Wien's Displacement Law - Derivation From Planck's Law

Derivation From Planck's Law

Wilhelm Wien first derived this law in 1893 by applying the laws of thermodynamics to electromagnetic radiation. A modern variant of Wien's derivation can be found in the textbook by Wannier.

Planck's law for the spectrum of black body radiation may be used to find the actual constant in the peak displacement law. Specifically, the spectral energy density (that is, the energy density per unit wavelength) is

Differentiating u(λ,T) with respect to λ and setting the derivative equal to zero gives

which can be simplified to give

By defining the dimensionless quantity

then the equation becomes

The numerical solution to this equation is

Solving for the wavelength λ in units of nanometers, and using kelvins for the temperature yields:

The frequency form of Wien's displacement law is derived using similar methods, but starting with Planck's law in terms of frequency instead of wavelength. The effective result is to substitute 3 for 5 in the equation for the peak wavelength. This is solved giving x = 2.82143937212...

Using the value 4 in this equation (midway between 3 and 5) yields a "compromise" wavelength-frequency-neutral peak, which is given for x = 3.92069039487....