Proving the Stefan–Boltzmann law by integrating the Planck spectrum

Step 1 of 3: The Planck energy spectrum

Intuition

Radiation in a cavity is like a 'photon gas' trapped in a box: count how many 'seats' (modes) exist at each frequency, multiply by the mean energy per 'seat' — like a restaurant's revenue: number of tables × average bill.

u(\nu, T)\,d\nu = \frac{8\pi h \nu^3}{c^3}\, \frac{d\nu}{e^{h\nu/k_BT} - 1}

Analysis

In a cavity of volume $V$ , the number of standing-wave states in $[\nu, \nu+d\nu]$ (counting 2 polarizations) is $8\pi V\nu^2 d\nu/c^3$ . Planck assigns each mode the mean energy $h\nu/(e^{h\nu/k_BT}-1)$ (Bose–Einstein statistics; photons are bosons). Multiplying the density of states by the mode energy and dividing by $V$ gives the spectral energy density $u(\nu,T)$ .