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Display information for equation id:math.2562.19 on revision:2562

* Page found: Das Photonengas im Strahlungshohlraum (eq math.2562.19)

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Das Photonengas im Strahlungshohlraum Eq: math.2569.19

Hash: 4b7fd3ee8c4338d32429996eb24edf56

TeX (original user input):

\begin{align}

& h\nu >>kT \\

& \frac{\partial u\left( \nu ,T \right)}{\partial \nu }=0=\frac{\partial }{\partial \nu }\frac{8\pi h}{{{c}^{3}}}\frac{{{\nu }^{3}}}{{{e}^{\frac{h\nu }{kT}}}}=\frac{8\pi h}{{{c}^{3}}}\frac{\partial }{\partial \nu }\left( {{\nu }^{3}}{{e}^{-\frac{h\nu }{kT}}} \right)=\frac{8\pi h}{{{c}^{3}}}\left( 3{{\nu }^{2}}{{e}^{-\frac{h\nu }{kT}}}-\frac{h}{kT}{{\nu }^{3}}{{e}^{-\frac{h\nu }{kT}}} \right) \\

& \Rightarrow 3{{\nu }^{2}}{{e}^{-\frac{h\nu }{kT}}}=\frac{h}{kT}{{\nu }^{3}}{{e}^{-\frac{h\nu }{kT}}} \\

& \Rightarrow \frac{3kT}{h}={{\nu }_{\max .}} \\

\end{align}

TeX (checked):

{\begin{aligned}&h\nu >>kT\\&{\frac {\partial u\left(\nu ,T\right)}{\partial \nu }}=0={\frac {\partial }{\partial \nu }}{\frac {8\pi h}{{c}^{3}}}{\frac {{\nu }^{3}}{{e}^{\frac {h\nu }{kT}}}}={\frac {8\pi h}{{c}^{3}}}{\frac {\partial }{\partial \nu }}\left({{\nu }^{3}}{{e}^{-{\frac {h\nu }{kT}}}}\right)={\frac {8\pi h}{{c}^{3}}}\left(3{{\nu }^{2}}{{e}^{-{\frac {h\nu }{kT}}}}-{\frac {h}{kT}}{{\nu }^{3}}{{e}^{-{\frac {h\nu }{kT}}}}\right)\\&\Rightarrow 3{{\nu }^{2}}{{e}^{-{\frac {h\nu }{kT}}}}={\frac {h}{kT}}{{\nu }^{3}}{{e}^{-{\frac {h\nu }{kT}}}}\\&\Rightarrow {\frac {3kT}{h}}={{\nu }_{\max .}}\\\end{aligned}}

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\begin{aligned} h ν > > k T \\ \frac{\partial u (ν, T)}{\partial ν} = 0 = \frac{\partial}{\partial ν} \frac{8 π h}{c^{3}} \frac{ν^{3}}{e^{\frac{h ν}{k T}}} = \frac{8 π h}{c^{3}} \frac{\partial}{\partial ν} (ν^{3} e^{- \frac{h ν}{k T}}) = \frac{8 π h}{c^{3}} (3 ν^{2} e^{- \frac{h ν}{k T}} - \frac{h}{k T} ν^{3} e^{- \frac{h ν}{k T}}) \\ \Rightarrow 3 ν^{2} e^{- \frac{h ν}{k T}} = \frac{h}{k T} ν^{3} e^{- \frac{h ν}{k T}} \\ \Rightarrow \frac{3 k T}{h} = ν_{\max .} \end{aligned}

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Identifiers

$h$

$ν$

$k$

$T$

$u$

$ν$

$T$

$ν$

$ν$

$π$

$h$

$c$

$ν$

$e$

$h$

$ν$

$k$

$T$

$π$

$h$

$c$

$ν$

$ν$

$e$

$h$

$ν$

$k$

$T$

$π$

$h$

$c$

$ν$

$e$

$h$

$ν$

$k$

$T$

$h$

$k$

$T$

$ν$

$e$

$h$

$ν$

$k$

$T$

$ν$

$e$

$h$

$ν$

$k$

$T$

$h$

$k$

$T$

$ν$

$e$

$h$

$ν$

$k$

$T$

$k$

$T$

$h$

$ν$

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