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Question #42830

Difficulty: 8/10 | 🟢 Free

The Question

A rigid insulating rod of length $2a$ has point charges $+q$ and $-q$ fixed at its ends, forming an electric dipole of moment $\vec{p}$ (where $p = 2qa$ ). The dipole is placed such that its center is fixed on the axis of a circular loop of radius $R$ ( $R \gg a$ ) carrying a steady counter-clockwise current $I$ . The loop lies in the $xy$ -plane with its center at the origin $(0,0,0)$ . The center of the dipole is located at coordinates $(0, 0, R)$ . The dipole is constrained to rotate in the plane $z = R$ with a constant angular velocity $\vec{\omega} = \omega \hat{k}$ . A uniform electric field $\vec{E} = E_0 \hat{i}$ is also present in the entire region. At the instant the dipole moment vector is aligned with the positive $y$ -axis ( $\vec{p} = p \hat{j}$ ), what is the net torque $\vec{\tau}$ acting on the dipole about its center? (Assume the magnetic field variation over the length of the dipole is small but non-zero, and account for all spatial components of the magnetic field).

Options

A.
$-\frac{3 \mu_0 I p a^2 \omega}{16\sqrt{2} R^2} \hat{i} - p E_0 \hat{k}$
B.
$- p E_0 \hat{k}$
C.
$+\frac{3 \mu_0 I p a^2 \omega}{16\sqrt{2} R^2} \hat{i} - p E_0 \hat{k}$
D.
$-\frac{3 \mu_0 I p a^2 \omega}{8\sqrt{2} R^2} \hat{j} - p E_0 \hat{k}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Dipole in Uniform Field|Magnetic Dipole Moment Orientation|Faraday’s Law Vector Form

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from pure electrostatics to electrodynamics by placing a rotating electric dipole in a non-uniform magnetic field.

The primary trap is the assumption that a magnetic field exerts no net torque on an electric dipole, leading students to only calculate the electrostatic torque $\vec{\tau} = \vec{p} \times \vec{E}$ (Option B).

Even if students realize the charges are moving and experience a Lorentz force, they often falsely assume the magnetic field on the axis is purely axial ( $\vec{B} = B_z \hat{k}$ ).

If $B_r = 0$ , the magnetic force is purely radial and produces zero torque.

The 'Lethal Collision' occurs because students fail to apply Gauss's Law for Magnetism ( $\nabla \cdot \vec{B} = 0$ ), which dictates that a spatially varying axial field must be accompanied by a radial field component.

This radial component interacts with the azimuthal velocity of the charges to produce a $z$ -directed force, creating a torque that tilts the dipole out of its plane of rotation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dipole in Uniform Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free