Two particles A and B, each carrying a charge Q, are held fixed with a separation d between them. A particle C having a mass m and charge –q, in kept at the middle point of line AB.It displaced through a distance x perpendicular to AB. Assume x << d. Then:

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Particle C may executes SHM with time period ${(\frac{{mπ}^{2} ε_{0} d^{2}}{qQ})}^{1 / 2}$

Particle C may execute SHM with time period ${(\frac{{mπ}^{3} ε_{0} d^{3}}{qQ})}^{1 / 2}$

Force experienced by C is proportional to x

Force experienced by C is proportional to d

answer is A, C.

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Detailed Solution

$\begin{array}{l} F = 2 \frac{k Q q}{x^{2} + \frac{d^{2}}{4}} . \frac{x}{{(x^{2} + \frac{d^{2}}{4})}^{\frac{1}{2}}} = \frac{16 kQqx}{d^{3}} = \frac{4 Qqx}{{πε}_{0} d^{3}} \\ a = - \frac{F}{m} = - \frac{4 Qqx}{{πε}_{p} {md}^{3}}; T = 2 π \sqrt{\frac{{πε}_{0} {md}^{3}}{4 Qq}} \end{array}$

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