicitly verified. Here we compute the
self-entropy of an electromagnetic $\delta$-function plate, which corresponds
to a perfectly conducting sheet in the strong coupling limit. The transverse
electric contribution to the self-entropy is negative, while the transverse
magnetic contribution is larger and positive, so the total self-entropy is
positive. However, this self-entropy vanishes in the strong-coupling limit. In
that case, it is the self-entropy of the nanoparticle that is just sufficient
to result in a nonnegative total entropy.
