Is a black body sphere inside a black body shell hotter than the shell?

Question

I am missing something very basic here. Let us assume the shell is at a fixed temperature T by a power generator. The sphere inside will radiate the same power it absorbs. The power radiated by the shell will be $4 \pi \sigma T_{shell}^4 R_{shell}^2$ and will be the same as that absorbed by the smaller sphere and radiated out, which should be $4 \pi\sigma T_{sphere}^4 R_{sphere}^2$, so the sphere should be hotter.

But wouldn't this violate the second law? Isn't this set up equivalent to a closed universe in thermal equilibrium in which the shell can only radiate inwards?

Answer 1

Not all the radiation from the outer shell reaches the inner shell. When you take into account the intensity distribution of radiation from the outer shell (Lambertian distribution, i.e. $\propto\cos\theta$) you will see that the amount of radiation for the inner to the outer shell is the same as in the other direction.

No violation of the second law.

Answer 2

There's no violation of the second law here. You have a system that is out of thermal equilibrium. That black bodies absorb and radiate is the driving mechanism that tries to move this system toward thermal equilibrium.

By way of analogy, suppose you are from a southern clime and take a trip at this time of year to a northern clime. You, as a southerner, aren't used to those cool fall northern temperatures. Your hotel, which is green-certified, has yet to turn on the heater. You're cold! The solution is simple: Toss an extra blanket on the bed, or lots of blankets if need be.

The outer shell in this problem acts as a blanket. It doesn't stop heat transfer from the sphere to the external environment (the rather cold 2.7 kelvin cosmic microwave background), but it does nicely slow down that transfer.