The formula for energy \(E\) of a photon gas at temperature \(T\) in a two-dimensional box at equilibrium with \(g_{2d}(\nu)\) denoting the density of states of photons is given below where symbols \(\nu\), \(h\) and \(k_{B}\) have their standard meaning. The specific heat (\(C_{V}\)) of this photon gas obeys
\[ \begin{aligned} E = &\int_{0}^{\infty} d\nu\, g_{2d}(\nu) \\ &\times \frac{h\nu}{\exp\left(\frac{h\nu}{k_{B}T}\right) - 1} \end{aligned} \]
GATE Physics - 2026,
Answer: B
The specific heat obeys \(C_{V} \propto T^{2}\).
In two dimensions the number of photon modes with frequency below \(\nu\) grows as the area of a disc in k-space, \(\propto \nu^{2}\), so the density of states is \(g_{2d}(\nu) \propto \nu\). Substituting \(x = h\nu/k_{B}T\):
\[ \begin{aligned} E &\propto \int_{0}^{\infty} \frac{\nu \cdot h\nu}{e^{h\nu/k_{B}T} - 1}\,d\nu \\ &\propto T^{3}\int_{0}^{\infty} \frac{x^{2}}{e^{x} - 1}\,dx \propto T^{3} \\ C_{V} &= \frac{dE}{dT} \propto T^{2} \end{aligned} \]
\(T^{3}\) is the three-dimensional result (from \(E \propto T^{4}\), the Stefan-Boltzmann law), and \(T^{4}\) confuses the 3D energy with its specific heat. \(C_{V} \propto T\) would be the one-dimensional photon gas.
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