Stefan-Boltzmann (L=σT⁴A)

Stefan-Boltzmann law: P = ε·σ·A·T⁴

The total power a body of area A radiates at absolute temperature T comes from P = ε·σ·A·T⁴. Here σ = 5.67·10⁻⁸ W/(m²·K⁴) is the Stefan-Boltzmann constant, and ε ∈ [0, 1] is the surface emissivity, with ε = 1 standing for a perfect black body. That fourth power on T is what makes the law so touchy. Double T and the radiated power jumps by a factor of 16. Look at the Sun: its surface near T ≈ 5778 K puts out roughly 63 MW/m², which is why a small shift in stellar temperature swings luminosity and color so much.

Applications

It shows up in planetary energy balance (climate models, a planet's equilibrium temperature), solar thermal efficiency, and infrared thermography for buildings and electronics. Engineers lean on it to design incandescent lamps, astronomers to work out stellar luminosity (L = 4π·R²·σ·T⁴), and spacecraft teams to handle cooling through radiative heat dissipation.

FAQ

What is emissivity ε? It's a dimensionless number from 0 to 1 that tells you how well a real surface radiates next to a perfect black body. Polished metals sit around ε ≈ 0.05, while matte black paint reaches ε ≈ 0.95.

Why is the law so sensitive to temperature? Blame the T⁴. Nudge T up by 10% and P climbs about 46%. That's how even slight global warming ends up with large radiative consequences.

Does T need to be in kelvin? Yes, absolute temperature is the only option. Plug in Celsius or Fahrenheit and the numbers come out physically wrong, because the law is built on thermodynamic temperature.