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Question #43872

Difficulty: 8/10 | 🟢 Free

The Question

A rigid circular loop of radius $R$ and mass $M$ carries a constant steady current $I$ in the $+\hat{\theta}$ direction (counter-clockwise when viewed from the $+z$ -axis). The loop is constrained to move along the $z$ -axis such that its center lies on the $z$ -axis and its plane is always parallel to the $xy$ -plane. The magnetic field in the region is given in cylindrical coordinates $(r, \theta, z)$ as:

$\vec{B}(r, \theta, z) = (\alpha z r - \gamma z r^3)\hat{r} + (B_0 - \alpha z^2 + 2\gamma z^2 r^2)\hat{k}$

where $\alpha$ , $\gamma$ , and $B_0$ are positive constants, and $\alpha > \gamma R^2$ . The self-inductance of the loop and any induced back-EMF are negligible. If the loop is given a small displacement from the origin along the $z$ -axis and released, it executes simple harmonic motion. What is the time period of these oscillations?

Options

A.
$\sqrt{\frac{2\pi M}{I R^2 \alpha}}$
B.
$\sqrt{\frac{2\pi M}{I R^2 (\alpha - 2\gamma R^2)}}$
C.
$\sqrt{\frac{2\pi M}{I R^2 (\alpha - \gamma R^2)}}$
D.
$\sqrt{\frac{\pi M}{I R^2 (\alpha - \gamma R^2)}}$

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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

Force on Current-Carrying Loop|SHM Definition via Differential Equation|Effective Spring Constant

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the magnetic dipole approximation.

Students frequently memorize the potential energy of a current loop as $U = -\vec{m} \cdot \vec{B}$ and evaluate the magnetic field only at the center of the loop.

In a highly non-uniform field where the field varies significantly across the loop's area, this approximation fails.

Applying the dipole approximation here evaluates $B_z$ at $r=0$ , missing the $\gamma$ term entirely and yielding an incorrect restoring force constant (Option A).

The correct approach requires either integrating the exact magnetic flux through the finite area of the loop ( $F_z = I \frac{\partial \Phi}{\partial z}$ ) or directly integrating the Lorentz force from the radial magnetic field component at the loop's perimeter.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Force on Current-Carrying Loop
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free