A certain charge Q is divided into two parts q and (Q − q). How should the charges Q and q be divided, so that q and (Q − q) placed at a certain distance apart experience maximum electrostatic repulsion ?

Let’s say the charge q and (Q - q) are at r distance from each other. This can be shown as

According to Coulomb’s law, force between both the parts can be given as

$\begin{array}{c}\mathrm{F}=\frac{\mathrm{kq}(\mathrm{Q}-\mathrm{q})}{{\mathrm{r}}^2}\\ \mathrm{F}=\frac{\mathrm{k}}{{\mathrm{r}}^2}\left(\mathrm{qQ}-{\mathrm{q}}^2\right)\end{array}$

As we know that $\frac{\mathrm{dF}}{\mathrm{dq}}=0$, for maximum force.

$\begin{array}{l}\Rightarrow \frac{\mathrm{dF}}{\mathrm{dq}}=\frac{\mathrm{d}}{\mathrm{dq}}\left[\frac{\mathrm{k}}{{\mathrm{r}}^2}\left(\mathrm{qQ}-{\mathrm{q}}^2\right)\right]=0\\ \Rightarrow \frac{\mathrm{k}}{{\mathrm{r}}^2}(\mathrm{Q}-2\mathrm{q})=0\Rightarrow \mathrm{Q}=2\mathrm{q}\end{array}$