I mentioned vacuum, because I want to discount the effects of conduction or convection. I simply want to know if some of the infrared radiation(IR) goes from the cooler body to the hotter body? How does each body know how much to radiate at any particular time? I assume that it ultimately comes down to temperature difference but how does the hotter body know what the temperature is of the cooler body and vice versa? We all know that both bodies will radiate IR at the 4th power of its temperature and obviously they will be eventually in equilibrium with each other, each of them then radiating an equal amount to each other.
Yes, both bodies radiate, including the cooler one. The warmer one puts out more power in its radiation, proportional to $T^4$ just as you say. The amount of radiation depends on absolute temperature, not temperature difference. But since the colder body has a lower absolute temperature, it radiates less than the hot one and the net result is that heat flows from the warmer body to the cooler one.
can a cooler body radiate Infrared radiation to a warmer body?
Yes, the cooler body will radiate, according to its temperature, as you've mentioned, and some of this radiation energy could be absorbed by the warmer body.
This will depend on the percentage of the cooler body radiation the warmer body is exposed to and on the ability of the warmer body to absorb this radiation (as opposed to reflecting or transmitting it).
obviously they will be eventually in equilibrium with each other, each of them then radiating an equal amount to each other.
This could be the case only if all radiation energy was bouncing between the two bodies and was not radiated away. If some of the energy did radiate away, it would be more difficult to predict how exactly the temperatures of the bodies would be changing without knowing all the relevant details, but, eventually, the temperature of both bodies would be zero.
How does each body know how much to radiate at any particular time?
They don't. If you assume that both bodies are "black", that is they radiate electromagnetic waves due to their temperature as described by the black body radiation equations, they do it because that's the way nature is.
[...] how does the hotter body know [..] the temperature [...] of the cooler body and vice versa?
Again: the bodies are independent of each other with regard to the emitted radiation. If the colder body is hot enough to emit IR radiation at all, it will do so regardless of other bodies around it.
I am agree with answers by @Ricky Tensor and @V.F., but more detailed answer is "there are several scenarios". It will depend on:
- How much total radiation emits each of bodies
- How much each body absorbs of enother body's radiation (how close they are, how reflective they are)
- What is the rate of cooling for each body (depends on their thermal transfer properties from inside to the surface)
For example if body A emits less radiation than absorbs radiation from hotter body B, them body A will not be cooling. Its temperature will be rizing instead until its radiation become equal to absorbed radiation.
Also if body A is cool but loses its temperature slowly, and body B is very hot but cools fast - there can be a moment than body B becomes cooler than A.
But in every case both bodies A and B lose their temperature slower than they do in absense of other body.
I believe your questions stem from a misunderstanding of the physics involved.
The IR a body radiates, depends on its absolute temperature, NOT on the temperature difference it might have, with respect to another body! Therefore, the bodies in question, do not know (nor do they need to know) the temperature of the other body.
Your statement about IR being dependent on the 4th power of its temperature, is almost correct. You did not say absolute temperature, and this might be the source of the confusion?
It is not obvious that both bodies would eventually reach the same temperature! This is due to the fact that they are not in an enclosed system.
Lets take the Sun and the Earth as an example.
The Sun is hotter, but it is radiating most of its energy away from the Earth.
Although the Earth is cooler, it also radiates IR due to its internal core temperature and the energy it receives from the Sun. A large part of this energy is also radiated away from the Sun.
Will both bodies eventually reach the same temperature? For reasons we all know, it is highly unlikely!