Kirchhoff's Law of Heat Radiation and its Theoretical Proof

Kirchhoff's law of thermal radiation connects a body's emissive power to its absorptive power, wavelength by wavelength, for a body in thermal equilibrium. It states: at a given temperature, the ratio of a body's emissive power to its coefficient of absorption equals the emissive power of a perfect blackbody at the same temperature, for every wavelength.

Since a body's emissive power can be expressed relative to a blackbody through its emissivity $e$ (Section 3.12.1), Kirchhoff's law can equivalently be stated as: for a body emitting and absorbing thermal radiation in thermal equilibrium, its emissivity equals its absorptivity -- symbolically, $a = e$, or wavelength-by-wavelength, $a(\lambda) = e(\lambda)$. In other words, a body that is a strong absorber at some wavelength is, at that same wavelength and temperature, equally strong an emitter -- and a poor absorber is equally a poor emitter.

Theoretical proof. Consider an ordinary body A and a perfect blackbody B, of identical geometric shape, placed together inside a closed enclosure. Once thermal equilibrium is reached, both A and B (and the enclosure) are at the same common temperature.

Let $R$ be the emissive power of body A, $R_B$ the emissive power of blackbody B, and $a$ the coefficient of absorption of body A. If $Q$ is the quantity of radiant heat incident on each body (per unit time), the quantity absorbed by body A is $Q_a = aQ$. Since both A and B remain at constant (equal) temperature, each must emit exactly as much energy as it absorbs, per unit time -- otherwise its temperature would drift. Because emissive power is heat radiated per unit area per unit time, this equilibrium condition for body A gives

For the perfect blackbody B, since $a_B = 1$ (it absorbs everything incident on it), the same equilibrium condition gives simply

Dividing the first equation by the second:

But by definition (Section 3.12.1), $e = R/R_B$ -- so $a = e$, which is exactly Kirchhoff's law, now proved from the requirement that both bodies stay in thermal equilibrium. …