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| [[Image:RWP-comparison.svg|thumb|300px|Comparison of Rayleigh–Jeans law with [[Wien approximation]] and [[Planck's law]], for a body of 8 mK [[temperature]].]]
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| In [[physics]], the '''Rayleigh–Jeans law''' attempts to describe the [[spectral radiance]] of [[electromagnetic radiation]] at all [[wavelengths]] from a [[black body]] at a given temperature through classical arguments. For wavelength ''λ'', it is:
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| :<math>B_\lambda(T) = \frac{2 c k T}{\lambda^4},</math>
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| where ''c'' is the [[speed of light]], ''k'' is the [[Boltzmann constant]] and ''T'' is the [[temperature]] in [[kelvin]]s. For [[frequency]] ''ν'', the expression is instead
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| :<math>B_\nu(T) = \frac{2 \nu^2 k T}{c^2}.</math>
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| The Rayleigh–Jeans law agrees with experimental results at large wavelengths (or, equivalently, low frequencies) but strongly disagrees at short wavelengths (or high frequencies). This inconsistency between observations and the predictions of [[classical physics]] is commonly known as the [[ultraviolet catastrophe]],<ref>''Astronomy: A Physical Perspective'', Mark L. Kutner pp. 15</ref><ref>''Radiative Processes in Astrophysics'', Rybicki and Lightman pp. 20–28</ref> and its resolution was a foundational aspect of the development of [[quantum mechanics]] in the early 20th century.
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| ==Historical development==
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| In 1900, the British physicist [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] derived the ''λ''<sup>−4</sup> dependence of the Rayleigh–Jeans law based on classical physical arguments.<ref>''Astronomy: A Physical Perspective'', Mark L. Kutner pp. 15</ref> A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir [[James Jeans]] in 1905. The Rayleigh–Jeans law revealed an important error in physics theory of the time. The law predicted an energy output that diverges towards [[Infinity#Physics|infinity]] as wavelength approaches zero (as frequency tends to infinity) and measurements of energy output at short wavelengths disagreed with this prediction.
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| ==Comparison to Planck's law==
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| In 1900 [[Max Planck]] empirically obtained an expression for [[black-body radiation]] expressed in terms of wavelength {{nowrap|''λ'' {{=}} ''c''/''ν''}} ([[Planck's law]]):
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| :<math>B_\lambda(T) = \frac{2 h c^2}{\lambda^5}~\frac{1}{e^\frac{hc}{\lambda kT}-1},</math>
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| where ''h'' is the [[Planck constant]] and ''k'' the [[Boltzmann constant]]. The Planck law does not suffer from an ultraviolet catastrophe, and agrees well with the experimental data, but its full significance (which ultimately led to quantum theory) was only appreciated several years later. Since,
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| :<math>e^x = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + \cdots. </math> | |
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| then in the limit of very high temperatures or long wavelengths, the term in the exponential becomes small, and the exponential is well approximated with the [[Taylor series|Taylor polynomial's]] first-order term,
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| :<math>e^{\frac{hc}{\lambda kT}} \approx 1 + \frac{hc}{\lambda kT}.</math> | |
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| So,
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| :<math>\frac{1}{e^\frac{hc}{\lambda kT}-1} \approx \frac{1}{\frac{hc}{\lambda kT}} = \frac{\lambda kT}{hc}.</math>
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| This results in Planck's blackbody formula reducing to
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| :<math>B_{\lambda}(T) = \frac{2ckT}{\lambda^4},</math>
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| which is identical to the classically derived Rayleigh–Jeans expression.
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| The same argument can be applied to the blackbody radiation expressed in terms of frequency {{nowrap|''ν'' {{=}} ''c''/''λ''}}. In the limit of small frequencies, that is <math> h \nu \ll kT </math>,
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| :<math>B_\nu(T) = \frac{2h\nu^3/c^2}{e^\frac{h\nu}{kT} - 1} \approx \frac{2h\nu^3}{c^2} \cdot \frac{kT}{h\nu} = \frac{2 \nu^2 kT}{c^2}.</math>
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| This last expression is the Rayleigh–Jeans law in the limit of small frequencies.
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| ==Consistency of frequency and wavelength dependent expressions==
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| When comparing the frequency and wavelength dependent expressions of the Rayleigh–Jeans law it is important to remember that
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| :<math>B_{\lambda}(T) \neq B_{\nu}(T)</math>
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| because <math>B_{\lambda}(T)</math> has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, '''per unit wavelength''', whereas <math>B_{\nu}(T)</math> has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, '''per unit frequency'''. To be consistent, we must use the equality
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| :<math>B_{\lambda} \, d\lambda = B_{\nu} \, d\nu</math>
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| where both sides now have units of energy emitted per unit time per unit area of emitting surface, per unit solid angle.
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| Starting with the Rayleigh–Jeans law in terms of wavelength we get
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| :<math>B_{\lambda}(T) = B_{\nu}(T) \times \frac{d\nu}{d\lambda}</math>
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| where
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| :<math>\frac{d\nu}{d\lambda} = \frac{d}{d\lambda}\left(\frac{c}{\lambda}\right) = -\frac{c}{\lambda^2}</math>.
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| This leads us to find:
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| :<math>B_{\lambda}(T) = \frac{2kT\left( \frac{c}{\lambda}\right)^2}{c^2} \times \frac{c}{\lambda^2} = \frac{2ckT}{\lambda^4}</math>.
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| ==Other forms of Rayleigh–Jeans law==
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| Depending on the application, the Planck Function can be expressed in 3 different forms. The first involves energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit frequency. In this form, the Planck Function and associated Rayleigh–Jeans limits are given by
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| :<math>B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1} \approx \frac{2c kT}{\lambda^4}</math>
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| or
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| :<math>B_\nu(T) = \frac{2h\nu^2/c^2}{e^\frac{h\nu}{kT} - 1} \approx \frac{2kT\nu^2}{c^2}</math>
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| Alternatively, Planck's law can be written as an expression <math>u(\nu,T) = \pi I(\nu,T)</math> for emitted power integrated over all solid angles. In this form, the Planck Function and associated Rayleigh–Jeans limits are given by
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| :<math>u(\lambda,T) = \frac{2\pi c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1} \approx \frac{2\pi ckT}{\lambda^4}</math>
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| or
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| :<math>u(\nu,T) = \frac{2\pi h\nu^2/c^2}{e^\frac{h\nu}{kT} - 1} \approx \frac{2 \pi kT\nu^2}{c^2}</math>
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| In other cases, Planck's Law is written as <math>\rho(\nu,T) = \frac{4\pi}{c} I(\nu,T)</math> for energy per unit volume (energy density). In this form, the Planck Function and associated Rayleigh–Jeans limits are given by
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| :<math>\rho(\lambda,T) = \frac{8 \pi c}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1} \approx \frac{8\pi kT}{\lambda^4}</math>
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| or
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| :<math>\rho(\nu,T) = \frac{8\pi h\nu^3/c^3}{e^\frac{h\nu}{kT} - 1} \approx \frac{8 \pi kT\nu^2}{c^3}</math>
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| ==See also==
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| * [[Stefan–Boltzmann law]]
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| * [[Wien's displacement law]]
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| * [[Sakuma–Hattori equation]]
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| ==References==
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| <references/>
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| ==External links==
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| * [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/rayj.html Derivation at HyperPhysics]
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| {{DEFAULTSORT:Rayleigh-Jeans law}}
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| [[Category:Foundational quantum physics]]
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Not much to write about me I think.
I enjoy of finally being a member of wmflabs.org.
I just wish I am useful in some way here.
Feel free to visit my web page :: www.hetilainat.fi