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A dipole moment $p$ and moment of inertia $I$ is placed in a uniform electric field $\mathbf{E}$. If it is displaced slightly from its stable equilibrium position, the period of oscillation of dipole is
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The correct answer is:
$2 \pi \sqrt{\frac{I}{p E}}$
Torque on electric dipole placed in uniform electric field $E$,
where, $p=$ electric dipole moment.
For small angle $\theta, \sin \theta=\theta...(i)$
$\therefore$ From Eq. (i), we have
$\tau=p E \theta...(ii)$
but $\tau=I \times \alpha...(iii)$
where $\alpha$ is angular acceleration.
From Eqs. (ii) and (iii), we have
$I \alpha=p E \theta$
$\alpha=\frac{p E}{I} \cdot \theta \Rightarrow \frac{\theta}{\alpha}=\frac{I}{p E}...(iv)$
$\therefore$ Time period for the oscillation of dipole,
$T=2 \pi \sqrt{\frac{\theta}{\alpha}}=2 \pi \sqrt{\frac{I}{p E}}$
where, $p=$ electric dipole moment.
For small angle $\theta, \sin \theta=\theta...(i)$
$\therefore$ From Eq. (i), we have
$\tau=p E \theta...(ii)$
but $\tau=I \times \alpha...(iii)$
where $\alpha$ is angular acceleration.
From Eqs. (ii) and (iii), we have
$I \alpha=p E \theta$
$\alpha=\frac{p E}{I} \cdot \theta \Rightarrow \frac{\theta}{\alpha}=\frac{I}{p E}...(iv)$
$\therefore$ Time period for the oscillation of dipole,
$T=2 \pi \sqrt{\frac{\theta}{\alpha}}=2 \pi \sqrt{\frac{I}{p E}}$
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