Black-body radiation: Difference between revisions
imported>Niek Sanders (Properties of Planck eqn.) |
imported>Niek Sanders (Sectioning. Added frequency formulation.) |
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Planck's blackbody equation describes the spectral exitance of an ideal blackbody. | Planck's blackbody equation describes the spectral exitance of an ideal blackbody. | ||
===Planck's Law: Wavelength=== | |||
Formulated in terms of wavelength: | |||
<math> | <math> | ||
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|} | |} | ||
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]]. | 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: | |||
{| class="wikitable" | |||
|- | |||
! 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]]. | Taking the first derivative leads to the wavelength with maximum exitance. This is known as the [[Wien Displacement Law]]. | ||
Revision as of 12:39, 18 September 2007
Planck's blackbody equation describes the spectral exitance of an ideal blackbody.
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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1eba9291ddfe915f2a665dac12ec9153fc8738d6)
where:
| Symbol | Units | Description |
|---|---|---|
| Input wavelength | ||
| Input temperature | ||
| Planck's constant | ||
| Speed of light in vacuum | ||
| Boltzmann constant |
![{\displaystyle [m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b649f5418f2ac84d66063f41319368b0926ccec4)
![{\displaystyle [K]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6c25071903364b1b51e69ffa99ec95f6d122d5)
![{\displaystyle [J*s]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/751c5bea56bc672d42ac27457e7e940b269e8990)
![{\displaystyle [{\frac {m}{sec}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c5921e89e1b41df1b07950bebb522fb96c81fd8)
![{\displaystyle [erg*K]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d64f4a0cf071f92fcf1ee72a31a5f47e39d8b51)
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.
![{\displaystyle [W/m^{2}*m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c244377e247036110f6b9089c4a81bac21dbf54)
![{\displaystyle [W/m^{2}*sr*m]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f0037b79d48b97852db90d92db992888f9cf7a9)
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 {hc}{KT}}-1)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f620a06c5d31382b64f4b4f291bebfffa4c84393)
where:
| Symbol | Units | Description |
|---|---|---|
| Input frequency |
![{\displaystyle [Hz]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33dac5031f1f01184208e90edb8d25bf37e2f399)
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 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 known 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 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 smaller as temperature increases. The total exitance increases with temperature.