Question

The frequency of oscillation of an electric dipole moment having dipole moment p and rotational inertia I, oscillating in a uniform elctric field E is given

• A. $\left(\frac{1}{2}\right)\sqrt{\frac{I}{pE}}$
• B. $\left(\frac{1}{2\pi }\right)\sqrt{\frac{pE}{I}}$
• C. $\left(2\pi \right)\sqrt{\frac{pE}{I}}$
• D. $\left(2\pi \right)\sqrt{\frac{I}{pE}}$

Answer 1

The correct option is B $\left(\frac{1}{2\pi }\right)\sqrt{\frac{pE}{I}}$

When the electric dipole is placed in an electric field in equilibrium position is displaced by small angle $\theta$, then it executes simple harmonic motion due to the restoring couple. The restoring couple will be given as:
$C=-pEsin\theta$
As $\theta$ is very small, so $sin\theta \simeq \theta$
$C=-pE\theta$
Since, $C=I\alpha =I\frac{d^2\theta }{d{t}^2}$
$\Rightarrow I\frac{d^2\theta }{d{t}^2}=-pE\theta \Rightarrow \frac{d^2\theta }{d{t}^2}=-\frac{pE}{I}\theta \Rightarrow \frac{d^2\theta }{d{t}^2}=-\frac{pE}{I}\theta \Rightarrow \frac{d^2\theta }{d{t}^2}=-{\omega }^2\theta$
This is the standard equation of SHM, where the angular frequency is given as:
$\omega =\sqrt{\frac{pE}{I}}$
Therefore, the frequency of oscillation,
$v=\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{\frac{pE}{I}}$