Planck's law

Remarks

Proof idea: in a cavity, the number of electromagnetic modes per unit volume and unit frequency is $8\pi \nu^2 / c^3$. If each mode of frequency $\nu$ can hold only energies $n h \nu$ with $n = 0, 1, 2, \dots$, the Boltzmann distribution gives the mean energy $h\nu / (e^{h\nu/(k_B T)} - 1)$ per mode. Multiplying gives the energy density $u_\nu$, and the radiance is $B_\nu = c \, u_\nu / (4\pi)$. The second form follows from $B_\lambda \, |d\lambda| = B_\nu \, |d\nu|$ with $\nu = c/\lambda$.

For $h\nu \ll k_B T$ the law reduces to the Rayleigh–Jeans law $B_\nu \approx 2 \nu^2 k_B T / c^2$, and for $h\nu \gg k_B T$ to Wien's approximation $B_\nu \approx (2h\nu^3/c^2) e^{-h\nu/(k_B T)}$. Integrating over frequency gives the Stefan–Boltzmann law, and locating the maximum gives Wien's displacement law. Max Planck found the law in 1900.

Going deeper

For a student

The law is the reference spectrum of radiative heat transfer: real surfaces are described by their emissivity relative to it. The two classical limits show where classical physics works: at low frequency equipartition is correct, and at high frequency the quantisation of energy suppresses emission. The same formula describes the cosmic microwave background, which fits a black-body spectrum at about $2.7$ K.