A spherical capacitor consists of a spherical conducting shell of radius a which is surrounded by a concentric outer spherical shell of radius b. The capacitance of the capacitor is

Let +Q be the charge on the inner shell and -Q on the outer shell. In the space between the two shells, the electric field is due to the inner shell alone. At a distance r from the centre of the inner shell, the electric field is
$E=\frac{1}{4\pi {\in }_0}\frac{Q}{r^2}$
Let V be the potential difference between the two shells. Then
$$\begin{gathered} V=-{\int }_b^{a}Edr={\int }_a^{b}Edr \\ V=\frac{Q}{4\pi {\in }_0}{\int }_a^b\frac{dr}{r^2} \\ =\frac{Q}{4\pi {\in }_0}{\left|-\frac{1}{r}\right|}_a^b \\ =\frac{Q}{4\pi {\in }_0}\left(\frac{1}{a}-\frac{1}{b}\right)=\frac{Q(b-a)}{4\pi {\in }_{0}ab} \\ Q=\left[\frac{4\pi {\in }_{0}ab}{(b-a)}\right]V \end{gathered}$$
Comparing this equation with Q = CV, we get

$C=4\pi {\in }_0\left(\frac{ab}{b-a}\right)$