FormulaProved

Stefan–Boltzmann law

Statement

M = σ·T⁴ — the total power radiated per unit area of an ideal black body is proportional to the fourth power of its absolute temperature, with σ = 5.670374×10⁻⁸ W·m⁻²·K⁻⁴.

Why is it true?

This is why a warm iron stays dull while a red-hot one glows — doubling the temperature multiplies the radiated power sixteenfold.

Stefan found it empirically in 1879; Boltzmann proved it thermodynamically in 1884. It's one of the few quantum-flavored laws stated before quantum mechanics existed — Planck later explained it via energy quantization.

Drag T from 1000 K to 6000 K — the area under the spectrum curve (which is M) grows as T⁴; the peak also shifts per Wien's law.

Applications

Earth's energy balance and the greenhouse effect, stellar temperature measurement, furnace and kiln design, thermal-imaging cameras, satellite radiative cooling.

Topics that use this theorem

Blackbody radiation and Planck’s law

Step-by-step proofs

Proving the Stefan–Boltzmann law by integrating the Planck spectrum

PhysicLabs, 2026

References

Josef Stefan (1879). Über die Beziehung zwischen der Wärmestrahlung und der Temperatur