Kirchhoff’s Law and Planck’s Law

Definition

Kirchhoff’s Law of Thermal Radiation states that for an arbitrary body emitting and absorbing thermal radiation in thermodynamic equilibrium, the emissivity is equal to the absorptivity at a given wavelength and temperature. Planck’s Law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature, providing a bridge between classical physics and quantum mechanics.

Main Content

1. Kirchhoff’s Law of Thermal Radiation

This law establishes that "good absorbers are good emitters." If a material absorbs a specific wavelength of radiation efficiently, it must also emit that same wavelength efficiently to maintain thermal equilibrium.
Mathematically, it is expressed as $\alpha_\lambda = \epsilon_\lambda$, where $\alpha$ is absorptivity and $\epsilon$ is emissivity.

2. Planck’s Law of Blackbody Radiation

Planck’s Law resolves the "ultraviolet catastrophe" by proposing that energy is emitted in discrete "quanta" or packets ($E=hf$), rather than continuous waves.
It provides a mathematical function that describes the intensity of radiation emitted by a black body as a function of frequency (or wavelength) and absolute temperature.

3. The Relationship Between the Laws

Kirchhoff’s Law acts as a localized observation of thermal behavior, while Planck’s Law provides the universal spectral distribution for an ideal radiator (the black body).
Together, they allow engineers to calculate the actual heat transfer rates of non-ideal objects (gray bodies) by comparing them to the ideal values predicted by Planck’s Law.

       Intensity (I)
          ^
          |      / \
          |     /   \  <-- Higher Temp
          |    /     \
          |   /       \ <-- Lower Temp
          |  /         \
          +----------------------> Wavelength (λ)
       (Distribution shift as temperature changes)

Working / Process

1. Identifying the Black Body

A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.
Because it is a perfect absorber, Kirchhoff’s Law dictates it must also be a perfect emitter ($\epsilon = 1$).

2. Applying Planck’s Distribution

Calculate the spectral radiance $B_\lambda(T)$ using the formula: $B_\lambda(T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}$.
This step determines the theoretical maximum radiation at any given temperature.

3. Calculating Real-World Emission

Since real objects are not perfect black bodies, we multiply the Planck intensity by the emissivity ($\epsilon < 1$) of the material.
By using Kirchhoff’s identity ($\epsilon = \alpha$), we can determine the emission based on the material's surface properties and its ability to absorb incoming light.

Advantages / Applications

Pyrometry: Used in infrared thermometers to measure the temperature of objects from a distance by detecting emitted radiation.
Solar Energy: Helps in designing selective surfaces for solar thermal collectors that absorb high-energy solar radiation but have low emissivity to prevent heat loss.
Astrophysics: Astronomers use these laws to determine the temperature of distant stars by analyzing the color and intensity of their light spectra.

Summary

Kirchhoff’s Law explains that the ability of a surface to absorb radiation is identical to its ability to emit it at thermal equilibrium, while Planck’s Law defines the precise distribution of energy across different wavelengths for an ideal black body. Together, these principles form the foundation of radiative heat transfer and spectroscopy.

Important terms to remember: Black Body, Emissivity, Absorptivity, Quanta, and Spectral Radiance.