GATE 2026 PH – Question 18
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 $$E=\int_0^\infty d\nu\,g_{2d}(\nu)\frac{h\nu}{\exp(h\nu/k_BT)-1}$$
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Correct answer: (B) $C_V\propto T^2$
Explanation
In two dimensions, the number of photon modes with frequency below $\nu$ is proportional to the area in $k$-space, $\pi k^2$, with $k=2\pi\nu/c$. So the density of states is
$$g_{2d}(\nu)\propto\nu .$$
**Energy:**
$$E=\int_0^\infty d\nu\,\nu\cdot\frac{h\nu}{e^{h\nu/k_BT}-1}\propto\int_0^\infty\frac{\nu^2\,d\nu}{e^{h\nu/k_BT}-1} .$$
Substitute $u=h\nu/k_BT$, so $\nu=k_BTu/h$ and $d\nu=k_BT\,du/h$:
$$E\propto(k_BT)^3\int_0^\infty\frac{u^2\,du}{e^u-1}\propto T^3 .$$
**Specific heat:**
$$C_V=\frac{\partial E}{\partial T}\propto T^2\quad(\text{option B}).$$
(In three dimensions the same argument gives the familiar $E\propto T^4$ and $C_V\propto T^3$.)