Say we have a square metal plate $A$, 1 $m^2$ on each side, of negligible thickness, in a vacuum with no other objects nearby and no other heat coming into the system. Say its temperature is at an equilibrium of 20°C.
Using the Stefan Boltzmann Law we calculate that $A$ will be radiating 418.8 W/$m^2$ or 837.6 W total -- or in other words, its internal heat source is constantly outputting 837.6 W in order to keep it at 20°C.
Now say we take another metal plate $B$ of identical size and shape, this one with no internal heat source. It starts off at just above 0K. We approach $B$ and place it right next to $A$ but with a little space. The only heat transfer is via radiation, and $A$ will start heating $B$. How will the system end up at equilibrium?
I can see one of two possibilities:
If we model energy in = energy out for $A$, $B$, and the system as a whole, with $A$ emitting all it receives in two directions with $a$ in each direction, and similarly for $B$ with $b$, then we have the following system of equations:
- $837.6 + b = 2a$ (energy into $A$ equals energy out of $A$)
- $a = 2b$ (energy into $B$ equals energy out of $B$)
- $837.6 = a + b$ (total energy into the system equals total energy out of the system)
Some simple math results in $a = 558.4$ W and $b = 279.2$ W.
In other words, $A$ will increase in temperature to 41.87°C while $B$ will end up with a temperature of -8.2°C.
Taking it from a different angle, $B$ will gradually heat up until we eventually reach a new equilibrium where both $A$ and $B$ reach 20°C, similarly as if they were touching and thus transferring heat by conduction. As both sides will be radiating 418.8 W out to space, total energy in still equals total energy out.
I cannot argue with the math and logic of #1, but #2 seems more intuitive and sensible in a "common-sense" type of reasoning, though I cannot explain it fully with equations (perhaps because it is wrong).
Further if #1 is the case, it seems strange that if we bring the two plates to touch into $AB$ we will end up $AB$ itself having 20°C (assuming good conductivity). Then if we separate them a little, $A$ heats up again. Then if we connect them again, $A$ cools back down. It seems a little odd, unfamiliar and non-intuitive, and as we could separate and re-attach the two plates with extremely minimal work, couldn't we take advantage of the new temperature gradient to do more work than that, thus violating some thermodynamic law?
My questions are:
- What is the case -- will the equilibrium be as in #1 or as in #2, or something else? And why?
- Have any experiments been done that actually show what will happen? For me this is the ultimate evidence.