B Where am I going wrong (energy transfer between black bodies)?

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The discussion revolves around a thought experiment involving two black bodies, BB1 and BB2, where BB1 radiates energy that is focused onto the smaller BB2. The key misunderstanding is the assumption that all energy from BB1 can be concentrated onto BB2, leading to the conclusion that BB2's temperature would be ten times that of BB1. However, this is incorrect due to the conservation of etendue and view factors, which limit how much energy can be focused onto a smaller area. The participants clarify that BB2 cannot radiate more energy than it receives from BB1, emphasizing the importance of these physical principles. The original poster acknowledges this mistake and expresses a desire to learn more about the conservation of etendue.
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Where am I going wrong (energy transfer between black bodies)?
I have a problem with a very basic ‘thought experiment’. I can’t see my mistake(s) - I’m pretty sure there must be at least one! So I’m accepting likely humiliation/embarrassment and asking if anyone can explain where I’m going wrong...

The surface of a black body (BB1) is at temperature T and radiates at R W/m².

An ‘optical’ system collects/redirects/focuses the radiated energy from 1m² of BB1 onto a smaller black body (BB2) of area 10⁻⁴m². We now have R watts directed onto 10⁻⁴m². That’s an incident intensity onto BB2 of 10⁴R W/m².

When BB2 reaches equilibrium, the power it receives (R watts) will be the same as the power it emits. So BB2 will emit R watts from an area of 10⁻⁴m². BB2’s surface is radiating at 10⁴R W/m².

Since a black body's radiated power/unit area is proportional to ##T_{abs}^4## this means BB2’s temperature must be 10T.

So energy is spontaneously flowing from an object at temperature T to one at temperature 10T. Err...
 
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You still only have 1 watt of radiated power to be transferred among bodies.
BB2 can't radiate more than it receives from the 1 m x 1 m of BB1 surface.
 
You appear to be assuming that all of the energy leaving a 1 square meter surface can be focused down to 1 square cm. It cannot. This is related to the conservation of etendue and view factors. I believe, without calculating it, that at most 1/10000 of the light can be focused that small.
 
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Lnewqban said:
You still only have 1 watt of radiated power to be transferred among bodies.
BB2 can't radiate more than it receives from the 1 m x 1 m of BB1 surface.
Not quite with you. The radiated power from 1m² BB1 is equal to the power received by (the much smaller) BB2. This, in turn is the same as the radiated power from BB2. Agreed,

But this means the power/m² radiated by BB2 is 10⁴ times the power/m² originally radiated by BB1. (Because BB2 is so much smaller than BB1.)

From the Stefan-Boltzmann law, this means BB2's temperature must be (⁴√(10⁴) =) 10 times that of BB1.
 
Dale said:
You appear to be assuming that all of the energy leaving a 1 square meter surface can be focused down to 1 square cm. It cannot. This is related to the conservation of etendue and view factors. I believe, without calculating it, that at most 1/10000 of the light can be focused that small.
Aha! I have indeed made that assumption. I'm not familiar with the conservation of etendue. I'll go and do some reading. Many thanks!

(Minor edit.)
 
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