Question

Two large conducting plates having surface charge densities $+\sigma$ and $-\sigma$ respectively, are fixed $‘d^{\prime }$ distance 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 figure. The charge $q$ is now released from rest with springs in natural length. Then, $q$ will [neglect gravity]

• A. perform SHM with angular frequency $\sqrt{\frac{2k}{m}}$
• B. perform SHM with amplitude $\frac{\sigma q}{2k{ϵ}_0}$
• C. not perform SHM, but will have a periodic motion, if charges are removed on plates as well as on $m$.
• D. remains stationary.

Answer 1

Answer: (A), (B)

The correct options are
A perform SHM with angular frequency $\sqrt{\frac{2k}{m}}$
B perform SHM with amplitude $\frac{\sigma q}{2k{ϵ}_0}$


The electrostatic force on charge is constant, hence it does not affect the time period of spring.
As two springs are on parallel $K_{eff}=k+k=2k$
$\therefore \omega =\sqrt{\frac{K_{eff}}{m}}=\sqrt{\frac{2k}{m}}$

Initially the charge is at rest, i.e. at extreme position.
Electrostatic force acting on $m$ is $qE=q(\frac{\sigma }{2{ϵ}_0}+\frac{\sigma }{2{ϵ}_0})$

The equilibrium position-
$\therefore 2k{x}_0=\frac{q\sigma }{{ϵ}_0}$ or $x_0=\frac{\sigma q}{2k{ϵ}_0}$
Amplitude is $x_0-0=\frac{\sigma q}{2k{ϵ}_0}$