# Wien approximation

Wien approximation

Wien's approximation (also sometimes called "Wien's law" or the "Wien distribution law") is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in 1896.cite book
author=J. Mehra, H. Rechenberg
year=1982
title=The Historical Development of Quantum Theory
volume=1
chapter=1
publisher=Springer-Verlag
location=New York
id=ISBN 0-387-90642-8
] cite book
author=R. Bowley, M. Sánchez
year=1999
title=Introductory Statistical Mechanics
edition=2nd edition
publisher=Clarendon Press
location=Oxford
id=ISBN 0-19-850576-0
] The equation does accurately describe the short wavelength (high frequency) spectrum of thermal emission from objects, but it fails to accurately fit the experimental data for long wavelengths (low frequency) emission.

The law may be written as

$I\left( u, T\right) = frac\left\{2 h u^3\right\}\left\{c^2\right\} e^\left\{-frac\left\{h u\right\}\left\{kT$ cite book
author=G. B. Rybicki, A. P. Lightman
year=1979
publisher=John Wiley & Sons
location=New York
id=ISBN 0-471-82759-2
]

where

:*$I\left( u, T\right)$ is the amount of energy per unit surface area per unit time per unit solid angle per unit frequency emitted at a frequency &nu;.:*$T$ is the temperature of the black body.:*$h$ is Planck's constant.:*$c$ is the speed of light.:*$k$ is Boltzmann's constant.

This equation may also be written as

$I\left(lambda, T\right) = frac\left\{2 h c^2\right\} \left\{lambda^5\right\} e^\left\{-frac\left\{hc\right\}\left\{lambda kT$ Equation derived using u=4&pi;/c; see Rybicki, Lightman (1979) reference.]

where $I\left(lambda, T\right)$ is the amount of energy per unit surface area per unit time per unit solid angle per unit wavelength emitted at a wavelength λ.

Relation to Planck's law

The Wien approximation was originally proposed as a description of the complete spectrum of thermal radiation, although it failed to accurately describe long wavelength (low frequency) emission. However, it was soon superseded by Planck's law, developed by Max Planck. Unlike the Wien approximation, Planck's law accurately describes the complete spectrum of thermal radiation. Planck's law may be given as

$I\left( u, T\right) = frac\left\{2 h u^3\right\}\left\{c^2\right\} frac\left\{1\right\}\left\{e^\left\{frac\left\{h u\right\}\left\{kT-1\right\}$

The Wien approximation may be derived from Planck's law by assuming $h u gg kT$. When this is true, then

$frac\left\{1\right\}\left\{e^\left\{frac\left\{h u\right\}\left\{kT-1\right\} approx e^\left\{-frac\left\{h u\right\}\left\{kT$

and so Planck's law approximately equals the Wien approximation at high frequencies.

The Rayleigh-Jeans law developed by Lord Rayleigh may be used to accurately describe the long wavelength spectrum of thermal radiation but fails to describe the short wavelength spectrum of thermal emission.

References

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