F = 2 h c2 -5 / (exp(hc/kT) - 1).
or, combining the constants:
F = c1 -5 / exp(c2 /T) - 1),
where c1 = 2 hc2 = 3.7419 × 10-5 erg cm2 s-1 [ in cm]
and c2 = hc/k = 1.4288 cm °K.
In order to find the total energy emitted by a body at a given temperature, it is necessary to make a transformation of variables, from to y, where y = h c/ kT. I want to emphasize that I will make no attempt to derive the correct value of the Stefan-Boltzmann constant as that requires knowledge of complex variable theory. Therefore, I am also going to ignore the fact that we should be intergrating the specific intensity, which differs by a factor of 4/c from our equation above. Here I will simply collect the constants, and concentrate on demonstrating that the total energy radiated varies as a power of the temperature.
Therefore, we wish to find:
f = 0 F d
Substituting y = h c/ kT into this equation, and finding that:
d = -[hc/kT] y-2 dy
f = [2 k5T5/h4 c3] [y5/(ey - 1)] [-[hc/kT] y-2 dy]
or, combining terms:
f = [2 k4T4/ h3c2] y3/(ey - 1)] dy
f = aT4 y3/(ey - 1)] dy
Integrated over all wavelengths. It can be shown that the remaining integral evaluates to a constant (4/15), which can then be combined with the constant already removed from the integral, leaving us with:
f = a´T4.
When the constant is properly evaluated, it is found to be = 5.67 × 10-5 erg cm-2 s-1 K-4 , which is known as the Stefan-Boltzmann constant.
This equation is known as the Stefan-Boltzmann Law.