Question

A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports as shown in figure. The capacitance C, of this spherical capacitor is:

• A. $\frac{4\pi {ϵ}_0{r}_1{r}_2}{r_1-r_2}$
• B. $\frac{4\pi {ϵ}_0(r_2-r_1)}{r_1{r}_2}$
• C. $\frac{r_1{r}_2}{4\pi {ϵ}_0(r_2-r_1)}$
• D. $\frac{(r_1-r_2)}{4\pi {ϵ}_0{r}_1{r}_2}$

Answer 1

The correct option is A $\frac{4\pi {ϵ}_0{r}_1{r}_2}{r_1-r_2}$


As shown in figure, +q charge spreads uniformly on inner surface of outer sphere of radius $r_1$. The induced charge -q spreads uniformly on the outer surface of inner sphere of radius $r_2$. The outer surface of outer sphere is earthen.
Due to electrostatic shielding
E = 0 for r < $r_2$ and E = 0 for r > $r_1$
In the space between the two spheres,
Potential difference between two spheres,
$V=\frac{q}{4\pi {ϵ}_0{r}_2}-\frac{q}{4\pi {ϵ}_0{r}_1}=\frac{q}{4\pi {ϵ}_0}[\frac{1}{r_2}-\frac{1}{r_1}]$
$V=\frac{q}{4\pi {ϵ}_0}\left(\frac{r_1-r_2}{r_1{r}_2}\right)...\left(i\right)$
Also, $C=\frac{q}{V}$
$\therefore C=\frac{4\pi {ϵ}_0{r}_1{r}_2}{r_1-r_2}using\left(i\right)$