Suppose you were inside a thick spherical shell of inner radius $R$, which was a perfect black body at some temperature T. What would be the power a sphere of radius $r$ would absorb located inside the shell?
I assume the location does not matter, given the spherical symmetry and using similar arguments to the gravitational force inside a shell. If the location does matter, let's place the sphere at the origin. I am tempted to think that you could solve this problem with the Stefan-Boltzmann law, thus I believe all the information necessary is provided.
I am almost lead to believe that the power absorbed is $P = \sigma_{SB} T^4 A_{abs}$, where $A_{abs}$ is the surface area of the absorbing sphere; but I would like to hear thoughts on how to approach this problem.
How I arrived at this result seems rather non-intuitive since the black body flux per unit area of the emitting shell is $\sigma_{SB} T^4$. So then $P = \sigma_{SB} T^4 A_{abs}$ would be the power emitted by the sphere if it was in thermal equilibrium with the shell. Which I suppose if you waited long enough and the shell was always held at a fixed T, the sphere would enter thermal equilibrium. Thus since the flux from the shell never changed, but it is clearly $P = \sigma_{SB} T^4 A_{abs}$ in thermal equilibrium, then this must have been the power absorbed the entire time.
While I am slightly satisfied with the logic, I am hoping for a more intuitive derivation, that doesn't invoke thermalization but rather talks directly about the flux from the shell at the location of the absorbing sphere.