## What is Kirchhoff’s Law?

To understand Kirchhoff’s law let us consider a small body of surface area As , emissivity ϵ, and absorptivity α at temperature T contained in a large isothermal enclosure at the same temperature, as shown in Figure 1. Recall that a large isothermal enclosure forms a blackbody cavity regardless of the radiative properties of the enclosure surface, and the body in the enclosure is too small to interfere with the blackbody nature of the cavity. Therefore, the radiation incident on any part of the surface of the small body is equal to the radiation emitted by a blackbody at temperature T. That is, G = Eb(T ) = σT 4, and the radiation absorbed by the small body per unit of its surface area is

Gabs = αG = ασT 4

The radiation emitted by the small body is

Eemit = ϵσT 4

Considering that the small body is in thermal equilibrium with the enclosure, the net rate of heat transfer to the body must be zero. Therefore, the radiation emitted by the body must be equal to the radiation absorbed by it:

As ϵσT 4 = AασT 4

Thus, we conclude that

ϵ(T ) = α(T )

That is, the total hemispherical emissivity of a surface at temperature T is equal to its total hemispherical absorptivity for radiation coming from a blackbody at the same temperature. This relation, which greatly simplifies the radiation analysis, was first developed by Gustav Kirchhoff in 1860 and is now called Kirchhoff’s law. Note that this relation is derived under the condition that the surface temperature is equal to the temperature of the source of irradiation, and the reader is cautioned against using it when considerable difference (more than a few hundred degrees) exists between the surface temperature and the temperature of the source of irradiation.