All Problems Thermal Radiation. Quantum Nature of Light.
Problem 5.255
Using Planck's formula, derive the approximate expressions for the space spectral density \(u_{\omega}\) of radiation (a) in the range where \(\hbar \omega \ll k T\) (Rayleigh-Jeans formula); (b) in the range where \(\hbar \omega \gg k T\) (Wien's formula).
(a) \(u_{\omega}=\left(k T / \pi^{2} c^{3}\right) \omega^{2}\) (b) \(u_{\omega}=\left(\hbar / \pi^{2} c^{3}\right) \omega^{3} \mathrm{e}^{-\hbar \omega / k T}\)