# Rayleigh-Jeans Law

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### Rayleigh-Jeans Law for the Low-frequency Limit of the Blackbody Spectrum

In the limit of low energies, low frequecies, and long wavelengths ($hv\ll kT$), we get the Rayleigh-Jeans law from the formula of the blackbody spectrum. It can be expressed in several ways:

\[

\begin{align*}

B_{\nu} & \sim \frac{2\nu^{2} kT}{c^{2}} \ \ \ {\rm erg \ cm^{-2} \ s^{-1} \ Hz^{-1} \ steradian^{-1}} \\

B_{\lambda} & \sim \frac{2c kT}{\lambda^{4}} \ \ \ {\rm erg \ cm^{-2} \ s^{-1} \ cm^{-1} \ steradian^{-1}} \\

B_{E} & \sim \frac{2E^{2} kT}{c^{2} h^{3}} \ \ \ {\rm erg \ cm^{-2} \ s^{-1} \ erg^{-1} \ steradian^{-1}}

\end{align*}

\]

where $\nu$, $\lambda$, and $E$ are frequency, wavelength, and energy respectively. $T$ is the temperature in Kelvin. Example units are given to make clear the physical quantity that $B$ respresents.

In order to get formulas for the energy density (energy per unit volume per unit steradian), multiply any of the above formulas by $1/c$. Then multiply by $4\pi$ if you need the total energy density of isotropic radiation, integrated over solid angle.