Two large conducting plates having surface charge densities $+ σ$ and $- σ$ respectively, are fixed ‘d’ distanced apart. A small test charge q of mass m is attached to two non-conducting identical springs of spring constant k as shown in the adjacent fig. The charge q is now released from rest with springs in natural length. Then q will [neglect gravity]

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

remain stationary

Perform SHM with amplitude $\frac{σ q}{2 k \in_{o}}$

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

answer is A, B.

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Detailed Solution

(a) The electrostatic force on charges constant hence it does not effect the time period of spring. As two springs are parallel $K_{e f f} = K + K = 2 K$ $∴ ω = \sqrt{\frac{2 k}{m}}$
(b) Initially the charges is at rest, i.e at extreme position. Electrostatic force on m is $E q = q (\frac{σ}{2 ε_{o}} + \frac{σ}{2 ε_{o}}) = \frac{q σ}{ε_{o}}$
The equilibrium position $2 k x_{o} = \frac{q σ}{ε_{o}}$
$x_{o} = \frac{q σ}{2 k ε_{o}}$