Black-body radiation

Planck's blackbody equation describes the spectral exitance of an ideal blackbody.

Planck's Law: Wavelength

Formulated in terms of wavelength:

$M(\lambda ,T)[{\frac {W}{m^{2}m}}]={\frac {2\pi hc^{2}}{\lambda ^{5}(\exp ^{\frac {hc}{\lambda KT}}-1)}}$

where:

Symbol	Units	Description
$\lambda$	$[m]$	Input wavelength
$T$	$[K]$	Input temperature
$h=6.6261\times 10^{-34}$	$[J*s]$	Planck's constant
$c=2.9979\times 10^{8}$	$[{\frac {m}{sec}}]$	Speed of light in vacuum
$k=1.3807\times 10^{-23}$	$[erg*K]$	Boltzmann constant

Note that the input $\lambda$ is in meters and that the output is a spectral irradiance in $[W/m^{2}*m]$. Omitting the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} term from the numerator gives the blackbody emission in terms of radiance, with units Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [W/m^2*sr*m]} where "sr" is steradians.

Planck's Law: Frequency

Formulated in terms of frequency:

$M(v,T)[{\frac {W}{m^{2}Hz}}]={\frac {2\pi hv^{3}}{c^{2}(\exp ^{\frac {hc}{KT}}-1)}}$

where:

Symbol	Units	Description
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [Hz]}	Input frequency

All other units are the same as for the Wavelength formulation. Again, dropping the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} from the numerator gives the result in radiance rather than irradiance.

Properties of the Planck Equation

Taking the first derivative leads to the wavelength with maximum exitance. This is known as the Wien Displacement Law.

A closed form solution exists for the integral of the Planck blackbody equation over the entire spectrum. This is the Stefan-Boltzmann equation. In general, there is no closed-form solution for the definite integral of the Planck blackbody equation ; numerical integration techniques must be used.

The relationship between the ideal blackbody exitance and the actual exitance of a surface is given by emissivity.

An ideal blackbody at 300K (~30 Celsius) has a peak emission 9.66 microns. It has virtually no self-emission before 2.5 microns, hence self-emission is typically associated with the "thermal" regions of the EM spectrum. However, the Sun has a peak emission around 0.49 microns which is in the visible region of spectrum.

The Planck equation has a single maximum. The wavelength with peak exitance becomes smaller as temperature increases. The total exitance increases with temperature.