Question:
27 similar drops of mercury are maintained at $10 \mathrm{~V}$ each. All these spherical drops combine into a single big drop. The potential energy of the bigger drop is _____ times that of a smaller drop.
Solution:
(243)
For self energy of sphere (conducting) $U=\frac{k q^{2}}{2 r}$
For small drop $\rightarrow U_{i}=\frac{k q^{2}}{2 r}$ ......(1)
After combine small drops volume remains same as bigger drop
$\therefore \quad \frac{4}{3} \pi r^{3} \times n=\frac{4}{3} \pi R^{3}$
$\mathrm{R}=(\mathrm{n})^{\frac{1}{3}} \mathrm{r} \quad \ldots \ldots \ldots(2)$ .....(2)
For large drop $\rightarrow U_{f}=\frac{k(n q)^{2}}{2 \times 3 R}$ ......(3)
From equation (1),(2),(3)
$\frac{U_{f}}{U_{i}}=(n)^{5 / 3}$
$\Rightarrow(27)^{5 / 3}$
\$\Rightarrow 2