Stefan-Boltzmann Law Calculator

Stefan–Boltzmann Law

The Stefan–Boltzmann law states that the total radiant exitance (power per unit area) emitted by a body is proportional to the fourth power of its absolute temperature: j = εσT⁴, where σ = 5.670374×10⁻⁸ W/m²·K⁴ is the Stefan–Boltzmann constant, ε is the surface emissivity (1 for an ideal blackbody, ~0.05 for polished metal, ~0.95 for human skin) and T is the absolute temperature in kelvin.

Derived in 1879 by Józef Stefan empirically and theoretically by Ludwig Boltzmann in 1884, it is a direct consequence of integrating Planck’s spectral radiation law over all wavelengths. Combined with Wien’s displacement law (λmaxT = 2898 µm·K) it completely characterizes blackbody emission.

Applications

Astrophysicists use it to compute the effective temperature of stars (the Sun’s 5778 K comes from this law); thermal engineers apply it in furnace, boiler and heat-exchanger design; building scientists rely on it for radiative heat balance and roof emissivity selection; thermography (FLIR cameras) inverts it to extract surface temperature from measured infrared flux.

FAQ

Why the fourth power? It comes from integrating Planck’s law ∫B(ν,T)dν over all frequencies — the dν³ volume element in phase space plus the kT scaling of photon energy yields T⁴.

What is net radiation between two surfaces? q = εσ(T₁⁴ − T₂⁴). A surface at 300 K losing heat to an environment at 290 K radiates approximately 56 W/m² net.

Does it apply to non-black bodies? Yes, by multiplying by emissivity ε (Kirchhoff’s law: emissivity equals absorptivity at the same wavelength and temperature).