Stefan-Boltzmann Law

The equation for blackbody radiation (the Planck equation) is:

F lambda = 2 h c² lambda ^-5 / (exp(hc/k lambda T) - 1).

or, combining the constants:

F lambda = c₁ lambda ^-5 / exp(c₂ / lambda T) - 1),

where c₁ = 2 hc² = 3.7419 × 10^-5 erg cm² s^-1 [ lambda 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 lambda to y, where y = h c/ lambda 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 integral ^inf F lambda d lambda

Substituting y = h c/ lambda kT into this equation, and finding that:

d lambda = -[hc/kT] y^-2 dy

we obtain

f = [2 k⁵T⁵/h⁴ c³] integral [y⁵/(e^y - 1)] [-[hc/kT] y^-2 dy]

or, combining terms:

f = [2 k⁴T⁴/ h³c²] integral y³/(e^y - 1)] dy

or:

f = aT⁴ integral y³/(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´T⁴.

When the constant is properly evaluated, it is found to be sigma = 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.

Note: The notation exp(x) is another way of writing e^x. In this case I used this form because of the complexity of the term in the exponent, the difficulty of properly creating it in HTML (the language used to write these pages), and the relatively low resolution of computer screens.

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.