Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 16.8-2

ENTROPY AND DISORDER; GENERALIZED CANONICAL FORMULATIONS

Physics — Statistical Mechanics · planck-radiation-law-rayleigh-jeans

Problem

Show that the energy per unit volume of electromagnetic radiation in the frequency range

d\omega

is given by the ``Planck Radiation Law''

\frac{U_\omega}{V}\,d\omega = \frac{\hbar\omega^3}{\pi^2 c^3}\left(e^{\beta\hbar\omega} - 1\right)^{-1}d\omega

and that at high temperature (

k_B T \gg \hbar\omega

) this reduces to the ``Rayleigh--Jeans Law''

\frac{U_\omega}{V}\,d\omega = \frac{\omega^2}{\pi^2 c^3}k_B T\,d\omega