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\begin{flushright}2002/08/26\end{flushright}
\begin{center}
\large{�ż�����2(���̶��顢����ô�����饹) �齬���� ά��}
\end{center}

\begin{enumerate}
\item ά��
\item ά��
\item ά��
\item ά��
\item \begin{enumerate}
      \item $r=\sqrt{x^2+y^2+z^2}$����
            \[
              \frac{\partial}{\partial x}\phi(r)=
              \frac{\partial r}{\partial x}\frac{\partial}{\partial r}\phi(r)=
              \frac{x}{r}\frac{\partial}{\partial r}\phi(r),
            \]
            \[
              \frac{\partial^2}{\partial x^2}\phi(r)=
              \left(\frac{1}{r}-\frac{x^2}{r^3}\right)
              \frac{\partial}{\partial r}\phi(r)+
              \frac{x^2}{r^2}\frac{\partial^2}{\partial r^2}\phi(r).
            \]
            $y,z$��ʬ�ˤĤ��Ƥ�Ʊ�͡������
            \[
              \triangle\phi(\bvec{x})=
              \left(\frac{\partial^2}{\partial x^2}+
                    \frac{\partial^2}{\partial y^2}+
                    \frac{\partial^2}{\partial z^2}\right)\phi(r)
            \]
            ����������Ф褤��

      \item \[
              \rho=-\varepsilon_0\triangle\phi=e^{-r/a}/(a^2 r)=
                   -\varepsilon_0\phi(r)/a^2
            \]
      \end{enumerate}
\item $\rho(r)=\rho_0\;(r<a),\;\;\rho(r)=0\;(r>a)$�Ȥ��롣
      I������8.4��ꡢ
      \[
        \phi(r)=\frac{1}{4\pi\varepsilon_0}\frac{e(r)}{r}+
                \frac{1}{\varepsilon_0}\int_r^\infty\rho(r')r'dr'.
      \]
      ��ä�
      \[
        \phi(r)=\frac{\rho_0 a^3}{3\varepsilon_0 r}\quad (r>a),\quad
        \phi(r)=\frac{\rho_0}{6\varepsilon_0}(3a^2-r^2)\quad (r<a).
      \]
      ���ͥ륮����
      \[
        W=2\pi\int_0^\infty\phi(r)\rho(r)r^2dr
         =\frac{4\pi\rho_0^2}{15\varepsilon_0}a^5
      \]
      �Ȥʤꡢ\S I.4������2�η�̤Ȱ��פ��롣
\item ��ɽ�̤���������(������$\delta S$)�˥�������ˡ§��Ŭ�Ѥ��ơ�
      $-E\delta S=\sigma\delta S/\varepsilon_0$����ä�
      ɽ���Ų�̩��$\sigma=-\varepsilon_0 E$��
      �ϵ��Ⱦ�¤�$R$�Ȥ��ơ�
      $Q=-4\pi R^2\varepsilon_0 E\simeq 4.6\times 10^5\mathrm{C}$��
\item $W=QV/2$.
\item $\bvec{E}$��������ʬ��Ϣ³�椨����Ⱦʬ�ȱ�Ⱦʬ��$\bvec{E}$����������
      �Ű̺���$V$�Ȥ���ȡ�$E=V/d$��$\varepsilon_i$���ΰ��
      $D_i=\varepsilon_i E=\varepsilon_i V/d$�����Ĥ��Ų�̩�٤�
      $\sigma_i=D_i=\varepsilon_i V/d$���Ų٤�
      $Q=lw\sigma_i+lw\sigma_2=lw(\varepsilon_1+\varepsilon_2)V/d$�椨��
      $C=Q/V=(\varepsilon_1+\varepsilon_2)lw/d$��

\item ���Υ���ǥ󥵡������̤�
      $C=[2l\varepsilon_1+(\varepsilon_2-\varepsilon_1)x]w/d$��
      $W=Q^2/(2C)$�椨��$F=-dW/dx=(\varepsilon_2-\varepsilon_1)Q^2 w/(2C^2 d)$
      �Ȥʤ롣(�ޤα��������Ϥ�Ư����)

\item ���Ĵ֤ε�Υ��Ȥ������ȸ�Ƕ��Ĥ��Ų٤��Ѳ����ʤ���
      ���Ĵ֤ε�Υ��Ȥ������ȸ�����̤��Ű̺��򤽤줾�졢
      $C_1,V_1,C_2,V_2$�Ƚ񤯤ȡ�$Q=C_1V_1=C_2V_2$��
      ���Ĥ����Ѥ�$A$�Ȥ���ȡ�$C_1=\varepsilon A/d_1$��
      $C_2=A/[d_1/\varepsilon+(d_2-d_1)/\varepsilon_0]$��
      ��äơ�
      \[
        V_2=\frac{C_1}{C_2}V_1=
            \left(1+\frac{\varepsilon}{\varepsilon_0}\frac{d_2-d_1}{d_1}
            \right).
      \]

\item  \begin{enumerate}
       \item $\sigma_P=\bvec{P}\cdot(\pm\hat{\bvec{z}})=\pm P\;(����/����)$.
             $\rho_P=-\nabla\cdot\bvec{P}=0$ (�����̤������)
       \item �����֤�$\rho_f=0$�椨�������֤�$\bvec{D}=0$��
             Ͷ���Τγ��Ǥ�$0=\bvec{D}=\varepsilon_0\bvec{E}$�椨��
             $\bvec{E}=0$��Ͷ���Τ���Ǥ�
             $0=\bvec{D}=\varepsilon_0\bvec{E}+\bvec{P}$��ꡢ
             $\bvec{E}=-\bvec{P}/\varepsilon_0
                      =-P\hat{\bvec{z}}/\varepsilon_0$��
       \item Ͷ���Τ�$z$�������濴��$z=0$�Ȥ��롣
             ���ťݥƥ󥷥���$z$�Τߤδؿ�$\phi(z)$�Ȥʤ롣
             $E_z=-d\phi/dz$��ꡢ
             \[
               \frac{d\phi}{dz}=\frac{P}{\varepsilon_0}\;\;\;(|z|<d),\quad
               \frac{d\phi}{dz}=0\;\;\;(|z|>d).
             \]
             $|z|<d$��$\phi(z)=Pz/\varepsilon_0+c_1$��
             $\phi(0)=0$�Ȥ��ơ�$\phi(z)=Pz/\varepsilon_0\;(|z|<d)$��
             $z>d$��$\phi(z)=c_2$��$z<-d$��$\phi(z)=c_3$��
             $z=\pm d$�Ǥ�$\phi$��Ϣ³����ꡢ$c_2=Pd/\varepsilon_0=-c_3$��
       \item ���Ĥ��Ų�̩��$\sigma=\pm P$��Ϳ�������Ĵ֤������ǵ�Υ$2d$��
             ʿ���ĥ���ǥ󥵡���Ʊ����
      \end{enumerate}
\item ά��
\item ��ή��$z$���˱�ä�ή��Ƥ����ΤȤ��롣$\bvec{x}'=(0,0,z')$�Ȥ���
      \[
        \bvec{B}(\bvec{x})=\frac{\mu_0 I}{4\pi}
                           \int\frac{dz'\hat{\bvec{z}}\times 
                                     (\bvec{x}-\bvec{x}')}
                                    {|\bvec{x}-\bvec{x}'|^3}
                          =\frac{\mu_0 I}{2\pi r}\hat{\bvec{\phi}}.
      \]
      ��������$r=\sqrt{x^2+y^2},
               \hat{\bvec{z}}\times (\bvec{x}-\bvec{x}')=r\hat{\bvec{\phi}}$��
\item ����ή��xyʿ�̤ˤ��ꡢ�����濴�����Ȥ��롣�濴�������
      $\bvec{x}=(0,0,z)$�Ǥμ���ϡ�
      \[
        \bvec{B}(\bvec{x})=\frac{\mu_0 I}{4\pi}
                           \int\frac{d\bvec{x}'\times 
                                     (\bvec{x}-\bvec{x}')}
                                    {|\bvec{x}-\bvec{x}'|^3}.
     \]
     $\bvec{x}'=(a\cos\theta,a\sin\theta,0)$��
     $d\bvec{x}'=(-a\sin\theta,a\cos\theta,0)d\theta$�Ƚ񤱤뤫�顢
     \[
       \bvec{B}(\bvec{x})=\frac{\mu_0 I}{2}\frac{a^2}{(a^2+z^2)^{3/2}}
                          \hat{\bvec{z}}
     \]
     �Ȥʤ롣
\item �ϵ��Ⱦ�¤�$R$�������ж˻ҥ⡼���Ȥ��礭����$m$�Ȥ���С�
      �����Ǥϡ�$B=\mu_0 m/(2\pi R^3)$����äơ�
      $m\simeq 8\times 10^{21}\mathrm{A}\cdot\mathrm{m^2}$��
      ��ƻ��Ǥϡ�$B=B(����)/2\simeq 0.3\mathrm{G}$��
\item ������ή�μ����ж˻ҥ⡼���Ȥ�
      $\bvec{m}=\pi a^2 I\hat{\bvec{z}}$��
      ����(�����ˤ���)$\bvec{m}$���Ĥ��뼧��ϡ��濴��(z��)���
      \[
        \bvec{B}=\frac{\mu_0 I}{2}\frac{a^2}{z^3}\hat{\bvec{z}}.
      \]
      ����������15��ꡢ
      \[
        \bvec{B}=\frac{\mu_0 I}{2}\frac{a^2}{(a^2+z^2)^{3/2}}\hat{\bvec{z}}
                \stackrel{z\rightarrow\infty}{\simeq}
                 \frac{\mu_0 I}{2}\frac{a^2}{z^3}\hat{\bvec{z}}
      \]
      �Ȥʤꡢ��η�̤Ȱ��פ��롣
\end{enumerate}

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\hfill {\tt http://www-het.phys.sci.osaka-u.ac.jp/tanaka/kougi.html}

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