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. $2\pi \sqrt{\frac{pE}{I}}$
• B. $2\pi \sqrt{\frac{I}{pE}}$
• C. $\pi \sqrt{\frac{pE}{I}}$
• D. $\pi \sqrt{\frac{I}{pE}}$

Answer 1

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

The stable equilibrium position of electric dipole in a uniform electric field is shown below.


Now, when displaced at an small angle $\theta$ from its mean position, the magnitude of restoring torque is,

${\tau }_{res}=-pE\sin \theta$


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

So, ${\tau }_{res}=-pE\theta$

The angular acceleration is,

$\alpha =\frac{{\tau }_{res}}{I}=-(\frac{pE}{I})\theta =-{\omega }^2\theta$

$\Rightarrow \omega =\sqrt{\frac{pE}{I}}$

So, time period of oscillation is

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

Hence, option $\left(b\right)$ is the correct answer.