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

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

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

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

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

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Solution

The correct option is B $\frac{Q_{1} Q_{2}}{{(\sqrt{Q_{1}} + \sqrt{Q_{2}})}^{2}}, n e g a t i v e$
Force on $Q_{2}$ is zero( $q$ should be -ve)
$\Rightarrow \frac{k Q_{1} Q_{2}}{R^{2}} = \frac{k q Q_{2}}{x^{2}} \Rightarrow \frac{x}{R} = \sqrt{\frac{q}{Q_{1}}}$
Force on $q$ is zero:
$\frac{k q Q_{1} q}{{(R - x)}^{2}} = \frac{k q Q_{2}}{x^{2}} \Rightarrow \frac{R - x}{x} = \frac{\sqrt{Q_{1}}}{\sqrt{Q_{2}}} \Rightarrow \frac{R}{x} = \frac{\sqrt{Q_{1}} + \sqrt{Q_{2}}}{\sqrt{Q_{2}}} \Rightarrow \frac{\sqrt{Q_{1}}}{\sqrt{q}} = \frac{\sqrt{Q_{1}} + \sqrt{Q_{2}}}{\sqrt{Q_{2}}}$

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

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Three charges +Q1,Q2, and q are placed on a straight line such that q is somewhere in equilibrium,what should be the magnitude and sign of charge q ?

The correct option is B Q1Q2(√Q1+√Q2)2,negativeForce on Q2 is zero(q should be -ve)⇒kQ1Q2R2=kqQ2x2⇒xR=√qQ1Force on q is zero: kqQ1q(R−x)2=kqQ2x2⇒R−xx=√Q1√Q2⇒Rx=√Q1+√Q2√Q2⇒√Q1√q=√Q1+√Q2√Q2⇒q=Q1Q2(√Q1+√Q2)2

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