# Modes inside a cavity and black body radiation

Consider a perfect conductor that encloses a spatial volume such as a parallelepiped or cylinder. If we solve Maxwell's equations inside that volume, seeking solutions that depends on time with a dependency of the form $e^{-i\omega t}$, we find that only TE and TM modes can exist inside of that volume (and no TEM modes). However both TE and TM modes have a cutoff frequency. This seems to imply that any EM wave inside the cavity cannot have any frequency and that it should be greater than a threshold.

However if we look at the problem from another perspective, the one of a black/grey body at a temperature $T > 0K$, we'd think that the walls are emitting EM waves without any cutoff frequency (and with a continuous spectrum).

I understand that the sum of two solutions to Maxwell's equations in the cavity is also a solution and I think that I could write any allowed EM wave as a sum of TE and TM modes, but if both TM and TE modes have a cutoff frequency, I don't see how I could obtain an EM wave with a lower frequency that the cutoff one.

Hence I don't see how to reconciliate the blackbody radiation with TE and TM modes inside a cavity. Where do I go wrong?

• Compare the energy of photons at the cut-off frequency with typical temperatures. Oct 21, 2015 at 23:21
• If I consider a circular cylindrical cavity with radius equal to 0.5 m, I get a cuttoff frequency of about 1.4 GHz for TM modes. That would be photons corresponding to about $1.52 \times 10^{-25}$ J. Or wavelength of about 3.02 meters... which is bigger than the radius of the cavity. Assuming the above is right, the main idea is that almost any wavelength produced by a black body at ~290 K (which has a peak at around $10 \mu m$) is allowed to exist in the cavity. Is this correct? (more to come) Oct 22, 2015 at 0:16
• Then what would happen if I reduce the size of the cavity enough so that microwaves can't exist there due to the cutoff frequency? The black body radiation wouldn't occur? Oct 22, 2015 at 0:16
• The confusion here comes from assuming that a hollow conductor is a black body. It's not. If that answers your question I'm happy to copy it into an answer, but I suspect you'll want more information. Please let me know. Oct 23, 2015 at 20:26
• @SebastianRiese it's not hard at all to make a cavity small enough that the cutoff frequency $\nu$ corresponds to a temperature $T = h \nu / k_b$ which is easily achieved in cryogenic systems. Oct 23, 2015 at 20:28