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Can thermal radiation from a cooler object (B, which emits longer wavelength radiation) ever ADD to the overall thermal energy level of a warmer object (A, which emits shorter wavelength radiation)? Subsidiary question: What exactly happens - at the molecular level - to the longer wavelength radiation from B as it arrives at A?

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The thermal radiation from B does indeed heat object A. The trouble is that A loses energy by thermal radiation faster than the thermal radiation from B can heat it, so the end result is that A cools down.

You can show this very easily. The Stefan-Boltzmann law tells us that the energy flux per unit area is proportional to $T^4$ so the rate of heat loss from A is:

$$ j_A = \sigma T_A^4 $$

If we take the extreme case where B is right next to A so there is no attenuation of the thermal radiation from it, the rate of heat loss by B and therefore the heat gain by A is:

$$ j_B = \sigma T_B^4 $$

So the net rate of heat loss by A is:

$$ j_{net} = \sigma \left(T_A^4 - T_B^4\right) $$

and since $T_A > T_B$ the net rate of heat loss is positive i.e. A is cooling down so B cannot raise the temperature of A.

You ask what happens at the molecular level. The answer is that heat absorption (and radiation) is a messy business and many different interactions can occur. The dominant one is usually that EM radiation makes electrons in the solid oscillate, and the electrons transfer this energy to the lattice by colliding with it. So the the EM energy ends up as lattice vibrations i.e. heat.

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Thanks for the comment, but wouldn't you get the same answer if there was NO 'heat' added to A from B? The equation shows it is the difference between A and B that lead to jnet, not that any 'heat' is sent to A from B. To make my point, if A was supplied by a fixed power supply (eg electricity) and therefore would not cool without the presence of B, would the addition of B then lead to an increase in temperature of A? As I understand the molecular energy transfer, only energy of a higher frequency (shorter wavelength) can impart an increase of energy to the lattice. –  Arfur B Feb 23 at 15:48

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