D-Dimensional Blackbody Cavity
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 36
25-8-2016
1655
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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