Question

For an electric dipole in a non-uniform electric field with dipole moment parallel to direction of the field, the force $F$ and torque $\tau$ on the dipole respectively are __________. Fill in the blank with the correct answer from the options given below.

• A. $F = 0$, $\tau = 0$
• B. $F \neq 0$, $\tau = 0$
• C. $F = 0$, $\tau \neq 0$
• D. $F \neq 0$, $\tau \neq 0$

Hint:

Remember: 1. In a uniform field: $F$ is always $0$, $\tau$ depends on the angle. 2. In a non-uniform field: $F$ is always $\neq 0$, $\tau$ depends on the angle. If $\vec{p}$ is parallel or anti-parallel to $\vec{E}$, torque is always zero!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Torque on a dipole depends on its orientation relative to the field.
Net force on a dipole in a field depends on whether the field is uniform or non-uniform.
Step 2: Detailed Explanation:
1. Torque (\( \tau \)): The formula for torque is \( \vec{\tau} = \vec{p} \times \vec{E} \), or \( \tau = pE \sin \theta \).
The question states the dipole moment is parallel to the field, meaning \( \theta = 0^\circ \).
Since \( \sin 0^\circ = 0 \), the torque \( \tau = 0 \).
2. Force (\( F \)): In a uniform field, the forces on \( +q \) and \( -q \) are equal and opposite, leading to zero net force.
However, in a non-uniform field, the electric field magnitude \( E \) varies with position.
Since the two charges of the dipole are at slightly different positions, they experience different magnitudes of electric force.
The net force is given by \( F = p \frac{dE}{dx} \).
Since the field is non-uniform, \( \frac{dE}{dx} \neq 0 \), hence the net force \( F \neq 0 \).
Step 3: Final Answer:
For a parallel dipole in a non-uniform field, force is non-zero and torque is zero.