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I am trying to calculate the rate at which photons of any particular frequency will pass through a volume in a room illuminated by black body radiation only.

I've found a couple of starting points but now I'm stuck putting it together.

I could start with $B_\lambda(\lambda, T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda k_\mathrm{B}T}} - 1}$ and pretend all the radiation is coming from one spot on the wall, then multiply it by the effective solid angle to the volume and the volume itself, but I'm not sure if this is reasonable.

Also since I'm counting photons it seems I want to take a definite integral over the relevant frequency range. In which case, setting $x=\lambda$, $k=2kc^2$, and $q=hc/(k_BT)$ I'll end up with this less-than-elegant answer:

enter image description here

I suspect there's an easier way. Is there a simple formula for the energy density of empty space when the volume under examination is "far" from the walls?

To restate the problem: take a small volume of space within a room whose walls are emitting black body radiation at a known temperature; the size and shape of the room is known (we can assume any shape which is convenient, if that matters) -- how many photons per second with wavelength $\lambda_{min}<\lambda<\lambda_{max}$ are passing through this volume?

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Just take the volume as your system and integrate the Plank distribution over the wavelengths/frequencies you are interested in (tip: don't use the wave-length but the angular frequency version, it is more concise). This works, because the Planck law describes an equilibrium setting, whether your volume is in equilibrium with the walls or the surrounding vacuum does not matter.

Then the average number of photons in the volume can easily be given, by the well known formulas for the number of photons in a cavity.

(This procedure obviously gets invalid if the volume becomes so small that a significant number of the photons have a wavelength larger than the sides of the box).

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  • $\begingroup$ Could you clarify "take the volume as your system" please? I think you've glossed over the essential core of my question :-) $\endgroup$ – spraff Nov 21 '15 at 1:36
  • $\begingroup$ I have to admit my answer does not give the quantity you ask for, but the number of photons "passing through" a volume in a given amount of time is just not a useful (or even meaningful) concept. Either you are interested in the photon flux through an area (and the net flux will be zero in equilibrium, and even this quantity is probably only available semi-classically), or you ask for the number of photons in the volume. I am not even sure one can rigorously define the quantity you ask for – photons are not located mass points that fly around in determined paths. $\endgroup$ – Sebastian Riese Nov 21 '15 at 1:46
  • $\begingroup$ The answer just gives a simple procedure calculating the average number of photons in a given box inside a larger cavity. BTW: I do not see how your calculation gives a photon flux rate either. Another side note: If I now understood your question correctly the title of the question is very confusing. $\endgroup$ – Sebastian Riese Nov 21 '15 at 1:48
  • $\begingroup$ Intuitively, if you put an object into this little volume, how many photons per second would hit it? $\endgroup$ – spraff Nov 21 '15 at 9:16
  • $\begingroup$ That is rather a question about the surface than about the volume of the object. And the answer is, as many as the object would radiate where it in equilibrium (because then it will emit as many photons as it absorbs). This number can also be easily calculated from the black body spectrum (similar to the way you derive the Stefan-Boltzmann law). $\endgroup$ – Sebastian Riese Nov 21 '15 at 16:19

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