Question

One thousand similar electrified raindrops merge into one bigger drop , so that their total charge remains unchanged. Find the total electrostatic energy of the bigger drop, assuming that the drops are spherical and that small drops are at large distances from one another.

• A. Increases by $50$ times
• B. Decreases by $50$ times
• C. Increases by $100$ times
• D. Decreases by $100$ times

Answer 1

The correct option is C Increases by $100$ times

If $q$ is the charge on one drop of radius $r$, then the total electrostatic energy of all $1000$ drops will be

$U_i=1000\left(\frac{k{q}^2}{2r}\right)....\left(1\right)$

Now, from the conservation of charge, charge of the bigger drop

$Q=1000q$

After merging drops into one, let the radius of bigger drop be $R$. Since, the volume of bigger drop is equal to the sum of volumes of $1000$ drops,

$\frac{4}{3}\pi R^3=1000\times \frac{4}{3}\pi r^3$

$\Rightarrow R=10r$

So, the energy of bigger drop will be:

$U_b=\frac{k{Q}^2}{2R}$

$\Rightarrow U_b=\frac{k(1000q{)}^2}{2\left(10\right)r}$

$\Rightarrow U_b={10}^5\left(\frac{k{q}^2}{2r}\right)....\left(2\right)$

So, from equation (1) and (2),

$U_b=100U_i$

So, energy increases by $100$ times.

Hence, option (c) is the correct answer.