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Question #88162

Difficulty: 5/10 | 🟢 Free

The Question

A small superconducting loop of radius $r$ and self-inductance $L$ is placed in a uniform external magnetic field $\vec{B} = B_0 \hat{k}$ . Initially, the loop lies in the $xy$ -plane with its area vector parallel to $\vec{B}$ , and it carries a steady current $I_0 = \frac{\pi r^2 B_0}{L}$ such that its magnetic moment is in the $+\hat{k}$ direction. The loop is then slowly rotated by $90^\circ$ about the $x$ -axis by an external agent. Calculate the mechanical work done by the external agent during this process.

Options

A.
$\frac{3 \pi^2 r^4 B_0^2}{2 L}$
B.
$\frac{\pi^2 r^4 B_0^2}{L}$
C.
$\frac{\pi^2 r^4 B_0^2}{2 L}$
D.
$\frac{2 \pi^2 r^4 B_0^2}{L}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Work against Conservative Force|Torque and Potential Energy in B-Field|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the application of the work-energy theorem in electromagnetism, specifically targeting the misconception that the potential energy of a magnetic dipole is always $U = -\vec{m} \cdot \vec{B}$ .

Students often incorrectly apply this formula to calculate work ( $W = \Delta U$ ) without realizing it is only valid for a rigid dipole with a constant magnetic moment.

In a superconducting loop, the dynamic constraint of flux conservation causes the induced current (and thus the magnetic moment) to vary with angle.

The rigorous approach requires integrating the angle-dependent torque $\tau(\theta) d\theta$ , which elegantly transforms into the change in the loop's self-energy $\Delta(\frac{1}{2}LI^2)$ .

Failing to account for the variable dipole moment leads to incorrect energy superpositions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Work against Conservative Force
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free