A certain charge Q is divided into two parts q and Q $-$ q. How the charge Q and q must be related, so that when q and (Q $-$ q) is placed at a certain distance apart experience maximum electrostatic repulsion?

The electrostatic force of repulsion between the charge q and (Q $-$ q) at separation r is given by

$F=\frac{1}{4\pi {\epsilon }_0}·\frac{q(Q-q)}{r^2}=\frac{1}{4\pi {\epsilon }_0}·\frac{qQ-q^2}{r^2}$

For maximum electrostatic force of repulsion, then  $\frac{\partial F}{\partial q}=0$

$\text{ i.e. } \frac{1}{4\pi {\epsilon }_0}·\frac{(Q-2q)}{r^2}=0$

$\text{ As, }1/4\pi {\epsilon }_0{r}^2\text{ is constant, therefore }$

$Q-2q=0\text{ or }Q=2q$