Last updated: 2026-06-15

Stefan–Boltzmann Law

Anything warmer than absolute zero glows — it radiates energy as electromagnetic waves. The Stefan–Boltzmann law tells you how much. It links the power a surface radiates to just one dominant variable, its absolute temperature, and the link is steep: the power climbs with the fourth power of temperature. This calculator turns a temperature, an emissivity and a surface area into the radiated power and the flux per square metre.

Where the formula comes from

A perfect emitter, a black body, radiates a spectrum fixed entirely by its temperature. If you add up Planck's spectrum across every wavelength, the total flux works out to be proportional to $T^4$. Writing the constant of proportionality as $\sigma$, the flux is $I = \sigma T^4$. Real surfaces radiate less than this ideal, so we scale by the emissivity $\varepsilon$ — a value from zero to one — giving $I = \varepsilon\sigma T^4$. Multiply by the area to get the total power.

Formula

The Stefan–Boltzmann constant is $\sigma = 5.670374419\times10^{-8}\ \text{W·m}^{-2}\text{·K}^{-4}$. The temperature must be absolute, because the $T^4$ term is meaningless on a scale whose zero is arbitrary.

Worked example

A black-body surface ($\varepsilon = 1$) of area $A = 1\ \text{m}^2$ at $T = 1000\ \text{K}$:

So each square metre of that surface pours out about 57 kW of radiant power.

Why the fourth power matters

The $T^4$ dependence is what makes radiation so temperature-sensitive. Doubling the absolute temperature does not double the power — it multiplies it by $2^4 = 16$. A heating element at 300 K and the same element at 600 K differ in radiated power by a factor of sixteen, which is why objects begin to glow visibly only once they are hot enough, and why small temperature errors translate into large power errors.

Limitations

The law gives the power emitted by a surface. The net radiative exchange with the surroundings subtracts what the surface absorbs from them, which is $\varepsilon\sigma A T_\text{surr}^4$ for surroundings at temperature $T_\text{surr}$. The single emissivity used here is also a simplification: real emissivity varies with wavelength, angle and surface finish. And radiation is only one of three heat-transfer routes — conduction and convection often dominate at everyday temperatures.