Four-point charges each of magnitude $1 μC$ (each positive) are fixed at the vertices of a square ABCD. The diagonals of the square intersect at O. Length of each diagonal is 2m. A particle of mass 2 g having a charge $+ 2 μC$ is released from a point 1cm from O and 99cm from A. The motion of the particle is SHM with period $\frac{π \sqrt{x}}{3}$ s. Find the value of x. (Neglect gravity).

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Detailed Solution

$r_{A} = 0.99 m; r_{B} = r_{D} = \sqrt{1^{2} + 0 . 01^{2}} m; r_{C} = 1.01 m; d = 0.01 m$

$F_{net} = \frac{kQq}{{r_{A}}^{2}} - \frac{kQq}{{r_{c}}^{2}} - \frac{kQq}{{r_{B}}^{2}} \times \frac{d}{r_{B}} \times 2$

$\Rightarrow {mω}^{2} d = \frac{kQq}{{r_{A}}^{2}} - \frac{kQq}{{r_{c}}^{2}} - \frac{kQq}{{r_{B}}^{2}} \times \frac{d}{r_{B}} \times 2$

$\Rightarrow ω = \sqrt{900 \times 0.02} = 3 \sqrt{2} rad / s$

$\Rightarrow T = \frac{2 π}{ω} = \frac{π \sqrt{2}}{3} = \frac{π \sqrt{x}}{3} \Rightarrow x = 2$

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