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.
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[edit] Planck's Law: Wavelength
Formulated in terms of wavelength:
where:
Symbol | Units | Description |
---|---|---|
λ | [m] | Input wavelength |
T | [K] | Input temperature |
[J * s] | Planck's constant | |
Speed of light in vacuum | ||
[erg * K] | Boltzmann constant |
Note that the input λ is in meters and that the output is a spectral irradiance in [W / m^{2} * m]. Omitting the π term from the numerator gives the blackbody emission in terms of radiance, with units [W / m^{2} * sr * m] where "sr" is steradians.
[edit] Planck's Law: Frequency
Formulated in terms of frequency:
where:
Symbol | Units | Description |
---|---|---|
v | [Hz] | Input frequency |
All other units are the same as for the Wavelength formulation. Again, dropping the π from the numerator gives the result in radiance rather than irradiance.
[edit] 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.
[edit] 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>.
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