1. Torque on an electric dipole in a uniform electric field

For a dipole of moment $p = qd$, placed in a uniform electric field $E$, the restoring torque when it is at an angle $\theta$ from stable equilibrium is

$\tau = -pE\sin\theta$

For a slight rotation, $\theta$ is small, so

$\sin\theta \approx \theta$

Hence,

$\tau \approx -pE\theta = -qdE\,\theta$

This is the form of restoring torque for angular SHM.

2. Moment of inertia of the dipole

The dipole consists of two particles, each of mass $m$, separated by distance $d$. About the center of mass, each mass is at distance $d/2$.

So the moment of inertia is

$I = m\left(\frac d2\right)^2 + m\left(\frac d2\right)^2$

$I = 2m\cdot \frac{d^2}{4} = \frac{md^2}{2}$

3. Equation of angular motion

Using rotational dynamics,

$I\ddot\theta = \tau$

So,

$\frac{md^2}{2}\ddot\theta = -qdE\theta$

Rearranging,

$\ddot\theta + \frac{2qE}{md}\theta = 0$

This is the standard SHM form:

$\ddot\theta + \omega^2\theta = 0$

Therefore,

$\omega^2 = \frac{2qE}{md}$

Thus,

$\boxed{\omega = \sqrt{\frac{2qE}{md}}}$

4. Checking options

• A: $\sqrt{\dfrac{qE}{md}}$ ❌
• B: $\sqrt{\dfrac{qE}{2md}}$ ❌
• C: $\sqrt{\dfrac{qE}{-2md}}$ ❌
• D: $\sqrt{\dfrac{2qE}{md}}$ ✅

So the correct option is D.