Planck's law of black-body radiation

Planck's law states that

${\displaystyle B_{\nu }(T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}},}$

where

B_ν(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency ν radiation per unit frequency at thermal equilibrium at temperature T.

h is the Planck constant;

c is the speed of light in a vacuum;

k is the Boltzmann constant;

ν is the frequency of the electromagnetic radiation;

T is the absolute temperature of the body.

For a black body surface the spectral radiance density (defined per unit of area normal to the propagation) is independent of the angle ${\displaystyle \theta }$ of emission with respect to the normal. However, this means that, following Lambert's cosine law, ${\displaystyle B_{\nu }(T)\cos \theta }$ is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor ${\displaystyle 1/\cos \theta }$ with respect to an area normal to the propagation direction. At oblique angles, the solid angle spans involved do get smaller, resulting in lower aggregate intensities.

Wien's displacement law

Wien's displacement law shows how the spectrum of black-body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of frequency.

A consequence of Wien's displacement law is that the wavelength at which the intensity per unit wavelength of the radiation produced by a black body is at a maximum, ${\displaystyle \lambda _{\max }}$, is a function only of the temperature:

${\displaystyle \lambda _{\max }={\frac {b}{T}},}$

where the constant b, known as Wien's displacement constant, is equal to 2.8977729(17)×10−3
 K m

Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by

${\displaystyle \nu _{\max }=T\times 58.8\ \mathrm {GHz} /\mathrm {K} }$.

Stefan–Boltzmann law

By integrating ${\displaystyle B_{\nu }(T)}$ over the frequency the integrated radiance ${\displaystyle L}$ is

${\displaystyle L={\frac {2\pi ^{5}}{15}}{\frac {k^{4}T^{4}}{c^{2}h^{3}}}{\frac {1}{\pi }}=:\sigma T^{4}{\frac {1}{\pi }}}$

by using ${\displaystyle \int _{0}^{\infty }dx\,{\frac {x^{3}}{e^{x}-1}}={\frac {\pi ^{4}}{15}}}$ with ${\displaystyle x\equiv {\frac {h\nu }{kT}}}$ and with ${\displaystyle \sigma \equiv {\frac {2\pi ^{5}}{15}}{\frac {k^{4}}{c^{2}h^{3}}}=5.670373\times 10^{-8}{\frac {W}{m^{2}K^{4}}}}$ being the Stefan–Boltzmann constant. The radiance ${\displaystyle L}$ is then

${\displaystyle \sigma T^{4}{\frac {\cos \theta }{\pi }}}$

per unit of emitting surface.

On a side note, at a distance d, the intensity ${\displaystyle dI}$ per area ${\displaystyle dA}$ of radiating surface is the useful expression

${\displaystyle dI=\sigma T^{4}{\frac {\cos \theta }{\pi d^{2}}}dA}$

when the receiving surface is perpendicular to the radiation.

By subsequently integrating over the solid angle ${\displaystyle \Omega }$ (where ${\displaystyle \theta <\pi /2}$) the Stefan–Boltzmann law is calculated, stating that the power j* emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:

${\displaystyle j^{\star }=\sigma T^{4},}$

by using

${\displaystyle \int \cos \theta \,d\Omega =\int _{0}^{2\pi }\int _{0}^{\pi /2}\cos \theta \sin \theta \,d\theta \,d\phi =\pi .}$