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

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$\frac{Q_{1} + Q_{2}}{2}, n e g a t i v e$

$\frac{Q_{1} Q_{2}}{{(\sqrt{Q_{1}} + \sqrt{Q_{2}})}^{2}}, n e g a t i v e$

$\frac{Q_{1} + Q_{2}}{2}, p o s i t i v e$

$\frac{Q_{1} Q_{2}}{{(\sqrt{Q_{1}} + \sqrt{Q_{2}})}^{2}}, p o s i t i v e$

answer is C.

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Detailed Solution

Force on $Q_{2}$ is zero (g should be negative).

$\frac{k Q_{1} Q_{2}}{R^{2}} = \frac{k q Q_{2}}{x^{2}} o r \frac{x}{R} = \sqrt{\frac{q}{Q_{1}}}$

Force on q is zero:

$\frac{k Q_{1} q}{{(R - x)}^{2}} = \frac{k q Q_{2}}{x^{2}}$

$o r \frac{R - x}{x} = \frac{\sqrt{Q_{1}}}{\sqrt{Q_{2}}}$

$o r \frac{R}{x} = \frac{\sqrt{Q_{1}} + \sqrt{Q_{2}}}{\sqrt{Q_{2}}} o r \frac{\sqrt{Q_{1}}}{\sqrt{q}} = \frac{\sqrt{Q_{1}} + \sqrt{Q_{2}}}{\sqrt{Q_{2}}}$

$o r q = \frac{Q_{1} Q_{2}}{{(\sqrt{Q_{1}} + \sqrt{Q_{2}})}^{2}}$

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