F = 2 h c^{2} ^{-5} / (exp(hc/kT) - 1).

or, combining the constants:

F = c_{1} ^{-5} / exp(c_{2} /T) - 1),

where c_{1} = 2 hc^{2} = 3.7419 × 10^{-5} erg cm^{2} s^{-1} [ in cm]

and c_{2} = 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

we obtain

*f* = [2 k^{5}T^{5}/h^{4} c^{3}] [y^{5}/(e^{y} - 1)] [-[hc/kT] y^{-2} dy]

or, combining terms:

*f* = [2 k^{4}T^{4}/ h^{3}c^{2}] y^{3}/(e^{y} - 1)] dy

or:

*f* = aT^{4} y^{3}/(e^{y} - 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´T^{4}.

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.

Introduction to Blackbody radiation

Wien Distribution

Wien Displacement Law

Stefan-Boltzmann Law

Constants

This VRML + HTML package on Blackbody emission was constructed by Karen M. Strom. Contact your local instructor for help if it is used as part of your class.