Question

The frequency of oscillation of an electric dipole of moment P and rotational inertia I for small amplitudes about its equilibrium position in a uniform electric field of strength E is ?

Answer 1

Dear student,
When the electric dipole is placed in an electric field in equilibrium position is displaced by small angle θ, then it executes simple harmonic motion due to the restoring couple. The restoring couple will be given as:
$C=-pE\sin \theta$
As θ is very small, so sinθ ≃θ
$$\begin{gathered} C=-pE\theta \\ \text{Since},C=I\alpha =I\frac{d^2\theta }{d{t}^2} \\ \Rightarrow I\frac{d^2\theta }{d{t}^2}=-pE\theta \\ \Rightarrow \frac{d^2\theta }{d{t}^2}=-\frac{pE}{I}\theta \\ \Rightarrow \frac{d^2\theta }{d{t}^2}=-\frac{pE}{I}\theta \\ \Rightarrow \frac{d^2\theta }{d{t}^2}=-{\omega }^2\theta \end{gathered}$$

This is the standard equation of SHM, where the angular frequency is given as:
$\omega =\sqrt{\frac{pE}{I}}$
Therefore, the frequency of oscillation,
$\nu =\frac{\omega }{2\pi }=\frac{1}{2\pi }\sqrt{\frac{pE}{I}}$
Regards