Kirchhoff’s Law of Thermal Radiation

Kirchhoff’s law of thermal radiation, derived from the principle of energy conservation, states that a body in thermal equilibrium has the same emissivity and absorptivity at a given wavelength and temperature. Otherwise, energy transfer would violate the second law of thermodynamics. This principle also applies to different wavelengths and polarization states.

[math]\epsilon\left(\lambda,T\right)=\alpha\left(\lambda,T\right)[/math]

This principle implies that a good absorber of infrared radiation is also a good emitter, and the first equation can be changed to the following.

[math]\epsilon+\rho+\tau=1[/math]

In practical terms, Kirchhoff’s thermal radiation law helps determine material properties for accurate infrared thermography. High-emissivity surfaces, which absorb most incident radiation, also radiate effectively, making them ideal for thermal imaging applications. Conversely, highly reflective surfaces with low emissivity are poor radiators, requiring compensation techniques in thermal measurement. Most bodies do not show transmissivity in infrared. Therefore, the following applies:

[math]\epsilon+\rho=1[/math]

To prove this, we allow the following thought experiment. Consider two bodies, A and B. Body A is a regular material body with an emissive power [math]M\left(\lambda,T\right)[/math] and an absorptivity [math]\alpha[/math], while the second Body B is a perfect black body with emissive power [math]M°\left(\lambda,T\right)[/math]​, meaning it absorbs all incident radiation. The radiant heat incident per unit time and area on both bodies is denoted as [math]\theta[/math].

For Body A, the heat it absorbs per unit time and area is given by [math]\alpha\theta[/math], while the heat it emits emissive power [math]M[/math]. Since the system is in thermal equilibrium and the temperature remains constant, the absorbed and emitted heat must be equal, leading to the equation [math]M=\alpha\theta[/math]. In the case of Body B, a perfect black body, it absorbs all incident radiation, meaning the heat absorbed is simply [math]\theta[/math], and the heat it emits is [math]M°[/math]. At thermal equilibrium, this results in the equation [math]M°=\theta[/math].

To express this relationship in terms of emissivity, we define a material’s emissivity as the ratio of its emissive power to that of a perfect black body at the same temperature. This result demonstrates that emissivity and absorptivity are equal by substituting, formally proving Kirchhoff’s law.

[math]\epsilon=\frac{M\left(\lambda,T\right)}{M°\left(\lambda,T\right)}=\frac{\alpha\theta\left(\lambda,T\right)}{\theta\left(\lambda,T\right)}=\alpha[/math]

This principle establishes that all bodies in thermal equilibrium emit radiation at the same efficiency as they absorb it, ensuring energy conservation in radiative heat exchange.

This principle is fundamental in infrared temperature measurement because it implies that a material’s ability to emit radiation is directly related to its ability to absorb it.