next up previous
Next: Maxwell方程式 Up: denjiki Previous: 磁性体

Faradayの電磁誘導の法則

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 電磁誘導
\begin{center}
\includ...
...bm{E} = -\frac{\partial\bm{B}}{\partial t}
\end{displaymath}
\end{itemize}
}}

[ 例題1 ] $ N$ 巻のループに発生する起電力

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

$\displaystyle B$ $\displaystyle =$ $\displaystyle B_0\sin\omega t$  
$\displaystyle \Phi$ $\displaystyle =$ $\displaystyle NBS\cos\theta$  


$\displaystyle V = -\frac{d\Phi}{dt}$ $\displaystyle =$ $\displaystyle -NB_0S\cos\theta \frac{d}{dt} \sin\omega t$  
  $\displaystyle =$ $\displaystyle -\omega NB_0S\cos\theta\cos\omega t$  

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 自己インダクタンス
\begin{center...
...th}
\Phi = N \int BdS = LI
\end{displaymath}}
\end{center}
\end{itemize}
}}

[ 例題2 ] 同軸線路

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

$\displaystyle 2\pi rB$ $\displaystyle =$ $\displaystyle \mu I$  
$\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{\mu I}{2\pi r}$  

単位長さたりの磁束

$\displaystyle \Phi$ $\displaystyle =$ $\displaystyle \frac{\mu I}{2\pi} \int_a^b \frac{dr}{r}$  
  $\displaystyle =$ $\displaystyle \frac{\mu I}{2\pi} \ln \frac{b}{a}$  
$\displaystyle L$ $\displaystyle =$ $\displaystyle \frac{\mu}{2\pi} \ln \frac{b}{a}$  

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item 磁気エネルギー
\begin{displaymat...
...int \frac{B^2}{2\mu} dV = \frac{1}{2} LI^2
\end{displaymath}
\end{itemize}
}}

[ 例題3 ] 同軸線路

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

$\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{\mu I}{2\pi r}$  
$\displaystyle U$ $\displaystyle =$ $\displaystyle \int \frac{B^2}{2\mu} dV$  
  $\displaystyle =$ $\displaystyle \frac{1}{2\mu} \frac{mu^2I^2}{4\pi^2} \int_a^b \frac{2\pi rdr}{r^2}$  
  $\displaystyle =$ $\displaystyle \frac{mu I^2}{4\pi} \int_a^b \frac{dr}{r}$  
  $\displaystyle =$ $\displaystyle \frac{mu I^2}{4\pi} \ln \frac{b}{a}$  
  $\displaystyle =$ $\displaystyle \frac{1}{2} LI^2$  
$\displaystyle L$ $\displaystyle =$ $\displaystyle \frac{\mu}{2\pi} \ln \frac{b}{a}$  

\fbox{\parbox{14.5cm}{
\begin{itemize}
\item Lorentz力
\par
電磁場中の荷電粒子...
...m{v}}{dt} = q(\bm{E} + \bm{v}\times\bm{B})
\end{displaymath}
\end{itemize}
}}

[ 例題4 ] 一様な電場・磁場中の運動

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


$\displaystyle \bm{E}$ $\displaystyle =$ $\displaystyle (0, E, 0)$  
$\displaystyle \bm{B}$ $\displaystyle =$ $\displaystyle (0, 0, B)$  

運動方程式

$\displaystyle m \frac{d\bm{v}}{dt} = q(\bm{E} + \bm{v}\times\bm{B})
$

\begin{displaymath}
\left\{
\begin{array}{ccl}
\vspace{1mm}
\displaystyle \f...
...m}
\displaystyle \frac{dv_z}{dt} &=& 0
\end{array}
\right.
\end{displaymath}

$\displaystyle v_z = v_{z0} \ ({\rm z}方向には等速運動)
$


$\displaystyle \frac{d^2v_x}{dt^2}$ $\displaystyle =$ $\displaystyle \frac{qB}{m} \frac{dv_y}{dt}$  
  $\displaystyle =$ $\displaystyle \left( \frac{qB}{m} \right)^2 \left( \frac{E}{B} - v_x \right)$  
  $\displaystyle =$ $\displaystyle -\omega_c^2 \left( v_x - \frac{E}{B} \right)$  

$\displaystyle \omega_c = \frac{qB}{m} \ : サイクロトロン振動数
$

\begin{displaymath}
\left\{
\begin{array}{ccl}
\vspace{1mm}
v_x &=& \display...
..._x}{dt} = -v_0\sin(\omega_ct + \varphi)
\end{array}
\right.
\end{displaymath}

\begin{displaymath}
\left\{
\begin{array}{ccl}
\vspace{1mm}
x &=& \displayst...
... \\
\vspace{1mm}
z &=& z_0 + v_{z0}t
\end{array}
\right.
\end{displaymath}

$\displaystyle r_L = \frac{v_0}{\omega_c} \ : {\rm Larmor}半径
$


next up previous
Next: Maxwell方程式 Up: denjiki Previous: 磁性体
Keiichi Takasugi
平成24年1月25日