JEE Main · 2021 · 20 Jul 2021 · Shift 1 · Electrostatics

At a fixed separation $r$ the repulsion between the two parts is

With $Q$ fixed, differentiating with respect to $q$ gives $\frac{dF}{dq}=\frac{Q-2q}{4\pi {\epsilon }_0{r}^2}$, which vanishes at $q=\frac{Q}{2}$, and $\frac{d^{2}F}{d{q}^2}=-\frac{2}{4\pi {\epsilon }_0{r}^2}<0$ confirms a maximum there. The force is largest when the charge is split into two equal halves, that is when $Q=2q$.