# 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.

## Contents

### Planck's Law: Wavelength

Formulated in terms of wavelength:

<math>

M(\lambda,T) [\frac{W}{m^2 m}] = \frac{ 2 \pi h c^2 }{ \lambda^5 ( \exp^{\frac{h c}{\lambda K T}} - 1 ) }

</math>

where:

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

<math>\lambda</math> | <math>[m]</math> | Input wavelength |

<math>T</math> | <math>[K]</math> | Input temperature |

<math>h = 6.6261 \times 10^{-34}</math> | <math>[J*s]</math> | Planck's constant |

<math>c = 2.9979 \times 10^{8}</math> | <math>[\frac{m}{sec}]</math> | Speed of light in vacuum |

<math>k = 1.3807 \times 10^{-23}</math> | <math>[erg*K]</math> | Boltzmann constant |

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

### Planck's Law: Frequency

Formulated in terms of frequency:

<math>

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

</math>

where:

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

<math>v</math> | <math>[Hz]</math> | Input frequency |

All other units are the same as for the Wavelength formulation. Again, dropping the <math>\pi</math> 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.^{[1]}^{[2]}

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.

### Citations

- ↑ Paez, G. and Strojnik, M. "Integrable and differentiable approxiations 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>.