Question

A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig.). Show that the capacitance of a spherical capacitor is given by
$c=\frac{4\pi {ϵ}_0{r}_1{r}_2}{r_1-r_2}$
where $r_1$ and $r_2$ are the radii of outer and inner spheres, respectively.

Answer 1

Radius of the outer shell = r1

Radius of the inner shell = r2

The inner surface of the outer shell has charge +Q.

The outer surface of the inner shell has induced charge −Q.

Potential difference between the two shells is given by,

$V=\frac{Q}{4\pi {\epsilon }_o{r}_2}-\frac{Q}{4\pi {\epsilon }_o{r}_1}$

Where,

${\epsilon }_o$= Permittivity of free space

$V=\frac{Q}{4\pi {\epsilon }_o{r}_2}-\frac{Q}{4\pi {\epsilon }_o{r}_1}$

$V=\frac{Q}{4\pi {\epsilon }_o}[\frac{1}{r_2}-\frac{1}{r_1}]$

$V=\frac{Q(r_1-r_2)}{4\pi {\epsilon }_o{r}_1{r}_2}$

Capacitance of the given system is given by,

$C=\frac{Q}{V}$

$C=\frac{4\pi {\epsilon }_o{r}_1{r}_2}{r_1-r_2}$

Hence, proved.