Physics Friday 33: Blackbody Radiation (Part 2)

Last week, in Part 1, we used a cubical cavity full of radiation at equilibrium to model a blackbody, and thus compute the specific intensity (radiated power per unit area per unit solid angle per unit frequency) Iν as a function of frequency:
,
where is the average energy per allowed radiation mode with the radiation at equilibrium temperature T. Using the classical Boltzmann probability distribution, we obtained the Rayleigh-Jeans law, which diverges from reality at high frequencies (and gives infinite total power radiated!). So, now we introduce the key assumption: that the energy in any given radiation mode in the cavity is not continuous, but instead can only take integer multiples of some quantity proportional to frequency: that is, for a mode of frequency ν, , with n a nonnegative integer, and h some proportionality constant.


We still have that the probability of a given energy is , but since the energy is discrete, this is a geometric distribution. Thus, with ,
.
and using the geometric series formula

with to find that the sum in the denominator to be , making the probability
, and giving us average energy:
.
A little calculus can be used to show that
, so that the sum above becomes , and so
.

Plugging into our specific intensity, we find
.
This last is Planck’s law, which fits the observed spectrum for blackbody radiation at all frequencies. The proportionality constant h is Planck’s constant (and thus its name).

We see that the spectrum given by Planck’s law has a frequency at which the specific intensity peaks; some calculus will show that this frequency is directly proportional to the temperature (with the constant of proportionality involving a numerical constant as the maximization equation can only be solved numerically). Further, it integrates over frequency to give a finite value. To find the total irradiance j (power emitted per unit of surface area of the blackbody), we need to integrate Iν over a half-sphere of solid angle (consider a small flat bit of surface) and frequency.
First, for the solid angle, we use spherical coordinates. We must combine Lambert’s cosine law and the fact to get

and thus
.
Using the u substitution , we find , , we find
.
Now, to find the remaining integral, I showed in Monday Math 29 that , we use x=4 to see that
. Now, , and we found in Monday Math 26 that , so , and thus
, where is known as the Stefan-Boltzmann constant, and the relation is called the Stefan-Boltzmann law.

As a final note, we note that the way Planck found this law, and the resulting rise of quantum mechanics, is often misrepresented in physics texts. You can read about the real story here.

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One Response to “Physics Friday 33: Blackbody Radiation (Part 2)”

  1. Physics Friday 35 « Twisted One 151’s Weblog Says:

    […] we found previously that the power radiated per unit of surface area for a blackbody is given by the Stefan-Boltzmann […]

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