Three charges $+ Q_{1} + Q_{2}$ , and q are placed on a straight line such that q is somewhere in between +Q₁ and +Q₂ . If this system of charges is in equilibrium, what should be the magnitude and sign of charge q?

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$\frac{Q_{1} + Q_{2}}{2}$ , positive

$\frac{Q_{1} Q_{2}}{{(\sqrt{Q_{1}} + \sqrt{Q_{2}})}^{2}}$ , negative

$\frac{Q_{1} Q_{2}}{{(\sqrt{Q_{1}} + \sqrt{Q_{2}})}^{2}}$ , positive

$\frac{Q_{1} + Q_{2}}{2}$ , negative

answer is C.

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Detailed Solution

Force on Q2 is zero (q should be negative)
Question Image

$\frac{{kQ}_{1} Q_{2}}{R^{2}} = \frac{{kqQ}_{2}}{x^{2}} ⇒ \frac{x}{R} = \sqrt{\frac{q}{Q_{1}}}$

Force on q is zero $\frac{{kQ}_{1} q}{(R - x)^{2}} = \frac{{kqQ}_{2}}{x^{2}} ⇒ \frac{R - x}{x} = \frac{\sqrt{Q_{1}}}{\sqrt{Q_{2}}}$
or $\frac{R}{x} = \frac{\sqrt{Q_{1}} + \sqrt{Q_{2}}}{\sqrt{Q_{2}}} or \frac{\sqrt{Q_{1}}}{\sqrt{q}} = \frac{\sqrt{Q_{1}} + \sqrt{Q_{2}}}{\sqrt{Q_{2}}}$