Question

A dipole of dipole moment 'P' and moment of inertia I is placed in a uniform electric field \(\overrightarrow{E}\). If it is displaced slightly from its stable equilibrium position, the period of oscillation of dipole is

• A. \(\sqrt\frac{PE}{1}\)
• B. \(2\pi\sqrt\frac{1}{PE}\)
• C. \(\frac{1}{2\pi}\sqrt\frac{PE}{1}\)
• D. \(\pi\sqrt\frac{1}{PE}\)

Answer 1

The Correct Option is B

The period of oscillation of a dipole in a uniform electric field is given by the formula: \[ T = 2\pi \sqrt{\frac{I}{PE}} \] Where \(T\) is the period of oscillation, \(I\) is the moment of inertia, \(P\) is the dipole moment, and \(E\) is the electric field. This equation represents simple harmonic motion where the restoring torque is proportional to the angular displacement.

The correct option is (B): \(2\pi\sqrt\frac{1}{PE}\)

Answer 2

The dipole in a uniform electric field \( \vec{E} \) experiences a torque given by: \[ \tau = - P E \sin \theta \] where \( P \) is the dipole moment, \( E \) is the electric field, and \( \theta \) is the angle between the dipole moment and the electric field. For small displacements, \( \sin \theta \approx \theta \), so the torque becomes: \[ \tau = - P E \theta \] This is the equation for simple harmonic motion, where the restoring torque is proportional to the angular displacement. The equation of motion is: \[ I \frac{d^2\theta}{dt^2} = - P E \theta \] This can be written as: \[ \frac{d^2\theta}{dt^2} + \frac{P E}{I} \theta = 0 \] This is a standard equation for simple harmonic motion with angular frequency \( \omega \), where: \[ \omega^2 = \frac{P E}{I} \] Thus, the angular frequency \( \omega \) is: \[ \omega = \sqrt{\frac{P E}{I}} \] The period of oscillation \( T \) is related to the angular frequency by: \[ T = \frac{2\pi}{\omega} \] Substitute \( \omega \): \[ T = 2\pi \sqrt{\frac{I}{P E}} \] Thus, the period of oscillation of the dipole is \( 2\pi \sqrt{\frac{I}{P E}} \).