Stefan-Boltzmann Law of Radiation

This section makes precise exactly how much thermal radiation a blackbody emits, as a function of its temperature.

History and statement. In 1879, Josef Stefan proposed an empirical relation, based on experimental observations, between the radiant power $R$ emitted per unit area by a perfect blackbody and its absolute temperature $T$. Five years later, Boltzmann derived the same relation theoretically from thermodynamics -- so the result is known as the Stefan-Boltzmann law. It states: the rate of emission of radiant energy per unit area (the power radiated per unit area) of a perfect blackbody is directly proportional to the fourth power of its absolute temperature.

where $\sigma$ is Stefan's constant, $\sigma = 5.67\times10^{-8}$ J m$^{-2}$s$^{-1}$K$^{-4}$ (equivalently W m$^{-2}$K$^{-4}$), with dimensions $[L^0M^1T^{-3}K^{-4}]$.

A crucial feature of this law is that the power radiated by a perfect blackbody depends only on its temperature -- not on its colour, its material, or the nature of its surface. (This is, of course, specific to the idealised perfect blackbody; Section 3.12.1's emissivity captures how real, non-ideal surfaces fall short of this.)

If a perfect blackbody of surface area $A$ at temperature $T$ radiates for a time $t$, the total energy emitted is $Q = A\sigma T^4 t$ (used directly in Example 3.6). For an ordinary (non-blackbody) surface with emissivity $e$, the energy radiated per unit area per unit time is reduced proportionally:

(used in Example 3.7 to find an unknown filament temperature from a known emissivity and power).

Radiating into a warmer or cooler surroundings. If a perfect blackbody at temperature $T$ sits in surroundings at a different absolute temperature $T_0$, it simultaneously radiates energy at rate $\sigma T^4$ per unit area and absorbs energy from the surroundings at rate $\sigma T_0^4$ per unit area (Prevost's theory of exchange, Section 3.12). Its net loss of energy per unit area per unit time is therefore

and for an ordinary body of emissivity $e$,

If instead $T < T_0$ (the body is cooler than its surroundings), this same expression, now negative, represents a net gain of thermal energy by the body per unit area per unit time (used in Example 3.8 to compare cooling rates of a sphere at two different temperatures against a common surrounding temperature, and in Example 3.9 to compare the total radiated powers -- and hence relative sizes -- of two blackbody stars). …