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Next: 磁性体 Up: denjiki Previous: Biot-Savartの法則

Ampèreの法則

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item Amp\\lq ereの法則
\begin{center}
\...
...ace{5mm} \nabla\times\bm{B} = \mu_0 \bm{J}
\end{displaymath}
\end{itemize}
}}

[ 例題1 ] 円筒内に分布した電流

\includegraphics[width=3cm]{cylinder.eps} \includegraphics[width=3.5cm]{outside.eps}

$ r>a$ のとき

$\displaystyle 2\pi rB = \mu_0J\pi a^2
$

$\displaystyle B = \frac{\mu_0Ja^2}{2r}
$

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

$ r<a$ のとき

$\displaystyle 2\pi rB = \mu_0J\pi r^2
$

$\displaystyle B = \frac{\mu_0J}{2} r
$

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 磁場中の電流が受ける力
\begin{eq...
...\bm{B} \\
d\bm{F} &=& Id\bm{r}\times\bm{B}
\end{eqnarray*}
\end{itemize}
}}

[ 例題2 ] 矩形電流にはたらく力

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

$\displaystyle \bm{B} = B(\cos\theta, 0, \sin\theta) \\
$

AB間

$\displaystyle \bm{r}$ $\displaystyle =$ $\displaystyle (x, a, 0)$  
$\displaystyle d\bm{r}$ $\displaystyle =$ $\displaystyle dx\bm{e}_x$  
$\displaystyle d\bm{F}$ $\displaystyle =$ $\displaystyle Idx\bm{e}_x\times\bm{B}$  
  $\displaystyle =$ $\displaystyle -IBdx\sin\theta\bm{e}_y$  
$\displaystyle \bm{F}_{AB}$ $\displaystyle =$ $\displaystyle -IB\sin\theta\bm{e}_y \int_a^{-a} dx$  
  $\displaystyle =$ $\displaystyle 2aIB\sin\theta\bm{e}_y$  

CD間

$\displaystyle \bm{r}$ $\displaystyle =$ $\displaystyle (x, -a, 0)$  
$\displaystyle \bm{F}_{CD}$ $\displaystyle =$ $\displaystyle -IB\sin\theta\bm{e}_y \int_{-a}^a dx$  
  $\displaystyle =$ $\displaystyle -2aIB\sin\theta\bm{e}_y$  

BC間

$\displaystyle \bm{r}$ $\displaystyle =$ $\displaystyle (-a, y, 0)$  
$\displaystyle d\bm{r}$ $\displaystyle =$ $\displaystyle dy\bm{e}_y$  
$\displaystyle d\bm{F}$ $\displaystyle =$ $\displaystyle Idy\bm{e}_y\times\bm{B}$  
  $\displaystyle =$ $\displaystyle IBdy(\sin\theta\bm{e}_x - \cos\theta\bm{e}_z)$  
$\displaystyle \bm{F}_{BC}$ $\displaystyle =$ $\displaystyle IB(\sin\theta\bm{e}_x - \cos\theta\bm{e}_z) \int_a^{-a} dy$  
  $\displaystyle =$ $\displaystyle -2aIB(\sin\theta\bm{e}_x - \cos\theta\bm{e}_z)$  

DA間

$\displaystyle \bm{r}$ $\displaystyle =$ $\displaystyle (a, y, 0)$  
$\displaystyle \bm{F}_{DA}$ $\displaystyle =$ $\displaystyle IB(\sin\theta\bm{e}_x - \cos\theta\bm{e}_z) \int_{-a}^a dy$  
  $\displaystyle =$ $\displaystyle 2aIB(\sin\theta\bm{e}_x - \cos\theta\bm{e}_z)$  

全体で

$\displaystyle \bm{F} = \bm{F}_{AB} + \bm{F}_{BC} + \bm{F}_{CD} + \bm{F}_{DA} = 0
$

\includegraphics[width=4cm]{torque.eps} \includegraphics[width=4cm]{torque2.eps}

力のモーメント

$\displaystyle \bm{N}$ $\displaystyle =$ $\displaystyle \bm{r}\times\bm{F}$  
$\displaystyle d\bm{N}$ $\displaystyle =$ $\displaystyle \bm{r}\times d\bm{F}$  

AB間

$\displaystyle d\bm{N}$ $\displaystyle =$ $\displaystyle -IB\sin\theta xdx\bm{e}_z$  
$\displaystyle \bm{N}_{AB}$ $\displaystyle =$ $\displaystyle -IB\sin\theta\bm{e}_z \int_a^{-a} xdx = 0$  

CD間

$\displaystyle d\bm{N}$ $\displaystyle =$ $\displaystyle -IB\sin\theta xdx\bm{e}_z$  
$\displaystyle \bm{N}_{CD}$ $\displaystyle =$ $\displaystyle -IB\sin\theta\bm{e}_z \int_{-a}^a xdx = 0$  

BC間

$\displaystyle d\bm{N}$ $\displaystyle =$ $\displaystyle -IBdy(y\cos\theta\bm{e}_x + a\cos\theta\bm{e}_y + y\sin\theta\bm{e}_z)$  
$\displaystyle \bm{N}_{BC}$ $\displaystyle =$ $\displaystyle -IB \left\{ (cos\theta\bm{e}_x + \sin\theta\bm{e}_z) \int_a^{-a} ydy + a\cos\theta\bm{e}_y \int_{-a}^a dy \right\}$  
  $\displaystyle =$ $\displaystyle 2a^2IB\cos\theta\bm{e}_y$  

DA間

$\displaystyle d\bm{N}$ $\displaystyle =$ $\displaystyle -IBdy(y\cos\theta\bm{e}_x - a\cos\theta\bm{e}_y + y\sin\theta\bm{e}_z)$  
$\displaystyle \bm{N}_{DA}$ $\displaystyle =$ $\displaystyle -IB \left\{ (cos\theta\bm{e}_x + \sin\theta\bm{e}_z) \int_{-a}^a ydy - a\cos\theta\bm{e}_y \int_{-a}^a dy \right\}$  
  $\displaystyle =$ $\displaystyle 2a^2IB\cos\theta\bm{e}_y$  

全体で

$\displaystyle \bm{N} = \bm{N}_{AB} + \bm{N}_{BC} + \bm{N}_{CD} + \bm{N}_{DA} = 0$ $\displaystyle =$ $\displaystyle 4a^2IB\cos\theta\bm{e}_y$  
  $\displaystyle =$ $\displaystyle \bm{m}\times\bm{B}$  
    $\displaystyle {\color{blue} 磁気モーメント}$  
    $\displaystyle \hspace{5mm} {\color{blue} \bm{m} = I\bm{S}}$  
    $\displaystyle \hspace{5mm} {\color{blue} \bm{S} = 4a^2\bm{e}_z}$  

これは$ y$ 軸のまわりに回転する力


next up previous
Next: 磁性体 Up: denjiki Previous: Biot-Savartの法則
Keiichi Takasugi
平成24年1月25日