Question

A sphercial capacitor is made of two conducting spherical shells of radii a and b. The space between the shells is filled with a dielectric of dielectric constant K up to a radius c as shown in figure (31-E27). Calculate the capacitance.

Answer 1

We have two capacitors: one made by the shells a and c and the other made by the shells b and c.

The capacitance of the capacitor C_ac is given by
$C_{\mathrm{ac}}=\frac{4\pi {\in }_{0}acK}{(c-a)}$

The capacitance of the capacitor C_cb is given by
$C_{\mathrm{cb}}=\frac{4\pi {\in }_{0}bcK}{K(b-c)}$

The two capacitors are in series; thus, the equivalent capacitance is given by
$$\begin{gathered} \frac{1}{C}=\frac{1}{C_{\mathrm{ac}}}+\frac{1}{C_{\mathrm{cb}}} \\ \Rightarrow \frac{1}{C}=\frac{(c-a)}{4\pi {\in }_{0}acK}+\frac{(b-c)}{4\pi {\in }_{0}cb} \\ \Rightarrow \frac{1}{C}=\frac{b(c-a)+Ka(b-c)}{K4\pi {\in }_{0}abc} \\ \Rightarrow C=\frac{K4\pi {\in }_{0}abc}{b(c-a)+Ka(b-c)} \end{gathered}$$