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I am following this video and; Eisberg and Resnick's Book for this derivation, for I cannot find other sources that go as in-depth as they do.

Question 1

Jean's cube, or the metallic cube, is assumed to be a perfect absorber. On this fact alone, authors state

Now assume that the walls of the cavity are uniformly heated to a temperature T. Then the walls will emit thermal radiation which will fill the cavity. The small fraction of this radiation incident from the inside upon the hole will pass through the hole. Thus the hole will act as an emitter of thermal radiation. Since the hole must have the properties of the surface of a blackbody, the radiation emitted by the hole must have a blackbody spectrum; but since the hole is merely sampling the thermal radiation present inside the cavity, it is clear that the radiation in the cavity must also have a blackbody spectrum. In fact, it will have a blackbody spectrum characteristic of the temperature T on the walls, since this is the only temperature defined for the system.

Now, I am not sure how having the absorption properties of a black body implies that it must also have emission properties of the black body as well. The closest answer to this was in the comments of this question's answer, which is basically "that's what experiments tell us."

Question 2

In deriving the spectrum of EM waves inside the cube, we assume that the waves can be broken up into three independent components. This seems logical, but given the complexity of Maxwell's equations, I have a hard time buying this. In the video @19:42 the diagram helps in clearing this up and is pretty satisfactory, but a mathematical proof would be better.

Question 3

Why did it seem reasonable at the time to use the Equipartition theorem when it clearly only adds to the kinetic energy of the system? There is no sensible way of talking about the kinetic energy of EM waves, even if it has a quadratic form $(\epsilon_0E^2/2)$

GedankenExperimentalist
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1 Answers1

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Now, I am not sure how having the absorption properties of a black body implies that it must also have emission properties of the black body as well.

Let me first point out that black body is an outdated way of deriving black body radiation spectrum - since these can be simply obtained by considering an ideal photon gas (see also detailed derivation on how quantization "saves" the black body spectrum from divergence.) In this sense, hole is emitting as a black body, because the radiation inside the cavity is in equilibrium - that is I would reason in the direction opposite to the quoted fragment. Indeed, this is how we assign the black body spectrum to objects from a heated slab of metal to stars.

Furthermore, a black body has a property that it absorbs all the radiation incident on it. It doesn't have any specific emission properties... However, an important implicit feature of derivations of a black body spectrum is that black body is in equilibrium with its surroundings. If this is the case for the hole in the cavity, then it will emit like a black body. If it is not the case (as in the case of a hot slab of metal, stars, etc.), it will emit approximately as a black body, because the whole is very small and the equilibrium radiation inside loses only small part of its energy via emission through the hole.

Question 2 is merely about solving the wave equation for electromagnetic field via separation of variables. It is really homework stuff, but I suppose that with the keywords in this paragraph one can find more detailed explanations on this site or elsewhere. Questions about resonators and cavities are likely to have more details.

Equipartition theorem is true for any quadratic Hamiltonian, and energy of the electromagnetic waves is a perfect example of it - it is just a matter of taking a Gaussian integral (see for more details this answer, already quoted above.)

Roger V.
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