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

We are tasked with analyzing the motion of an electric dipole in a uniform electric field and determining the time period $ T $ of its oscillations. The solution proceeds as follows:

1. Electric Dipole in a Uniform Field:
The dipole moment is denoted by $ \overrightarrow{P} $, and the electric field is $ \overrightarrow{E_0} $. The angle between the dipole moment and the electric field is $ \theta $.

2. Torque on the Dipole:
The torque $ \tau $ acting on the dipole is given by:

$ \tau = -(PE_0)\theta $

This equation holds when $ \theta $ is small, as the torque is proportional to the angular displacement $ \theta $.

3. Moment of Inertia:
The moment of inertia $ I $ of the dipole depends on the mass $ m $ and the length $ l $ of the dipole. For a dipole consisting of two point masses separated by a distance $ l $, the moment of inertia is:

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

4. Time Period of Oscillation:
For small angular displacements, the motion of the dipole is simple harmonic. The time period $ T $ of oscillation is given by:

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

Here, $ \kappa $ is the restoring torque constant, which is equal to $ PE_0 $. Substituting $ I = \frac{ml^2}{2} $ and $ \kappa = qE_0 $, we get:

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

Simplify the expression:

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

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

Final Answer:
The time period of oscillation of the dipole is:

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