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Question #42369

Difficulty: 8/10 | 🟢 Free

The Question

A rigid circular loop of mass $m$ and radius $R$ is made of a uniform thin wire carrying a steady current $I$ . The loop is placed in a gravity-free space with a uniform magnetic field $\vec{B}$ of magnitude $B_0$ present throughout. Initially, the loop is held at rest such that its magnetic dipole moment vector is perpendicular to the direction of $\vec{B}$ . The loop is then released and is free to rotate about its center of mass. Assume the current $I$ is maintained constant by an external source, and ignore self-inductance. What is the maximum angular speed acquired by the loop?

Options

A.
$2 \sqrt{\frac{\pi I B_0}{m}}$
B.
$\pi \sqrt{\frac{2 I B_0}{m}}$
C.
$\sqrt{\frac{2 \pi I B_0}{m}}$
D.
$2 \sqrt{\frac{2 \pi I B_0}{m}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Torque about non-fixed axes|Force on Current-Carrying Loop|Magnetic Dipole Moment Orientation|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the dynamic coupling between a rotating magnetic dipole and a uniform magnetic field, specifically targeting two lethal misconceptions.

The primary trap (Option B) arises if students assume the magnetic dipole moment remains static in its orientation relative to the torque calculation, leading them to incorrectly apply constant-acceleration kinematics ( $\omega^2 = 2\alpha\theta$ ) using the initial maximum torque.

In reality, the torque $\tau = \mu B \sin\theta$ is non-linear, requiring an energy conservation approach.

A secondary physical trap (Option C) tests the rigid body mechanics of non-fixed axes: students often blindly use the symmetry-axis moment of inertia ( $mR^2$ ) instead of the diametric moment of inertia ( $mR^2/2$ ) about which the loop actually rotates.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Torque about non-fixed axes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free