Question

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

• A. \( \frac{2\pi}{\frac{q m l}{q E_0}} \)
• B. \( \frac{1}{2\pi} \frac{q^2 m l}{q E_0} \)
• C. \( \frac{1}{2\pi} \frac{q m l}{2 q E_0} \)
• D. $T = 2\pi \sqrt{\frac{ml}{2qE_0}}$

Hint:

For oscillations of a dipole in a uniform electric field, the time period depends on the moment of inertia of the dipole and the torque due to the electric field.

Answer 1

The Correct Option is D

The analysis concerns the oscillatory motion of an electric dipole within a uniform electric field and the derivation of its time period, denoted by $ T $. The process is outlined below:

1. Dipole Configuration:
An electric dipole with dipole moment $ \overrightarrow{P} $ is placed in a uniform electric field $ \overrightarrow{E_0} $. The angle between $ \overrightarrow{P} $ and $ \overrightarrow{E_0} $ is $ \theta $.

2. Torque Calculation:
The torque $ \tau $ experienced by the dipole is given by:

$ \tau = -(PE_0)\theta $

This relation is valid for small angular displacements, indicating a linear dependence of torque on angular displacement.

3. Moment of Inertia:
The moment of inertia $ I $ of the dipole, which depends on its mass $ m $ and length $ l $, is calculated for a dipole composed of two point masses separated by $ l $ as:

$ I = m \left( \frac{l}{2} \right)^2 \cdot 2 = \frac{ml^2}{2} $

4. Derivation of Time Period:
For minor angular deviations, the dipole's motion approximates simple harmonic motion. The oscillation time period $ T $ is determined by:

$ T = 2\pi \sqrt{\frac{I}{\kappa}} $

Here, $ \kappa $ represents the restoring torque constant, equaling $ PE_0 $. Substituting $ I = \frac{ml^2}{2} $ and $ \kappa = qE_0 $, we obtain:

$ T = 2\pi \sqrt{\frac{\frac{ml^2}{2}}{qE_0}} $

Simplification yields:

$ T = 2\pi \sqrt{\frac{mI^2}{2qIE_0}} $

$ T = 2\pi \sqrt{\frac{ml}{2qE_0}} $

Conclusion:
The time period of the dipole's oscillation is:

$ \boxed{T = 2\pi \sqrt{\frac{ml}{2qE_0}}} $