Question

An electric dipole of mass \( m \), charge \( q \), and length \( l \) is placed in a uniform electric field \( \mathbf{E} = E_0 \hat{i} \). When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be:

• A. \( 2\pi \sqrt{\frac{ml}{qE_0}} \)
• B. \( \frac{1}{2\pi} \sqrt{\frac{2ml}{qE_0}} \)
• C. \( \frac{1}{2\pi} \sqrt{\frac{ml}{2qE_0}} \)
• D. \( 2\pi \sqrt{\frac{ml}{2qE_0}} \)

Hint:

For small oscillations of an electric dipole in a uniform electric field, treat the motion as simple harmonic motion and use the standard formula for the time period of oscillation.

Answer 1

The Correct Option is D

For an electric dipole in a uniform electric field, the torque \( \tau \) is given by: \[ \tau = -pE_0 \sin \theta \] where \( p = ql \) is the dipole moment and \( E_0 \) is the electric field strength. For small oscillations, the equation of motion is: \[ I \frac{d^2\theta}{dt^2} = -pE_0 \sin \theta \] Approximating \( \sin \theta \approx \theta \) for small angles, we get a simple harmonic oscillator equation: \[ I \frac{d^2\theta}{dt^2} = -pE_0 \theta \] The time period \( T \) of oscillation for a dipole is: \[ T = 2\pi \sqrt{\frac{I}{pE_0}} \] where \( I = \frac{1}{2} m l^2 \) is the moment of inertia of the dipole about its center. Substituting \( I \) and \( p = ql \), we get the time period: \[ T = 2\pi \sqrt{\frac{ml}{2qE_0}} \] Thus, the correct answer is \( 2\pi \sqrt{\frac{ml}{2qE_0}} \).