KCET · Physics · Electrostatics
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
- A \(\sqrt{\frac{p E}{I}}\)
- B \(2 \pi \sqrt{\frac{I}{p E}}\)
- C \(\frac{1}{2 \pi} \sqrt{\frac{p E}{I}}\)
- D \(\pi \sqrt{\frac{I}{p E}}\)
Answer & Solution
Correct Answer
(B) \(2 \pi \sqrt{\frac{I}{p E}}\)
Step-by-step Solution
Detailed explanation
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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