Question

Two spherical conductors $A_1$ and $A_2$ of radii $r_1$ and $r_2(r_2>r_1)$ are placed concentrically in air. $A_1$ is given a charge +Q while $A_2$ is earthed. Then the equivalent capacitance of the system is:

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

Answer 1

Answer: (A)

By Gauss's law the electric field at a distance r in between the shells is
$E.4\pi r^2=\frac{Q}{{ϵ}_0}\Rightarrow E=\frac{Q}{4\pi {ϵ}_0{r}^2}$
The potential difference between shells is ${\int }_0^{V}dV=-{\int }_{r_2}^{r_1}Edr$
or $V=-\frac{Q}{4\pi {ϵ}_0}{\int }_{r_2}^{r_1}\frac{dr}{r^2}=\frac{Q}{4\pi {ϵ}_0}(\frac{1}{r_1}-\frac{1}{r_2})$
or $V=\frac{Q}{4\pi {ϵ}_0}.\frac{r_2-r_1}{r_1{r}_2}$
$\therefore$ Capacitance, $C=\frac{Q}{V}=\frac{4\pi {ϵ}_0{r}_1{r}_2}{r_2-r_1}$