Recently I came up with an idea of a system that apparently is able to transfer heat from a colder body to a hotter one. Obviously, this violates the 2nd rule of thermodynamics, so I would be grateful if you could point out a flaw in my understanding :)
Below is a diagram of a system. There are 2 perfectly black balls (initially both at 100K), inside 2 perfectly reflective spheres, the whole system is in a vacuum. However, parts of the spheres were removed and replaced with lenses which focus electromagnetic radiation emitted by 2 black balls.
Lenses are chosen so that they focus all the incoming radiation from one ball on another ball. No heat escapes the system since both spheres are perfectly reflective. Now, the left ball transfers a big fraction of its own radiation to the right ball, but receives little in return from the right ball. Therefore, we can deduce that there should be a heat transfer from left ball to the right one.
We can deduce the equilibrium temperature by equating heat transfers:
Where F1 is a fraction of left sphere occupied by the lens; T1 is the temperature of left ball. And F2 is a fraction of right sphere occupied by the lens; T2 is the temperature of right ball.
Equilibrium temperature does not depend on initial conditions, so if we start with both balls at 100K the equilibrium should be like 95K on left and 105K on right sphere. Also by choosing the right lens to be very small (F2 goes to 0) we can extract almost all the heat from the left ball so say 1K and 199K in equilibrium, that would be very useful!
Could you please point out a flaw in my reasoning?