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 $C=\frac{4\pi {\epsilon }_0{r}_1{r}_2}{r_1-r_2}$,where $r_1\text{and}{r}_2$ are the radii of outer and inner spheres, respectively.

Answer 1

Step 1: Find the potential difference between the shells
Given
Radius of outer shell $=r_1$
Radius of inner shell $=r_2$
The inner surface of the outer shell has charge $+Q$
The inner shell has charge $-Q$.

As the potential between two shells is given by

$V=\frac{Q}{4\pi {\epsilon }_0{r}_2}-\frac{Q}{4\pi {\epsilon }_0{r}_1}$

And we know that ${\epsilon }_0$ is permittivity of free space

$V=\frac{Q}{4\pi {\epsilon }_0}[\frac{1}{r_2}-\frac{1}{r_1}]$

$V=\frac{Q(r_1-r_2)}{4\pi {\epsilon }_0{r}_1{r}_2}$


Step 2: Find the capacitance

$\text{Formula used}:-C=\frac{Q}{V}$

Capacitance of the given system is given by,
$C=\frac{Q}{V}$

$C=\frac{4\pi {\epsilon }_0{r}_1{r}_2}{r_1-r_2}$