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Display information for equation id:math.2158.27 on revision:2158
* Page found: Wellenausbreitung in Materie (eq math.2158.27)
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Hash: d76f9d801df2f4d3a74919d7f850139a
TeX (original user input):
\begin{align}
& {{k}^{2}}=\frac{{{\omega }^{2}}}{{{c}^{2}}}\left( {{n}^{2}}-{{\gamma }^{2}}+2in\gamma \right)\approx \frac{{{\omega }^{2}}}{{{c}^{2}}}\varepsilon \mu \frac{i}{\omega \tau } \\
& \Rightarrow {{n}^{2}}-{{\gamma }^{2}}\approx 0 \\
& n\gamma \approx {{n}^{2}}\approx {{\gamma }^{2}}\approx \frac{\varepsilon \mu }{2\omega \tau }\Rightarrow n=\gamma =\sqrt{\frac{\varepsilon \mu }{2\omega \tau }} \\
& \tan \phi =\frac{\gamma }{n}\approx 1\Rightarrow \phi \approx \frac{\pi }{4} \\
\end{align}
TeX (checked):
{\begin{aligned}&{{k}^{2}}={\frac {{\omega }^{2}}{{c}^{2}}}\left({{n}^{2}}-{{\gamma }^{2}}+2in\gamma \right)\approx {\frac {{\omega }^{2}}{{c}^{2}}}\varepsilon \mu {\frac {i}{\omega \tau }}\\&\Rightarrow {{n}^{2}}-{{\gamma }^{2}}\approx 0\\&n\gamma \approx {{n}^{2}}\approx {{\gamma }^{2}}\approx {\frac {\varepsilon \mu }{2\omega \tau }}\Rightarrow n=\gamma ={\sqrt {\frac {\varepsilon \mu }{2\omega \tau }}}\\&\tan \phi ={\frac {\gamma }{n}}\approx 1\Rightarrow \phi \approx {\frac {\pi }{4}}\\\end{aligned}}
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<math xmlns="http://www.w3.org/1998/Math/MathML" class="mwe-math-element mwe-math-element-inline"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mtable displaystyle="true"><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><msup><mi>k</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">−</mo><msup><mi>γ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">+</mo><mn>2</mn><mi>i</mi><mi>n</mi><mi>γ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo stretchy="false">≈</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><msup><mi>ω</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow><mrow data-mjx-texclass="ORD"><msup><mi>c</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup></mrow></mfrac></mrow><mi>ε</mi><mi>μ</mi><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ω</mi><mi>τ</mi></mrow></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mo stretchy="false">⇒</mo><msup><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">−</mo><msup><mi>γ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">≈</mo><mn>0</mn></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>n</mi><mi>γ</mi><mo stretchy="false">≈</mo><msup><mi>n</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">≈</mo><msup><mi>γ</mi><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo stretchy="false">≈</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ε</mi><mi>μ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>ω</mi><mi>τ</mi></mrow></mrow></mfrac></mrow><mo stretchy="false">⇒</mo><mi>n</mi><mo stretchy="false">=</mo><mi>γ</mi><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><msqrt><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>ε</mi><mi>μ</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mn>2</mn><mi>ω</mi><mi>τ</mi></mrow></mrow></mfrac></mrow></msqrt></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd><mtd class="mwe-math-columnalign-l"><mi>tan</mi><mo>⁡</mo><mi>ϕ</mi><mo stretchy="false">=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>γ</mi></mrow><mrow data-mjx-texclass="ORD"><mi>n</mi></mrow></mfrac></mrow><mo stretchy="false">≈</mo><mn>1</mn><mo stretchy="false">⇒</mo><mi>ϕ</mi><mo stretchy="false">≈</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>π</mi></mrow><mrow data-mjx-texclass="ORD"><mn>4</mn></mrow></mfrac></mrow></mtd></mtr><mtr><mtd class="mwe-math-columnalign-r"></mtd></mtr></mtable></mstyle></mrow></math>
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