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A black body absorbs and then re-emits all incident EM radiation. By definition it has an absorptivity and emissivity of 1, and a transmissivity and reflectivity of 0. The Planck black body equation describes the spectral exitance of an ideal black body. The study of black-body radiation was an integral step in the formulation of quantum mechanics.

### Planck's Law: Wavelength

Formulated in terms of wavelength:

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

where:

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

Note that the input ${\displaystyle \lambda }$ is in meters and that the output is a spectral irradiance in ${\displaystyle [W/m^{2}*m]}$. Omitting the ${\displaystyle \pi }$ term from the numerator gives the blackbody emission in terms of radiance, with units ${\displaystyle [W/m^{2}*sr*m]}$ where "sr" is steradians.

### Planck's Law: Frequency

Formulated in terms of frequency:

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

where:

Symbol Units Description
${\displaystyle v}$ ${\displaystyle [Hz]}$ Input frequency

All other units are the same as for the Wavelength formulation. Again, dropping the ${\displaystyle \pi }$ from the numerator gives the result in radiance rather than irradiance.

### Properties of the Planck Equation

Taking the first derivative of the Planck's law wavelength equation leads to the wavelength with maximum exitance as a function of temperature. This is known as Wien's displacement law:

${\displaystyle \lambda _{max}={\frac {2898{\mbox{ }}\mu {\mbox{m-K}}}{T}}}$

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.[1][2]

The ratio of the actual exitance of a surface to that of an ideal blackbody is the surface's emissivity, which is always less than or equal to 1.

An ideal blackbody at 300K (27 Celsius) has a peak emission at 9.66 microns. It has virtually no self-emission below 2.5 microns, hence self-emission is typically associated with the "thermal" regions of the EM spectrum. However, the Sun can be characterized as a 5900K blackbody and has a peak emission at around 0.49 microns, which is in the visible region of the electromagnetic spectrum.

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

### Citations

1. Paez, G. and Strojnik, M. "Integrable and differentiable approximations to the generalized Planck's equations." Proceedings of SPIE. Vol 3701, pp 95-105. DOI=10.1117/12.352985
2. Lawson, Duncan. "A closer look at Planck's blackbody equation." Physics Education 32.5 (Sept. 1997): 321-326. IOP. 19 Sept. 2007 <http://stacks.iop.org/0031-9120/32/321>.