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- TL;DR Summary
- 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...

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...