today in educational posts: black bodies and energy radiation, now with more math

A black body, for those who don't recall, exactly what it sounds like -- a perfectly black body that absorbs any electromagnetic radiation, and a perfect thermal emitter. The properties of the emitted radiation depend only on the temperature of the black body.

The energy density is thus also dependent on the frequency of the emitted light & the temperature of the black body, and is given by:

$u_{\nu} = \frac{8\pi\nu^{2}}{c^{3}}\frac{h\nu}{e^{\frac{h\nu}{kT} -1}}$
where u_ν is the energy density per unit volume, measured in erg-seconds per cm³. k is Boltzmann's constant, which is 1.4 * 10^-16 ergs/K in CGS. h is, of course, Planck's constant, and is 6.6 * 10^-27 erg-seconds.

The units for u_ν may be confusing, as it seems that it should be in ergs/cm³, rather than erg-seconds/cm³. However, it is important to note that u_ν is the energy density per frequency. As frequency is measured in Hz (s^-1), we then end up with (s^-1)^-1 - or, simply, s.

Intensity of a black body is simply the energy emitted per unit area per frequency per solid angle. Solid angle is a two-dimensional angle, usually used in spheres. In SI, this is measured in steradians, but in CGS, it is unitless. In order to figure out the number of steradians in a sphere, we look at a differential element of solid angle, dΩ.

Thus,

$\mathrm{d}\Omega = \mathrm{d}\theta * \cos{(90 - \phi)}}$
Since

$\cos{(90 - \phi)} = \sin{\phi}$
$\mathrm{d}\Omega = \mathrm{d}\theta * \sin{\phi}$
In order to determine the total number of steradians in a sphere, we must integrate over the entirety of the sphere.

$\int_0^{2\pi} \int_0^{\pi} (\theta * \sin{\phi})\,\mathrm{d}\theta\,\mathrm{d}\phi = 4\pi$
Therefore, there are 4π steradians in a sphere, regardless of radius.

As we have already determined the he energy emitted per unit area per frequency, one must then divide by 4π to determine the intensity, which gives:

$B_{\nu} = \frac{u_{\nu}}{4\pi} = \frac{2\nu^{2}}{c^{3}}\frac{h\nu}{e^{\frac{h\nu}{kT} -1}}$

In order to find the flux, one must multiply the intensity times the velocity with which the particles travel. This will give us the rate of flow - or flux - per solid angle. As the black body is emitting electromagnetic radiation, the velocity of the particles is c. In addition, since there are 4π steradians in a sphere, one must multiply the flux per solid angle by 4π to determine the flux through the entire sphere. Thus,

$F_{\nu} = c\frac{B_{\nu}}{4\pi} = \frac{\nu^{2}}{2\pi c^{2}}\frac{h\nu}{e^{\frac{h\nu}{kT} -1}}$

Many thanks to my classmates, Joanna Robaszewski and Cassi Lochhaas, for their help.

Flat | Top-Level Comments Only

I enjoyed reading this! Your sense of humor definitely adds to this post.

Jackie

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