(a) Show that the number of photons in equilibrium at...

(a) Show that the number of photons in equilibrium at temperature $T$ in a cavity of volume $V$ is $N=V(k T / \hbar c)^3$ times a numerical constant.
(b) Use this result to obtain a qualitative expression for the heat capacity of a photon gas at constant volume.

Key Concepts

Heat Capacity

Heat capacity is a thermodynamic measure of how much energy is needed to raise the temperature of a system by a given amount. In the context of a photon gas, understanding how the total energy of the photon population changes with temperature (often scaling as T^4 for the energy density) allows one to qualitatively determine the heat capacity at constant volume, reflecting the underlying statistical behavior of the photons.

Bose-Einstein Statistics

Bose-Einstein statistics describe the distribution of bosons over various energy states at thermal equilibrium. For photons, this statistical framework is simplified by the fact that their chemical potential is zero, allowing the derivation of the equilibrium distribution and the associated thermodynamic properties such as number density and energy density.

Black-Body Radiation

Black-body radiation refers to the electromagnetic radiation emitted by an idealized object that absorbs all incident radiation, regardless of frequency or angle. The study of black-body radiation lays the foundation for understanding the equilibrium properties of a photon gas inside a cavity, including the derivation of temperature-dependent quantities like energy density and photon number.

Photon Gas

A photon gas is a collection of photons in thermal equilibrium, exhibiting properties typical of a gas of massless bosons. It is characterized by a spectrum described by Planck's law and shows that the photon number is not conserved, which is reflected in the zero chemical potential in its Bose-Einstein distribution.

Density of States

The density of states represents the number of available quantum states per unit energy or frequency interval for photons in a cavity. It is crucial for determining how the number of photons scales with temperature and volume by integrating over all possible modes, which leads to expressions like N = V (kT/?c)^3 multiplied by a numerical constant.

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