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Date: 29-8-2016
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D-Dimensional Blackbody Cavity
Consider a D-dimensional hypercube blackbody cavity. What is the energy density as a function of temperature? It is not necessary to derive the multiplicative constant. Assume that the radiation is in quanta of energy E(ω) = hω.
SOLUTION
For a photon gas the average number of photons per mode ω is given by
(1)
The energy
(2)
where V is the volume of the hypercube:
Substituting x = hω/τ into (2), we obtain
(3)
The energy density is simply
(4)
For D = 3 we recover the Stefan–Boltzmann law:
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