Question

Two large non-conducting plates having surface charge densities $+\sigma$ and $-\sigma$ respectively, are fixed $d$ distance apart. A small test charge $q$ of mass $m$ is attached to two non-conducting springs, each of spring constant $k$ as shown in the figure. The sum of lengths of both springs in undeformed state is $d$. The charge $q$ is released from rest with both the spring undeformed. Then charge $q$ will -

$($Neglect gravity$)$

• A. Performs SHM with angular frequency $\sqrt{\frac{2k}{m}}$ and amplitude $\frac{\sigma q}{k{ϵ}_0}$.
• B. Performs SHM with angular frequency $\sqrt{\frac{2k}{m}}$ and amplitude $\frac{\sigma q}{2k{ϵ}_0}$.
• C. Does not perform SHM, but will execute periodic motion.
• D. Remains stationary.

Answer 1

The correct option is B Performs SHM with angular frequency $\sqrt{\frac{2k}{m}}$ and amplitude $\frac{\sigma q}{2k{ϵ}_0}$.

When the charge $q$ is released, let us suppose the equilibrium is attained at $x$ distance from the initial position.

Then, $\frac{q\sigma }{{ϵ}_o}=2kx$

$\Rightarrow x=\frac{\sigma q}{2k{ϵ}_o}...\left(1\right)$

Now, suppose, the charge is further displaced by $dx$ from equilibrium position.

Then, net force,

$F=2k(x+dx)-\frac{\sigma q}{{ϵ}_o}$

$\Rightarrow F=2k(\frac{\sigma q}{2k{ϵ}_o}+dx)-\frac{\sigma q}{{ϵ}_o}\left[\text{From}\right(1\left)\right]$

$\Rightarrow F=2kdx...\left(2\right)\left[\text{Toward opposite of displacement}dx\right]$

Hence, $F$ is restoring in nature and as $F\propto dx$, the charge perform SHM.

From $\left(2\right)$,

$2kdx=m{\omega }^{2}dx$

$\Rightarrow \omega =\sqrt{\frac{2k}{m}}$

As charge starts from rest, the initial position is one of the extreme positions.

Thus, the amplitude of the SHM is $\frac{\sigma q}{2k{ϵ}_o}\left[\text{From}\right(1\left)\right]$.