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A.2 Self Capacitance of a Sphere with a Dielectric Shell

 
  
Figure A.1: Conducting sphere with dielectric shell.
\includegraphics{sphere_shell.eps}

The capacitance of a conducting sphere with a dielectric shell, as shown in Fig. A.1, is calculated using the boundary condition
\begin{gather}\boldsymbol{D_1} = \boldsymbol{D_2} \quad\rightarrow\quad
\epsil...
...}\qquad
\boldsymbol{E}=\frac{q}{4\pi\epsilon r^2}\boldsymbol{e_r}.
\end{gather}
The potential of the sphere is calculated by a path integral from the surface of the sphere to infinity over the electric field.
\begin{align}\varphi =&\int_a^{\infty}\boldsymbol{E}\,d\boldsymbol{r}=\int_a^bE_...
...2}\right)=
\frac{q(b-a(1-\epsilon_r))}{4\pi\epsilon_0\epsilon_r ab}
\end{align}
And hence the capacitance of a conducting sphere with a dielectric shell is
\begin{gather}C=\frac{4\pi\epsilon_0\epsilon_r ab}{b-a(1-\epsilon_r)}.
\end{gather}




Christoph Wasshuber