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Display information for equation id:math.2564.19 on revision:2564

* Page found: Das Photonengas im Strahlungshohlraum (eq math.2564.19)

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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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hν>>kT∂u(ν,T)∂ν=0=∂∂ν8πhc3ν3ehνkT=8πhc3∂∂ν(ν3e−hνkT)=8πhc3(3ν2e−hνkT−hkTν3e−hνkT)⇒3ν2e−hνkT=hkTν3e−hνkT⇒3kTh=νmax⁡.
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data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ν</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>h</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></msup></mrow></mfrac></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>h</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>ν</mi></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>8</mn><mi>π</mi><mi>h</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>3</mn><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></mrow></msup><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>h</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mn>3</mn><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></mrow></msup><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>h</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow><msup><mi>ν</mi><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></msup><msup><mi>e</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo stretchy="false">−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>h</mi><mi>ν</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>k</mi><mi>T</mi></mrow></mrow></mfrac></mrow></mrow></mrow></msup></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>3</mn><mi>k</mi><mi>T</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>h</mi></mrow></mfrac></mrow><mo stretchy="false">=</mo><msub><mi>ν</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>max</mo><mo>&#x2061;</mo><mo stretchy="false">.</mo></mrow></mrow></msub></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>

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