Stefan-Boltzmann Law

Every surface warmer than absolute zero radiates, and the power it sends out climbs with the fourth power of its absolute temperature. Double the kelvin temperature and the radiated power grows sixteenfold. The fourth power is not arbitrary: the number of photons a hot body emits per second scales roughly as $T^3$, and the average energy each one carries scales as $T$, so the product goes as $T^4$. The Stefan–Boltzmann constant σ = 5.670374419 × 10⁻⁸ W/(m²·K⁴) is exact in the 2019 SI, and emissivity ε runs from 0 to 1, where 1 is a perfect black body.

Work a person. A clothed adult presents about 1.8 m² of surface at roughly 28 °C (301 K) with an emissivity near 0.98. The gross emission is $0.98 \times 5.67 \times 10^{-8} \times 1.8 \times 301^4 \approx 823\ \text{W}$ — which is absurd, since no one eats 823 W. The resolution is in the next paragraph but one, and it is the whole practical lesson of this equation.

Josef Stefan found the fourth-power rule empirically in 1879, fitting it to John Tyndall's measurements of glowing platinum wire. Ludwig Boltzmann derived it from thermodynamics five years later, treating radiation as a gas that exerts pressure, and the joint name has stuck since. It sat as an empirical law with a thermodynamic argument behind it until 1900, when Planck's radiation formula produced it by integration over all wavelengths — one of the first things anyone checked about the new quantum picture. On the neighbouring astronomical page it does its most famous work: a star's luminosity is $4\pi R^2 \sigma T^4$, so measuring brightness and surface temperature gives the radius of an object no telescope can resolve.

Nothing radiates into a void, and forgetting that is the standard error. That person is standing in a room whose walls are at 20 °C, and those walls are radiating back. What you feel is the net exchange, $\varepsilon\sigma A(T^4 - T_{surr}^4)$, which here comes to about 84 W — a believable figure and one-tenth of the gross. Use the bare form when the surroundings really are cold, as with a spacecraft radiator or a clear night sky, and use the difference form for anything in a room. It also explains why a 20 °C room feels cold beside a single-glazed window and comfortable beside an insulated wall at the same air temperature: the window's inner surface is colder, so you lose more by radiation while the thermometer reports nothing amiss.

Two further traps. Because of the fourth power, a Celsius temperature is not merely inaccurate, it is destroyed: entering 100 instead of 373.15 K is wrong by a factor of $(373.15/100)^4 \approx 194$. And emissivity is a property of the surface, not the substance, and not of its colour to the eye. Polished aluminium sits near 0.05, the same aluminium oxidised near 0.2, and almost every paint — white, black, or anything between — sits around 0.9 in the thermal infrared, because visible colour says nothing about behaviour at 10 μm. This is why an infrared thermometer pointed at shiny bare pipe reads far too low: the instrument assumes an emissivity, usually 0.95, and the metal is not obliging. A strip of matte tape on the pipe fixes the reading.