Question

An electric dipole of dipole moment $p$ is placed in a uniform electric field $E$ in stable equilibrium position. Its moment of inertia about the centroidal axis is $I$. If it is displaced slightly from its mean position, find the period of small oscillations.

• A. $\pi \sqrt{\frac{I}{pE}}$
• B. $2\pi \sqrt{\frac{I}{pE}}$
• C. $\frac{1}{2\pi }\sqrt{\frac{pE}{I}}$
• D. $2\pi \sqrt{\frac{IE}{p}}$

Answer 1

The correct option is B $2\pi \sqrt{\frac{I}{pE}}$


When the dipole displaced at an angle $\theta$ from its mean position, the magnitude of restoring torque will be

$\tau =-pE\sin \theta ...\left(1\right)$

For very small angular displacement,
$\sin \theta \approx \theta$

From equation $\left(1\right)$,

$\tau =-pE\theta$

We know that, the angular acceleration is
$\alpha =\frac{\tau }{I}$

Substituting the value of $\tau$ in the above expression,
$\alpha =-\left(\frac{pE}{i}\right)\theta =-{\omega }^2\theta$
where, $\omega$ is angular frequncy of the oscillation,

Therefore,
${\omega }^2=\frac{pE}{I}$

So, time period, $T$ of oscillation,

$T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{I}{pE}}$

Hence, option (a) is correct answer.