Two large non-conducting plates having surface charge densities $+ σ$ and $- σ$ , respectively, are fixed ‘d’ distance apart. A small test charge q of mass m is attached to two non-conducting identicals springs, each of spring constant 2K as shown in the fig. The charge q is now released from rest with springs in natural length. Then q will (neglect Gravity)

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Perform SHM with angular frequency $2 \sqrt{\frac{K}{m}}$

Remain stationary

Perform SHM with amplitude $\frac{σq}{4 K \in_{0}}$

Not perform SHM, but will have a periodic motion if charges are removed on plates as well as on m

answer is A, B.

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Detailed Solution

Eelctrostatic force doesn’t effect time period $w = \sqrt{\frac{keff}{m}} = \sqrt{\frac{4 k}{m}} = 2 \sqrt{\frac{K}{m}}$
Charge is at rest at extreme position $F = qE = q [\frac{σ}{2 \in_{0}} + \frac{σ}{2 \in_{0}}] = \frac{qσ}{\in_{0}}$
$4 {kx}_{0} = \frac{qσ}{\in_{0}}$
Amplitude $x_{0} = \frac{qσ}{4 k \in_{0}}$