Derive energy density proportional to emitted power per unit area

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The following derives the relation that for a blackbody radiation the energy density is proportional to the energy emitted per unit area over unit time.

The average energy density ##d\psi## is obtained by dividing the radiant energy ##dE## received by the surface ##dB## in 1 second by the cylindrical volume ##dV## occupied by the radiation in 1 second.

I don't understand why ##d\psi## is uniform anywhere inside the cylinder, in particular, at points near the surface ##dS##. At a point far away from ##dS##, the point is always the same distance ##\rho## away from every part of ##dS##. But this is not true for a point near ##dS##. The point would be closer to some parts of ##dS##. Then the energy density ##d\psi## at the point should not be uniform but vary depending on how far the point is from the center of the surface ##dS##.

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Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

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