Problem 5.259

Making use of Planck's formula, derive the expressions determining the number of photons per $1 \mathrm{~cm}^{3}$ of a cavity at a temperature $T$ in the spectral intervals $(\omega, \omega+d \omega)$ and $(\lambda, \lambda+d \lambda)$ .

Reveal Answer

n_{\omega} d \omega=\frac{1}{\pi^{2} c^{3}} \frac{\omega^{2} d \omega}{e^{\hbar \omega / h T}-1}, \quad n_{\lambda} d \lambda=\frac{8 \pi \lambda^{-4} d \lambda}{e^{2 \pi \hbar c / h T \lambda}-1}

Thermal Radiation. Quantum Nature of Light.