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Although I know many arguments against concentrating blackbody radiation to create a spot hotter than the blackbody, I encountered this confusing counter-example.

Consider a thin sphere blackbody of the radius $R$ locked up in a sphere mirror. And there is another blackbody of ball shape with the radius of $r$, at the center of the sphere blackbody.

Now, by doing some calculation based on the Stefan-Boltzmann law, the outgoing radiation power of the ball blackbody is given as

$$r^2 T_{ball}^4$$

ignoring all the irrelevant constants, and the all the inward radiation from the sphere blackbody that is absorbed by the ball blackbody is

$$R^2 T_{sphere}^4 (1-\sqrt{1-(r/R)^2}).$$

(This results from the geometrical calculations from the ratio of solidangles.)

And since $$ \sqrt{1-x} > 1-x $$ for all $0<x<1$, we have $$T_{sphere} > T_{ball}.$$ Now, this result is very confusing since two balckbodies never have the same equilibrium temperature. If you placed a hotter ball blackbody at the center from the beginning, you end up with reversed temparature distribution at the end. You may be able to build a permanent engine from this setting. Obviously there is a critical flaw in this reasoning but I cannot see it. Any kind of help of advice is appreciated.

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  • $\begingroup$ you have the good answer in a similar but different question $\endgroup$ – user46925 Feb 19 '16 at 4:01
  • $\begingroup$ You are way over-thinking this. The second law of thermodynamics in the Clausius formulation takes care of this: heat can only flow from hot to cold (unless something else happens). $\endgroup$ – CuriousOne Feb 19 '16 at 8:00
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Actually the temperature is same because of the Lambert's cosine law. If you apply that law, you will figure out that why is the temperature same.

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