Black Body Radiation II

In the last 2 sessions we have established that the Cavity Radiation spectrum is identical to the black body radiation spectrum and this is expressed as

ρ(ν)dν=constant. R(ν)dν at some temperature T deg K at thermal equilibrium.

The cavity radiation is due to the energized atomic oscillators and at ε is a standing wave configuration ( assuming no dissipation at walls)

To calculate the energy density of the cavity radiation in an infinitesimal frequency interval dν we need

(i) The number of modes in that interval

(ii) The average total energy in each mode

so ρ_T(ν)= # of modes X average energy/volume

To count the modes for simplicity we assume a cubic cavity and we will ses that the final result is independent of the geometry of the cavity.

As a warm up exercise we first compute the modes for 1 dimension ( line cavity of length L). We solve the 1 d wave equation with the boundary condition of nodes at the walls at x=0 and x=L. The boundary conditions restricts the modes to be such that the wave vector k=(nπ/L).

from standard relations between frequency and wave vectors its easy to see that the number of modes in the frequency interval dν is dN=(2L/C) X dν

This is easily generalized to 3d with a cube of side L and
assuming independent propagation in (x, y,z) directions due to parallel walls and

n²=n²_x + n²_y
+ n²_z

Using the same construction as 1d we mark the n's on a 3 d lattice and we see that
the number of modes = volume in lattice space. To find the number of modes in the frequency interval dν it is required to find the volume between two spheres of radius N and N + dN in the first octant as all
n > 0.

N= (1/8)(4π/3)n³=(π/6)(2L/c)³ ν³

so that dN=N(ν)dν=(4π/c)³ L³ν²dν X 2

for the 2 states of polarization of EMW

so the density is dN/L³ = (8π/c³)ν² dν

This is to be multiplied by the average total energy of each mode which may have any energy continuously between 0 and infinity.

Since the standing waves are in Ε with each other the equipartition theorem may be applied to find the average total energy to be E=kT.
(potential and kinetic)

So ρ(ν)dν=kT (8π/c³)ν² dν

This is the famous Rayleigh Jeans law and shows a quadratic dependence on the frequency. It is quite obvious that the law predicts an indefinite increase in the
energy density with frequency. It describes the black body spectrum at low frequencies but breaks down at high frequencies ( ultraviolet direction). Hence this came to be known as the "Ultraviolet Catastrophe"

It actually marks the failure of classical physics comprising of Mechanics and
EM.