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Imagine a spherical oven with radius R. The inside of the spherical wall is at temperature T (everywhere) and is assumed emit perfect black body radiation (everywhere).

(1) What is the energy density in the interior of the oven?

(2) Is the energy density homogeneous or does it depend on the radius?

Klaus-Frank

I wonder whether the radiation density of the universe can be described in this way, i.e., as result of a "hot" horizon. Of course, in this case there is the additional issue that the horizon ("oven wall") is growing with time.

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closed as off-topic by StephenG, Aaron Stevens, Yashas, user191954, ahemmetter Apr 30 at 7:47

This question appears to be off-topic. The users who voted to close gave this specific reason:

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If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Homework-type questions are generally off-topic without reasonable effort being made by the OP. It does not have to be homework as-in-school, it can be also cover self-study and curiosity type questions. $\endgroup$ – StephenG Apr 29 at 21:29
  • $\begingroup$ I was wondering wether the radiation density of the universe can be understood as result of the temperature of the horizon. I added the explanation. $\endgroup$ – Klaus-Frank Apr 30 at 4:37
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It's uniform, of course. If it weren't uniform, you could quickly seal off two portions with different radiation densities, and use the difference to extract work. Extracting work from a bath of uniform temperature violates the second law of thermodynamics.

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The answer to (1) is that the energy density is $4\sigma T^4/c$, where $\sigma$ is the Stefan-Boltzmann constant, $T$ is the absolute temperature, and $c$ is the speed of light. A derivation can be found here. Note that the oven radius $R$ is irrelevant as far as the energy density is concerned.

The Stefan-Boltzmann constant can be expressed in terms of more fundamental physical constants as

$$\sigma=\frac{2\pi^5k^4}{15c^2h^3},$$

where $k$ is Boltzmann’s constant and $h$ is Planck’s constant.

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