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Display information for equation id:math.2156.34 on revision:2156
* Page found: Mikroskopisches Modell der Polarisierbarkeit (eq math.2156.34)
(force rerendering)Occurrences on the following pages:
Hash: 47a7c13985c64ad02e9143fdc673efc5
TeX (original user input):
\begin{align}
& \bar{P}={{\varepsilon }_{0}}n\alpha {{{\bar{E}}}_{a}}={{\varepsilon }_{0}}n\alpha \left( \bar{E}+\frac{1}{3{{\varepsilon }_{0}}}\bar{P} \right) \\
& \bar{P}={{\varepsilon }_{0}}{{\chi }_{e}}\bar{E} \\
& \Rightarrow {{\chi }_{e}}=\frac{n\alpha }{1-\frac{1}{3}n\alpha } \\
& n\alpha =\frac{{{\chi }_{e}}}{1+\frac{1}{3}{{\chi }_{e}}}=\frac{\varepsilon -1}{1+\frac{\varepsilon -1}{3}}=3\frac{\varepsilon -1}{\varepsilon +2} \\
\end{align}
TeX (checked):
{\begin{aligned}&{\bar {P}}={{\varepsilon }_{0}}n\alpha {{\bar {E}}_{a}}={{\varepsilon }_{0}}n\alpha \left({\bar {E}}+{\frac {1}{3{{\varepsilon }_{0}}}}{\bar {P}}\right)\\&{\bar {P}}={{\varepsilon }_{0}}{{\chi }_{e}}{\bar {E}}\\&\Rightarrow {{\chi }_{e}}={\frac {n\alpha }{1-{\frac {1}{3}}n\alpha }}\\&n\alpha ={\frac {{\chi }_{e}}{1+{\frac {1}{3}}{{\chi }_{e}}}}={\frac {\varepsilon -1}{1+{\frac {\varepsilon -1}{3}}}}=3{\frac {\varepsilon -1}{\varepsilon +2}}\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>P</mi><mo>¯</mo></mover></mrow></mrow><mo>=</mo><msub><mi>ε</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mi>n</mi><mi>α</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>E</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>a</mi></mrow></msub><mo>=</mo><msub><mi>ε</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><mi>n</mi><mi>α</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>E</mi><mo>¯</mo></mover></mrow></mrow><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>3</mn><msub><mi>ε</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub></mrow></mrow></mfrac></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>P</mi><mo>¯</mo></mover></mrow></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>P</mi><mo>¯</mo></mover></mrow></mrow><mo>=</mo><msub><mi>ε</mi><mrow data-mjx-texclass="ORD"><mn>0</mn></mrow></msub><msub><mi>χ</mi><mrow data-mjx-texclass="ORD"><mi>e</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>E</mi><mo>¯</mo></mover></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><msub><mi>χ</mi><mrow data-mjx-texclass="ORD"><mi>e</mi></mrow></msub><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>n</mi><mi>α</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>−</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></mfrac></mrow><mi>n</mi><mi>α</mi></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi>n</mi><mi>α</mi><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msub><mi>χ</mi><mrow data-mjx-texclass="ORD"><mi>e</mi></mrow></msub></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></mfrac></mrow><msub><mi>χ</mi><mrow data-mjx-texclass="ORD"><mi>e</mi></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ε</mi><mo>−</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>+</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ε</mi><mo>−</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>3</mn></mrow></mfrac></mrow></mrow></mrow></mfrac></mrow><mo>=</mo><mn>3</mn><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ε</mi><mo>−</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ε</mi><mo>+</mo><mn>2</mn></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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