If two objects close together in a vacuum are in thermal equilibrium with their surroundings, but are of different sizes so they are actually emitting radically different amounts of energy, is it still the case that they will have no effect upon each other?
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$\begingroup$ As the answer below says, you need to consider "the wall" that surrounds both objects. So imagine a closed perfectly reflecting cavity and within the two objects. Any radiation emitted by any of the objects hit either the other object or the cavity. Reflections from the cavity will again hit either of the two objects. All in all an object will absorb the same amount of energy as it emits (independent of the form of the cavity). $\endgroup$– daniSep 17, 2014 at 13:05
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Essentially from the definition of thermodynamic equilibrium yes. The Zeroth Law of Thermodynamics states that if a body $A$ is in thermodynamic equilibrium with $B$ and $B$ is in thermodynamic equilibrium with $C$ then $A$ is in thermodynamic equilibrium with $C$ and we say that $A$, $B$ and $C$ have the same temperature. This is precisely the setup you have described, with the surroundings playing the role of $B$, so the two objects must be in thermal equilibrium with each other.
A nice way to see why this will work in the case you have described, assuming both objects to be black bodies, is to notice that when calculating the energy emitted by one object in the direction of the other we only need to consider the cross section of the object first object, as that is the area which can draw line of sight to the second object. The intensity of radiation emitted from this surface depends only on the temperature, by the Stefan-Boltzmann Law. However the first object also blocks radiation from the surroundings from reaching the second object. The area casting this shadow is the cross sectional area of the first object and the intensity and spectrum of the radiation is again that of thermal radiation for a given temperature. But since the surroundings and the first object are in thermal equilibrium, they have the same temperature, and so the radiation being blocked and the radiation being emitted are identical! Consequently the two objects have no effect on each other.
answered Sep 17, 2014 at 11:17