A point charge \(Q\) is placed at the center of the line joining two equal point charges \(+q\) and \(+q\). The value of \(Q\) if the system of the charges is equilibrium, is

The forces on charge \(Q\) due to the other two charges will be equal and opposite and hence it will be in equilibrium. Force on \(+\mathrm{q}\) due to the other two charges should also be equal and opposite. Their magnitudes will be equal if

\(
\begin{aligned}
& \frac{1}{4 \pi \varepsilon_0} \cdot \frac{\mathrm{q}^2}{\mathrm{r}^2}=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{\mathrm{qQ}}{\left(\frac{\mathrm{r}}{2}\right)^2}=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{4 \mathrm{qQ}}{\mathrm{r}^2} \\
& \therefore \mathrm{q}=4 \mathrm{Q} \text { or } \mathrm{Q}=\frac{\mathrm{q}}{4}
\end{aligned}
\)
\(\mathrm{Q}\) must be negative, hence \(\mathrm{Q}=-\frac{\mathrm{q}}{4}\)