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Display information for equation id:math.1444.168 on revision:1444

* Page found: Materie in elektrischen und magnetischen Feldern (eq math.1444.168)

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TeX (original user input): 

\begin{align}
& {{\varepsilon }_{0}}\oint\limits_{\partial V(r\acute{\ })}{{}}d\bar{f}\cdot \bar{E}\left( \bar{r},t \right)=\int_{V(r\acute{\ })}^{{}}{{}}\frac{Q}{\frac{4}{3}\pi {{R}^{3}}}=\frac{r{{\acute{\ }}^{3}}}{{{R}^{3}}}Q \\
& \Rightarrow 4r{{\acute{\ }}^{2}}\pi {{\varepsilon }_{0}}\left| \bar{E}\left( \bar{r},t \right) \right|=\frac{r{{\acute{\ }}^{3}}}{{{R}^{3}}}Q \\
& \Rightarrow \left| \bar{E}\left( \bar{r},t \right) \right|=\frac{r\acute{\ }}{4\pi {{\varepsilon }_{0}}{{R}^{3}}}Q \\
\end{align}

TeX (checked): 

{\begin{aligned}&{{\varepsilon }_{0}}\oint \limits _{\partial V(r{\acute {\ }})}{}d{\bar {f}}\cdot {\bar {E}}\left({\bar {r}},t\right)=\int _{V(r{\acute {\ }})}^{}{}{\frac {Q}{{\frac {4}{3}}\pi {{R}^{3}}}}={\frac {r{{\acute {\ }}^{3}}}{{R}^{3}}}Q\\&\Rightarrow 4r{{\acute {\ }}^{2}}\pi {{\varepsilon }_{0}}\left|{\bar {E}}\left({\bar {r}},t\right)\right|={\frac {r{{\acute {\ }}^{3}}}{{R}^{3}}}Q\\&\Rightarrow \left|{\bar {E}}\left({\bar {r}},t\right)\right|={\frac {r{\acute {\ }}}{4\pi {{\varepsilon }_{0}}{{R}^{3}}}}Q\\\end{aligned}}

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MathML (41.894 KB / 4.713 KB) :

ε 0 ⁢ ∮ ∂ ⁡ V ⁢ ( r ⁢ ´ ) 𝑑 f ¯ ⋅ E ¯ ⁢ ( r ¯ , t ) = ∫ V ⁢ ( r ⁢ ´ ) Q 4 3 ⁢ π ⁢ R 3 = r ⁢ ´ 3 R 3 ⁢ Q ⇒ 4 ⁢ r ⁢ ´ 2 ⁢ π ⁢ ε 0 ⁢ | E ¯ ⁢ ( r ¯ , t ) | = r ⁢ ´ 3 R 3 ⁢ Q ⇒ | E ¯ ⁢ ( r ¯ , t ) | = r ⁢ ´ 4 ⁢ π ⁢ ε 0 ⁢ R 3 ⁢ Q missing-subexpression subscript 𝜀 0 subscript contour-integral 𝑉 𝑟 ´ absent ⋅ differential-d ¯ 𝑓 ¯ 𝐸 ¯ 𝑟 𝑡 subscript 𝑉 𝑟 ´ absent 𝑄 4 3 𝜋 superscript 𝑅 3 𝑟 superscript ´ absent 3 superscript 𝑅 3 𝑄 missing-subexpression ⇒ absent 4 𝑟 superscript ´ absent 2 𝜋 subscript 𝜀 0 ¯ 𝐸 ¯ 𝑟 𝑡 𝑟 superscript ´ absent 3 superscript 𝑅 3 𝑄 missing-subexpression ⇒ absent ¯ 𝐸 ¯ 𝑟 𝑡 𝑟 ´ absent 4 𝜋 subscript 𝜀 0 superscript 𝑅 3 𝑄 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle{{\varepsilon}_{0}}% \oint\limits_{\partial V(r\acute{\ })}{{}}d\bar{f}\cdot\bar{E}\left(\bar{r},t% \right)=\int_{V(r\acute{\ })}{{}}\frac{Q}{\frac{4}{3}\pi{{R}^{3}}}=\frac{r{{% \acute{\ }}^{3}}}{{{R}^{3}}}Q\\ &\displaystyle\Rightarrow 4r{{\acute{\ }}^{2}}\pi{{\varepsilon}_{0}}\left|\bar% {E}\left(\bar{r},t\right)\right|=\frac{r{{\acute{\ }}^{3}}}{{{R}^{3}}}Q\\ &\displaystyle\Rightarrow\left|\bar{E}\left(\bar{r},t\right)\right|=\frac{r% \acute{\ }}{4\pi{{\varepsilon}_{0}}{{R}^{3}}}Q\\ \end{aligned}}}

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ε0∮∂V(r´)df¯⋅E¯(r¯,t)=∫V(r´)Q43πR3=r´3R3Q⇒4r´2πε0|E¯(r¯,t)|=r´3R3Q⇒|E¯(r¯,t)|=r´4πε0R3Q

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Identifiers

  • ε0
  • V
  • r
  • ´
  • d
  • f¯
  • E¯
  • r¯
  • t
  • V
  • r
  • ´
  • Q
  • π
  • R
  • r
  • ´
  • R
  • Q
  • r
  • ´
  • π
  • ε0
  • E¯
  • r¯
  • t
  • r
  • ´
  • R
  • Q
  • E¯
  • r¯
  • t
  • r
  • ´
  • π
  • ε0
  • R
  • Q

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