First slide

Electrostatic force

Question

Three charges +Q₁, +Q₂, 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?

Moderate

A

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

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

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

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

Solution

Force on Q₂ is zero (q should be negative)
$\frac{k Q_{1} Q_{2}}{R^{2}} = \frac{k q Q_{2}}{x^{2}} or \frac{x}{R} = \sqrt{\frac{q}{Q_{1}}}$
Force on q is zero:
$\begin{array}{l} \frac{k Q_{1} q}{(R - x)^{2}} = \frac{k q Q_{2}}{x^{2}} \\ or \frac{R - x}{x} = \frac{\sqrt{Q_{1}}}{\sqrt{Q_{2}}} \\ or \frac{R}{x} = \frac{\sqrt{Q_{1}} + \sqrt{Q_{2}}}{\sqrt{Q_{2}}} or \frac{\sqrt{Q_{1}}}{\sqrt{q}} = \frac{\sqrt{Q_{1}} + \sqrt{Q_{2}}}{\sqrt{Q_{2}}} \\ or q = \frac{Q_{1} Q_{2}}{{(\sqrt{Q_{1}} + \sqrt{Q_{2}})}^{2}} \end{array}$

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