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\begin{eqnarray*}
\phi\left(r\right) & = &  D_0 \left[ e^{- 2 \alpha (r - r_0)} - 2 e^{- \alpha (r - r_0)} \right] \qquad r < r_c \\
E\left(r\right) & = & \phi\left(r\right)  - \phi\left(R_c\right) - \left(r - R_c\right) \left.\frac{d\phi}{d r} \right|_{r=R_c}       \qquad r < R_c 
\end{eqnarray*}                           

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