# Blackbody radiation - Radiative flux derivation of the Stefan-Boltz.

1. Jun 28, 2013

### klawlor419

I am trying to resolve some long standing problems I have encountered with blackbody radiation. Namely, the derivation of the radiation energy flux equation $$J=\sigma_{B} T^4$$.
I understand the derivation of the energy density of photons in "a box". $$U/V=const. T^4$$
I do not understand the jump from particles in a box to blackbody radiation flux.

I get the radiation inside a box argument with the cavity modes and etc. That is fine. But what I do not understand is the jump from that model to an actual blackbody. I also understand that a blackbody is purely a thought experiment. Meaning there is no such thing as a perfect blackbody I could hold in my hands.

My problem is that I do not completely see the relevance of the photons in a box to the way in which a blackbody radiates. Why should the particle in a box model be representative of the way things radiate?

In Schroeder's Thermal Physics text he presents a derivation of the radiation flux. The argument makes perfect sense to me. And you can derive the following type of expression for the flux by considering an volume element in a solid angle containing photons in the direction of the by weighting the result with the probability of passing through the effective hole size. You end up with something like this,
$$cu_\omega\cos{\theta}\frac{d\Omega}{4\pi}$$
Where $$\theta$$ goes from $$0 \text{ to } \pi/2$$, which gives you the $$J=cU/4V$$ result.

He then discusses the significance of the second law in determining the equivalence of these two problems, radiation in a box and blackbody radiation. The second law is violated in the following experiment unless a blackbody absorbs precisely as a hole in a box filled with radiation.

Put a black surface (blackbody) directly outside of the hole in a box or spherical cavity. If the two objects are held at the same temperature there should be no net radiation transfer the between, meaning that neither of the two objects temperatures should increase else the second law is violation.

Is this the only evidence of equivalence we have for these two problems? What are the cavity modes in the blackbody?

To get back to my original question, why should the equivalence of the two problems lead us to transfer radiative flux through the hole to the blackbody itself? Is there a derivation of the radiative flux that involves purely a blackbody?

My work is leading me to one issue that the integration range for the theta values in the following is from 0 to Pi by standard definitions of spherical coordinates, when it has to be from 0 to Pi/2 to give the correct answer.

Thanks ahead of time.

2. Jun 28, 2013

### klawlor419

Btw by "long standing problems" I mean nothing more than concepts I have yet to understand. I don't mean to sound as if the problem of blackbody radiation is not resolved.

3. Jun 29, 2013

### klawlor419

What happens to the photons when they are absorbed on the surface of a black body?