Three point charges $\displaystyle \mathrm{Q}_{1}, \mathrm{Q}_{2}$ and $\displaystyle \mathrm{Q}_{3}$ are located in $\displaystyle \mathrm{x}-\mathrm{y}$ plane at points $\displaystyle (-\mathrm{d}, 0),(0,0)$ and $\displaystyle (\mathrm{d}, 0)$ respectively. $\displaystyle \mathrm{Q}_{1}$ and $\displaystyle \mathrm{Q}_{3}$ are identical and $\displaystyle \mathrm{Q}_{2}$ is positive. What will be the nature and value of $\displaystyle \mathrm{Q}_{1}$ so that the potential energy of the system is zero ?

Nature of \(\displaystyle \mathrm{Q}_{1}\) Value of \(\displaystyle \mathrm{Q}_{1}\) Nature of \(\displaystyle \mathrm{Q}_{1}\) will be negative. Let, \(\displaystyle \mathrm{Q}_{1}=\mathrm{Q}_{3}=\mathrm{q}\) \[\begin{aligned} & \frac{1}{4 \pi \varepsilon_{0}}\left[\frac{\mathrm{q} \mathrm{Q}_{2}}{\mathrm{d}}+\frac{\mathrm{q} \mathrm{q}}{2 \mathrm{d}}+\frac{\mathrm{Q}_{2} \mathrm{q}}{\mathrm{d}}\right]=0 \\ & \frac{1}{4 \pi \varepsilon_{0} \mathrm{d}}\left[\mathrm{q} \mathrm{Q}_{2}+\frac{\mathrm{q}^{2}}{2}+\mathrm{Q}_{2} \mathrm{q}\right]=0 \\ & 2 \mathrm{q} \mathrm{Q}_{2}+\frac{\mathrm{q}^{2}}{2}=0 \\ & 2 \mathrm{q} \mathrm{Q}_{2}=-\frac{\mathrm{q}^{2}}{2} \\ & \mathrm{Q}_{1}=\mathrm{q}=-4 \mathrm{Q}_{2} \end{aligned} \]