Recently I suggested an argument which might lead one to believe that
the volumetric heat capacity of an EM field in a cavity was
insensitive to the cavity dimensions (without much success). But
there is an even more basic question about the effects of confinement
of the field.
Starting with infinite free space, we have any allowed plane wave mode
-- any wavevector. Now, introduce a rectangular cavity, however
large. Immediately we have restricted our continuous spectrum of
modes to a discrete spectrum. This is a very drastic change! Since,
with the exception of a set of measure zero, all of the original modes
have become disallowed, perhaps the heat capacity of the EM field in
any cavity is zero? :-)
We expect the heat capacity of the field per volume of a sufficiently
large cavity to be arbitrarily close to that of free space. Yet the
radical qualitative change of the field persists. How do we
understand this? I don't think it's sufficient merely to say that the
modes get very close together and therefore approximate a continuum
(you were going to say that, weren't you?). "Very close together"
is still far away from a continuum, always comprising a subset of
measure zero.
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