Question

Water from a tap, maintained at a constant potential V, is allowed to fall by drops of radius r through a small hole into a hollow conducting sphere of radius R standing on an insulating stand until it fills the entire sphere. Find the potential of the hollow conductor after it is completely filled with water.

Answer 1

Potential of a drop

$V=\frac{1}{4\pi {\epsilon }_0{\epsilon }_r}\frac{q}{r}$

$q=4\pi {\epsilon }_0{\epsilon }_{r}rV$

Therefore, total charge

$Q=nq=4\pi {\epsilon }_0{\epsilon }_{r}r{V}_n$

(n = total number of drops)

Also $\frac{4}{3}\pi R^3=n\times \frac{4}{3}\pi r^{3}orn=\frac{R^3}{r^3}$

Hence potential of the hollow conductor

$V^{{}^{\prime }}=\frac{1}{4\pi {\epsilon }_0{\epsilon }_r}\frac{Q}{R}=\frac{1}{4\pi {\epsilon }_0{\epsilon }_r}\times \frac{4\pi {\epsilon }_0{\epsilon }_{r}Vn}{R}$

$=\frac{r}{R}\times V\times \left(\frac{R^3}{r^3}\right)$

$=\left({\frac{R}{r}}^2\right)V$