Received heat by radiation from the surroundings

Question

In Stefan's law, the radiated energy of a black body with a temperature of $T$ is
$q = sigma T^4$
Where $sigma$ is Stefan's constant.
However, if it is surrounded by a medium then the net radiated energy is
$q = sigma(T^4 -t^4)$
Where $t$ is the temperature of the surroundings.
So my question is, Why the energy received from the surrounding is $sigma t^4$?
Even that air doesn't have a definite area, volume..etc. why do we assume it follows Stefan's law?

Floris · Accepted Answer

Typically for a situation like this we don't look at the air close by, but "the object at the other end of the air" (assuming that the emissivity of the air is not very large - that it is mostly transparent to EM radiation of interest).

But if you put any black body inside a container (which could be as large as the universe), there is radiation going out, and radiation coming back. The formula you give assumes that all of the surroundings are at the same temperature $t$; more realistically, you "see" a mixture of objects of different temperature, and receive "some" flux from each, proportional to the solid angle they subtend.

This is how Earth can have a temperature that is greater than the temperature of the universe, but less than the temperature of the Sun (which we can see, but it only fills part of our sky).

Does that help at all?

Received heat by radiation from the surroundings

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