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Next: Poisson方程式 Up: denjiki Previous: Coulombの法則

Gaussの法則

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item Gaussの法則
\begin{center}
\inc...
...{\epsilon_0} \\
\nabla\cdot\bm{D} &=& \rho
\end{eqnarray*}
\end{itemize}
}}

[ 例題1 ] 導体表面の電場

\includegraphics[width=4cm]{metal.eps}

$\displaystyle 導体内で \hspace{5mm} \phi$ $\displaystyle =$ $\displaystyle const$  
$\displaystyle \bm{E}$ $\displaystyle =$ 0  


$\displaystyle \int \bm{E}\cdot d\bm{S}$ $\displaystyle =$ $\displaystyle \frac{1}{\epsilon_0} \int \sigma dV$  
$\displaystyle \rotatebox{90}{=} \hspace{6mm}$   $\displaystyle \hspace{7mm} \rotatebox{90}{=}$  
$\displaystyle ES \hspace{4mm}$   $\displaystyle \hspace{5mm} \frac{\sigma S}{\epsilon_0}$  
$\displaystyle \bm{E}$ $\displaystyle =$ $\displaystyle \frac{\sigma}{\epsilon_0}$  

[ 例題2 ] 半径$ a$ の球内に電荷が一様に分布

\includegraphics[width=3.5cm]{outside.eps}

$ r>a$ のとき

$\displaystyle E4\pi r^2 = \frac{\rho}{\epsilon_0} \frac{4\pi}{3} a^3
$

$\displaystyle E = \frac{\rho a^3}{3\epsilon_0r^2}
$

$\displaystyle \phi = - \int_{\infty}^r Edr = \frac{\rho a^3}{3\epsilon_0}
$

\includegraphics[width=3cm]{inside.eps}

$ r<a$ のとき

$\displaystyle E4\pi r^2 = \frac{\rho}{\epsilon_0} \frac{4\pi}{3} r^3
$

$\displaystyle E = \frac{\rho r}{3\epsilon_0}
$


$\displaystyle \phi$ $\displaystyle =$ $\displaystyle \phi(a) - \int_a^r Edr$  
  $\displaystyle =$ $\displaystyle \frac{\rho a^2}{3\epsilon_0} - \frac{\rho}{3\epsilon_0} \int_a^r rdr$  
  $\displaystyle =$ $\displaystyle \frac{\rho a^2}{3\epsilon_0} - \frac{\rho}{3\epsilon_0} \left( \frac{r^2}{2} - \frac{a^2}{2} \right)$  
  $\displaystyle =$ $\displaystyle \frac{\rho}{6\epsilon_0} (3a^2 - r^2)$  


next up previous
Next: Poisson方程式 Up: denjiki Previous: Coulombの法則
Keiichi Takasugi
平成24年1月25日