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Question #42963

Difficulty: 8/10 | 🟢 Free

The Question

A small solid diamagnetic sphere of mass $M$ and volume $V$ is attached to one end of a massless rigid rod of length $L$ . The other end of the rod is pivoted at the origin $O$ , allowing the sphere to rotate freely in the $xy$ -plane. The system is placed in a static, non-uniform magnetic field given by $\vec{B} = B_0 (1 + \alpha x) \hat{i} - B_0 \alpha y \hat{j}$ , where $B_0$ and $\alpha$ are positive constants. The induced magnetic dipole moment of the sphere is $\vec{m} = -\frac{\chi V}{\mu_0} \vec{B}$ , where $\chi > 0$ is the magnitude of its diamagnetic susceptibility. If the sphere is projected from the position $(L, 0)$ with an initial angular velocity $\omega_0 \hat{k}$ , what is its angular velocity $\omega$ when it crosses the position $(-L, 0)$ ? Assume no gravity and no other resistive forces.

Options

A.
$\sqrt{\omega_0^2 + \frac{4\chi V B_0^2 \alpha}{\mu_0 M L}}$
B.
$\sqrt{\omega_0^2 + \frac{8\chi V B_0^2 \alpha}{\mu_0 M L}}$
C.
$\sqrt{\omega_0^2 - \frac{4\chi V B_0^2 \alpha}{\mu_0 M L}}$
D.
$\sqrt{\omega_0^2 + \frac{2\chi V B_0^2 \alpha}{\mu_0 M L}}$

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

Diamagnetic Property of Materials|Non-Uniform Circular Motion|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize rigid body dynamics with the energetics of induced magnetic dipoles in non-uniform fields.

The primary 'Lethal Trap' lies in the potential energy formulation.

Students frequently misapply the potential energy formula for a permanent dipole ( $U = -\vec{m} \cdot \vec{B}$ ) to an induced dipole.

For an induced dipole, the magnetic moment is proportional to the field ( $\vec{m} \propto \vec{B}$ ), meaning the work done to establish the dipole moment must be accounted for, introducing a crucial factor of $1/2$ ( $U = -\frac{1}{2}\vec{m} \cdot \vec{B}$ ).

Failing to recognize this distinction leads directly to Option B.

Additionally, students must correctly evaluate the spatial gradient of $B^2$ along the constrained circular trajectory rather than mistakenly integrating a 1D force, testing their vector calculus application in non-uniform circular motion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Diamagnetic Property of Materials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free