Derivation of Radiation Laws

I will be deriving the Stefan-Boltzmann Law of Radiation:

as well as the Planck Law of Radiation:

I've already derived the equation of the thermal average energy in a mode (see previous post), so now to find a description of the mode with which we are going to determine the energy,  let us take a look at these energy levels within a cavity in the form of a hollow conducting cube of edge L. From E&M, we know that there are three field components, (x,y,z), and the corresponding fields within the cavity are of the form:

We know that the divergence of the field must be zero, and as such the fields are dependent upon each other:

Defining the polarization direction to be the direction of E-naught, we can choose two directions that are both perpendicular to n-hat to satisfy the above equation. 

Now substituting one of our field vectors into the laplacian wave equation we can find the frequency of the mode in terms of our triplet of integers within the n-hat vector (n-sub x,y,z)

and if we define a n to be:

then the frequencies of the modes are of the form:

and thus our total energy of all the photons within the cavity is just a sum of the average energy per mode across all modes:

Positive values for the x,y,z components of n are sufficient to describe all modes of the field vectors, therefore if we replace the sum by an integral over the volume dn, we would only be interested in the positive octant of the resulting volume, multiplied by a factor of two from our previously identified polarization direction; since there are two sets of possible modes for each nsub(xyz):

The last equation is an expression of the Stefan-Boltzmann law of radiation.

Now in order to derive the Planck radiation law, it is necessary to define the term "spectral density." Spectral density ( u sub omega) is the energy per unit volume per unit frequency range. We can find it using one of the above equations:

This law gives the frequency distribution of thermal radiation. Now it is useful to introduce the term "black body." An object is said to radiate as a black body if the radiation emission is characteristic of a thermal equilibrium distribution. Going back to our cavity problem, if this cavity were to have a small hole in it, that hole would radiate as a black body. 

In effort to manipulate the previous expression of the Stefan-Boltzman law, it is necessary now to introduce the energy flux density term J sub U. Energy flux density is defined as the rate of energy emission per unit area.

where the x(g) is a geometrical factor that arises form the angular distribution of the radiant energy flux.  The ratio of the amount of flux through the solid angle dΩ is the ratio dΩ/4π, where 4π is the total solid angle Ω. Also, since blackbody objects follow Lambert's cosine law in that the radiant intensity is directly proportional to the cosine of the angle theta away from the solid angle, we can find the geometrical factor and derive the Stefan-Boltzman constant:

Sources: Thermal Physics 2nd edition by Charles Kittel and Herbert Kroemer.