  Main Article Discussion Related Articles  [?] Bibliography  [?] External Links  [?] Citable Version  [?] This editable Main Article is under development and subject to a disclaimer. [edit intro]

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:

$M(\lambda ,T)[{\frac {W}{m^{2}m}}]={\frac {2\pi hc^{2}}{\lambda ^{5}(\exp ^{\frac {hc}{\lambda KT}}-1)}}$ 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 $\pi$ term from the numerator gives the blackbody emission in terms of radiance, with units $[W/m^{2}*sr*m]$ 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
$v$ $[Hz]$ Input frequency

All other units are the same as for the Wavelength formulation. Again, dropping the $\pi$ 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 can be characterized as a 5900K blackbody and 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 shorter as temperature increases. The total exitance increases with temperature.