Stefan-Boltzmann Law

Insert area and absolute temperature into P = σ·A·T⁴, with the Stefan-Boltzmann constant σ = 5.670×10⁻⁸ W/(m²·K⁴). The temperature must be in kelvin. Filament example: A = 10 mm² = 10⁻⁵ m² at T = 2500 K gives P = 5.67×10⁻⁸ × 10⁻⁵ × 2500⁴ ≈ 22 W, since 2500⁴ = 3.9×10¹³. For real, non-ideal surfaces additionally multiply by the emissivity ε (0 to 1). Compute the fourth power step by step or with the power key and check the powers of ten at the end; they are the most common error source.

Because small temperature changes cause huge power changes. If the absolute temperature doubles, the radiation grows by a factor of 2⁴ = 16. Just 19 % more temperature already doubles the power, since 1.19⁴ ≈ 2. That is why thermal radiation dominates all other transport paths at high temperatures: a hotplate at 500 °C (773 K) radiates per area about 48 times more than the same plate at room temperature (293 K), calculation: (773/293)⁴ ≈ 48. Conversely, for astronomy this means: a star with twice the surface temperature shines 16 times brighter at the same size, which is why hot blue stars are far more luminous than red ones of similar radius.

The law describes radiation as a function of absolute temperature, which starts at absolute zero, where a body radiates nothing. The Celsius scale instead places its zero arbitrarily at the freezing point of water; a body at 0 °C still radiates strongly (about 315 W/m²). Because of the fourth power, Celsius errors are catastrophic: for 25 °C, computing with T = 25 instead of 298 K yields a value too small by the factor (298/25)⁴ ≈ 20,000, and for negative Celsius values the fourth power would even mask the sign problem. So always convert: T in K = temperature in °C + 273.15.

Stars radiate to a good approximation like black bodies, so L = σ·4πR²·T⁴ with the stellar radius R. For the Sun (R = 6.96×10⁸ m, T = 5778 K) this gives L = 5.67×10⁻⁸ × 6.09×10¹⁸ × 1.115×10¹⁵ ≈ 3.8×10²⁶ W. Astronomers use the relation in both directions: from measured luminosity and spectral temperature follows the radius R = √(L/(4πσT⁴)), which is how giant stars and white dwarfs were recognised as such. The temperature itself comes from the Wien displacement law using the colour of the star (λ_max·T = 2.898×10⁻³ m·K). Together the two laws make radius, temperature and power determinable from light alone.

Because the surroundings radiate back. A body at temperature T emits σ·A·T⁴ but simultaneously receives σ·A·T_U⁴ from surrounding surfaces at temperature T_U. Net, it only loses P = ε·σ·A·(T⁴ − T_U⁴). Human example: skin at 306 K in a room at 293 K gives a net of about 135 W over 1.7 m²; without the back radiation it would be an unrealistic 845 W. For small temperature differences the net loss grows approximately linearly with ΔT (Newton law of cooling). This is why we feel cold near cold windows despite warm room air: the cold pane radiates less back, so our net loss towards it increases.