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

 

  

Figure A.1: Conducting sphere with dielectric shell.

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
$$\begin{gather}C=\frac{4\pi\epsilon_0\epsilon_r ab}{b-a(1-\epsilon_r)}. \end{gather}$$