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Question #43019

Difficulty: 8/10 | 🟢 Free

The Question

A fixed large circular coil of radius $R$ carries a steady current $I_1$ . A small rigid circular wire loop of mass $m$ and radius $r$ ( $r \ll R$ ) carries a steady current $I_2$ . The small loop is suspended from a frictionless horizontal axis passing through its highest point and lying in its own plane. The center of the small loop is located on the central axis of the large coil at a distance $x = \sqrt{3}R$ from the center of the large coil. The central axis of the large coil is horizontal. In the equilibrium position, the small loop hangs vertically such that the magnetic field of the large coil is perpendicular to the plane of the small loop. The currents are directed such that the magnetic torque opposes any angular deflection. Find the time period $T$ of small oscillations of the small loop.

Options

A.
$2 \pi \sqrt{\frac{3 m r}{2 m g + \frac{\mu_0 \pi I_1 I_2 r}{8 R}}}$
B.
$2 \pi \sqrt{\frac{2 m r}{2 m g + \frac{\mu_0 \pi I_1 I_2 r}{8 R}}}$
C.
$2 \pi \sqrt{\frac{4 m r}{2 m g + \frac{\mu_0 \pi I_1 I_2 r}{8 R}}}$
D.
$2 \pi \sqrt{\frac{3 m r}{2 m g + \frac{\mu_0 \pi I_1 I_2 r}{R}}}$

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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

Magnetic Field on Axis vs. Center|Physical Pendulum Inertia

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of electromagnetism and rigid body dynamics.

The primary trap lies in the calculation of the moment of inertia for the physical pendulum.

Students frequently default to $I = mr^2$ (treating the loop as a point mass) or $I = 2mr^2$ (incorrectly applying the perpendicular axis theorem).

Because the loop swings out of its plane, the relevant centroidal moment of inertia is about an axis in the plane of the loop ( $I_{cm} = \frac{1}{2}mr^2$ ), making the correct pivot inertia $\frac{3}{2}mr^2$ via the parallel axis theorem.

Additionally, it requires correctly evaluating the Biot-Savart field on the axis of a coil rather than at its center, penalizing students who blindly use the standard $B = \mu_0 I / 2R$ formula.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Field on Axis vs. Center
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free