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\begin{document}
\pagestyle{empty}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_1=-(a/l)q$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline300: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_2=-q_1=(a/l)q$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline302: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline308: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}F & = & \frac{q q_1}{\left( l-a^2/l \right)^2} + \frac{q q_2}{l^2} =
\nonumber \\ 
 &  = & q \left( \frac{-(a/l)q}{\left( l-a^2/l \right)^2} + 
\frac{(a/l)q}{l^2} \right)     
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}U & = & -\int_\infty^l F dx = -q \int_\infty^l 
\left( \frac{-(a/x)q}{( x-a^2/x)^2} + 
\frac{(a/x)q}{x^2} \right) =
\nonumber \\ 
& = & q^2 a \left[ \int_\infty^l \frac{1}{x (x -a^2/x)^2} dx 
- \int_\infty^l \frac{1}{x^3} dx \right].
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}U & = & q^2 a \left[ \left. - \frac{1}{2( x^2-a^2) } \right| _\infty ^l 
+  \left. \frac{1}{2 x^2} \right| _\infty ^l  \right] = 
\nonumber \\ 
& = & \frac{-q^2 a}{2 (l^2-a^2) } + \frac{q^2a}{2 l^2} =
\nonumber \\ 
& = & \frac{-q^2 a^3}{2 l^2(l^2-a^2)} 
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\rho(\bf{r})= N r^2 b^5 e^{-b^2r^2} cos^2\theta   \nonumber
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}q_{lm} = \int Y_{lm}^*(\theta,\phi) r^l \rho({\bf r }) d^3 r.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}q_{lm} = N \int Y_{lm}^*(\theta,\phi) r^{l+4} b^5 
 e^{-b^2r^2} cos^2\theta dr d(cos \theta)d\phi.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\int_0 ^{2\pi}  e^{im\phi} d\phi = 2 \pi \delta_{m,0}, 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}q_{lm} = 2\pi N \sqrt{\frac{2l+1}{4 \pi}} \int d(cos \theta) 
cos^2 \theta P_l(cos \theta)
\int _0 ^\infty dr r^{l+4} b^5 e^{-b^2r^2}.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f_l= \int _0 ^\infty dr r^{l+4} b^5 e^{-b^2r^2}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline312: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}P_2 = 3/2 cos^2 \theta - 1/2 \Longrightarrow cos^2\theta = \frac{2P_2 +1}{3}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}q_{lm} & = & 2\pi N \sqrt{\frac{2l+1}{4 \pi}} f_l \int d(cos \theta) 
  P_l(cos \theta)\frac{ 2P_2 + P_0}{3} =
\nonumber \\ 
& = & \frac{2\pi N}{3} \sqrt{\frac{2l+1}{4 \pi}} f_l \left(2 \delta_{l,0}
+ (4/5) \delta_{l,2} \right),   
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_{00}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline314: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_{20}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline316: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}q_{00} & = & \frac{2\pi N}{3} \frac{1}{\sqrt{\pi}} f_0
\\ 
q_{22} & = & \frac{8\pi N}{15} \sqrt{\frac{5}{4\pi}} f_2
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline318: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline320: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}q_{00} & = & \frac{\pi N}{2}  
\\ 
q_{22} & = & \pi N \sqrt{\frac{5}{4}} \frac{1}{b^2}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\Phi & = & 
\sum_{l,m} Y_{lm}(\theta, \phi) q_{lm} \frac{4 \pi}{2l+1} \frac{1}{r^{l+1}}
\nonumber \\ 
& = & 
Y_{00} q_{00} \frac{4 \pi}{r} + Y_{20} q_{20} \frac{4 \pi}{5 r^3} 
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf r} \equiv {\bf x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline328: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\Phi = \int \frac{\rho d^3 x'}{\mid {\bf x} - {\bf x'} \mid}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\frac{1}{\mid {\bf x} - {\bf x'} \mid} = 
\sum_{l,m} \frac{r_<^l}{r_>^{l+1}} \frac{4 \pi}{2l+1} Y_{lm}(\theta,\phi) 
Y_{lm}^*(\theta',\phi') 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}r_< & = & r^\prime,  \; \; r_> = r \; \; for \, r \gg r^\prime  
\nonumber \\  
r_< & = & r,  \; \; r_> = r^\prime \; \; for \, r \ll r^\prime
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline330: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bf{B}=B_o \bf{i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline338: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Phi_M$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline340: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bf{H}=- {\bf \nabla} \Phi_M$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline342: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf \nabla} \cdot \bf{H}=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline344: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nabla^2 \Phi_M=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline346: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$H_\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline348: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline350: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\Phi_M = -B_0 r cos\theta + \sum_{l=0}^\infty \frac{A_l}{r^{l+1}}P_l(cos\theta)
\label{rb}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Phi_M = \sum_{l=0}^\infty \left( B_l r^l+ \frac{C_l}{r^{l+1}}
\right)P_l(cos\theta)
\label{arb}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\Phi_M = \sum_{l=0}^\infty  D_l r^l P_l(cos\theta)
\label{ar}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\left. \frac{\partial \Phi_M^{out}}{\partial \theta}  \right|_{r=b} & = &  
\left. \frac{\partial \Phi_M^{in}}{\partial \theta}  \right|_{r=b}  
\\ 
\left. \frac{\partial \Phi_M^{out}}{\partial r}  \right|_{r=b} & = &  
\left. \mu \frac{\partial \Phi_M^{in}}{\partial r}  \right|_{r=b}  
\\ 
\left. \frac{\partial \Phi_M^{out}}{\partial \theta}  \right|_{r=a} & = &  
\left. \frac{\partial \Phi_M^{in}}{\partial \theta}  \right|_{r=a}  
\\ 
\left. \mu \frac{\partial \Phi_M^{out}}{\partial r}  \right|_{r=a} & = &  
\left. \frac{\partial \Phi_M^{in}}{\partial r}  \right|_{r=a}  
\\ 
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}A_1 - b^3B_1 -C_1 & = & b^3 B_0
\\ 
2 A_1 + \mu b^3 B_1 -2 \mu C_1 & = & -b^3 B_0
\\ 
a^3 B_1 + C_1 -a^3D_1 & = & 0
\\ 
\mu a^3 B_1 - 2 \mu C_1 -a^3 D_1 & = & 0.
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$D_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline364: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}D_1= -\frac{9\mu}{(2\mu+1)(\mu+2) -2(a^3/b^3)(\mu-1)^2} B_0
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\Phi_M & = & -
\frac{9\mu}{(2\mu+1)(\mu+2) -2(a^3/b^3)(\mu-1)^2} B_0 r cos\theta,
\\ 
{\bf B} & = & - \nabla \Phi_M =
\frac{9\mu}{(2\mu+1)(\mu+2) -2(a^3/b^3)(\mu-1)^2} B_0 {\bf k}  
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu\rightarrow \infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline370: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\stackrel{{\textstyle lim}}{{\scriptstyle \mu \rightarrow \infty}} 
{\bf B} = \frac{9}{2 \mu (1-(a^3/b^3)}  
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}E_x = E_o e^{i(kz-\omega t)} 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}k=(1+i) \left( \frac{K \omega}{c^2} \right)  
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf B}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline374: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf E}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline378: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\nabla \times {\bf E} = - \frac{1}{c} \frac{\partial {\bf B}}{\partial t} 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\nabla \times {\bf E} = - {\bf j} \left( - E_o (ik)  e^{i(kz-\omega t)}
\right)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\bf B} & = & - c \int \nabla \times {\bf E} \, dt = 
\nonumber \\ 
& = & - c {\bf j} E_o (ik)  e^{ikz} \int e^{-i \omega t} \, dt =
\nonumber \\   
& = & c \, {\bf j} \, E_o \frac{ik}{i \omega}  e^{i(kz-\omega t)}
\nonumber \\ 
& = & {\bf j} \, E_o \frac{ck}{\omega}  e^{i(kz-\omega t)}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf j}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline386: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}


\end{document}
