Question

Define electric dipole and dipole moment. An electric dipole of dipole moment \( 2 \times 10^{-8} \, \text{C} \cdot \text{m} \) is inclined at an angle of 30° from a uniform electric field of \( 2 \times 10^5 \, \text{V/m} \). Find the potential energy of the dipole and the moment of the couple acting on it.

Hint:

The potential energy of an electric dipole in a uniform electric field is given by \( U = - p E \cos(\theta) \), and the moment of the couple is \( \tau = p E \sin(\theta) \).

Answer 1

Step 1: Electric Dipole and Dipole Moment.
An electric dipole consists of two equal and opposite charges separated by a small distance. The dipole moment \( \vec{p} \) is given by: \[ \vec{p} = q \times \vec{d} \] where: - \( q \) is the charge, - \( \vec{d} \) is the displacement vector between the charges.
Step 2: Potential Energy of the Dipole.
The potential energy \( U \) of a dipole in a uniform electric field \( E \) is given by: \[ U = - \vec{p} \cdot \vec{E} \] where \( \vec{p} \) is the dipole moment and \( \vec{E} \) is the electric field. For \( \theta = 30^\circ \), \[ U = - p E \cos(\theta) \] Substitute the values: \[ U = - (2 \times 10^{-8} \, \text{C} \cdot \text{m}) \times (2 \times 10^5 \, \text{V/m}) \times \cos(30^\circ) \] \[ U = - 2 \times 10^{-8} \times 2 \times 10^5 \times \frac{\sqrt{3}}{2} \] \[ U = - 1.732 \times 10^{-3} \, \text{J} \]
Step 3: Moment of the Couple.
The moment \( \tau \) of the couple acting on the dipole is given by: \[ \tau = p E \sin(\theta) \] Substitute the values: \[ \tau = (2 \times 10^{-8} \, \text{C} \cdot \text{m}) \times (2 \times 10^5 \, \text{V/m}) \times \sin(30^\circ) \] \[ \tau = 2 \times 10^{-8} \times 2 \times 10^5 \times \frac{1}{2} \] \[ \tau = 2 \times 10^{-3} \, \text{N} \cdot \text{m} \]