3.3. Four Laws of Light

Kirchoff's Law

This radiative transfer law is very important when considering energy balance. It states that at thermal equilibrium, the emissivity ($ϵ$) of a body or surface equals its absorptivity ($\alpha$).

Mathematically, we can conceptualize Kirchoff's law as

$$\frac{E}{E_b}=ϵ=\alpha$$

The radiant energy emitted from a real surface is represented as E (W/m²), while that of a blackbody (a theoretical condition) is given by E_B (W/m²).

Simply put, a surface at steady state temperature will absorb light equally, as well as it emits light. Though light is directional, surfaces exchange photons in both directions.

Planck's Law

This law is generalized to mean that all objects have some internal temperature, and given that temperature, they all glow. Max Planck was able to establish the dependence of the spectral emissive energy of a blackbody for all wavelengths of light ($E_{\lambda ,b}$), given a known equilibrium temperature of the blackbody.

$$E_{\lambda ,b}=\frac{2\pi h{c}^2}{{\lambda }^5[e^{\frac{hc}{\lambda KT}}-1]}$$

Wien's Displacement Law

The Wien's displacement law provides us with expected values for the most probable wavelengths in the Bose-Einstein distribution of blackbody radiation. The law implies that the distribution of photons emitted from a surface at any temperature will have the same form or shape as a distribution of photons emitted from a surface at any other temperature. Mathematically,

$${\lambda }_{max}=\frac{2.8978\times {10}^{6}nmK}{T(K)}$$

Stefan-Boltzmann Law

The Stefan-Boltzmann Law states that the radiative energy emitted by a surface is proportional to the fourth power of the surface's absolute temperature. The Stefan-Boltzmann Law shows that if you were to integrate all the energies from the wavelengths in Planck's Law, you have an analytical solution of the form below:

$$E_b=\underset{0}{\overset{\text{∞}}{{\displaystyle \int }}}{E}_{\lambda ,b}\text{d}\lambda =\sigma T^4$$

where $\sigma$ = Stefan-Boltzmann constant $5.67\times {10}^{-}8W/m^2{K}^4$

Keep in mind that this is for the surface of the emitter. A surface like the Sun will have a very high value for the energy density on the surface, which then decreases in proportion with the Inverse Square Law from the last page (over the 93 million miles distance to Earth's surface).