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Black Body Radiation :

The frequency $\nu=\omega/2\pi$ is given by

$\displaystyle \nu = \frac{c}{2L}\sqrt{n_x^2+n_y^2+n_z^2}$

For a given frequency, the equation above represents a sphere of radius $R = 2L\nu/c$ in the three dimensional space of

and

and each value of $\nu$ represents a distinct point in this space. Since

can only take integral values, the number of points per unit volume is one. If we treat $\nu$ as a continuous variable, the number of modes for frequency less than some given $\nu$ is given by

$\displaystyle N(\nu)$	$\displaystyle =$	$\displaystyle 2\times \frac{1}{8} \frac {4\pi R^3}{3}$
	$\displaystyle =$	$\displaystyle \frac{8\pi V}{3 c^3}\nu^3$

where

is the volume of the cavity. In the above, the factor of 1/8 comes because we are restricted to the positive octant as

can only be positive. The factor of 2 takes into account the fact that there are two transverse modes. The number of modes in the frequency interval $\nu$ and $\nu+d\nu$ is

$\displaystyle N(\nu)d\nu$	$\displaystyle =$	$\displaystyle N(\nu+d\nu)-N(\nu)$
	$\displaystyle =$	$\displaystyle \frac{8\pi V}{3 c^3}[(\nu+d\nu)^3-\nu^3]$
	$\displaystyle =$	$\displaystyle \frac{8\pi V}{c^3}\nu^2d\nu$