# Black-body radiation

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:

where:

Symbol | Units | Description |
---|---|---|

Input wavelength | ||

Input temperature | ||

Planck's constant | ||

Speed of light in vacuum | ||

Boltzmann constant |

Note that the input is in meters and that the output is a spectral irradiance in . Omitting the term from the numerator gives the blackbody emission in terms of radiance, with units where "sr" is steradians.

### Planck's Law: Frequency

Formulated in terms of frequency:

where:

Symbol | Units | Description |
---|---|---|

Input frequency |

All other units are the same as for the Wavelength formulation. Again, dropping the 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:

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

- ↑ 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
- ↑ 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>.