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Blackbody Radiation

1. The problem statement, all variables and given/known data

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/radpow.html#c1

What I don't understand is the second part, with the angles. The more I think about it, the less it makes sense.

2. Relevant equations

R = c/4 U

3. The attempt at a solution

Whenever I did a surface integral in the past, for example, for ∫∫ j.dS in electromagnetism, I multiplied by a factor of cos θ. The flux orthogonal to the surface being considered is always the smallest. However, in the construction given on hyperphysics for blackbody radiation, the flux orthogonal to the surface is greater than the flux at an angle.

Furthermore, the angle seems arbitrarily defined because there are two degrees of freedom not counting the x-axis.

Lastly, I don't understand why there is a "longer element" and a "reduced cross section" when the effective volume of the slanted surface element is the same.

I've been looking everywhere for a better reason behind the "c/4" factor, but all I found are these:

http://sci-fix.blogspot.com/2010/07/greenhouse-effect.html

"The c/4 factor is because Planck's (and Rayleigh-Jeans) law stands for density (unit volume) of radiant energy, but what is actually measured is radiant emittance or spectral radiance (per unit surface), which depends as well on speed of outgoing radiation."

http://www.cdeep.iitb.ac.in/nptel/Core%20Science/Engineering%20Physics%202/Slides/Module-5/Lec-23/lec23_8.html [Broken]

"We need to average over all angles. In computing the radiant power, we get a factor of cos2 θ which averages to 1/2."

I've been at this for hours. Someone please tell me how to -properly- get the c/4 factor.
 
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