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Question #42624

Difficulty: 8/10 | 🟢 Free

The Question

A small magnetic dipole of magnetic moment $M$ and moment of inertia $J$ is suspended by a torsional fiber of torsional constant $C$ . The equilibrium position of the dipole is aligned with the $x$ -axis. A long straight wire parallel to the $y$ -axis is located at $z = -d$ and carries a constant current $I_0$ in the $+y$ direction. A fixed circular conducting loop of radius $a$ and resistance $R$ is placed in the $x$ - $y$ plane, centered at the origin. The dipole is given a small angular displacement $\theta$ in the $x$ - $z$ plane and released. Assuming weak damping, what is the number of oscillations $N$ (where $N = \frac{\omega t}{2\pi}$ ) after which the amplitude of oscillation reduces to $1/e$ of its initial value?

Options

A.
$\frac{4 a^2 R}{\pi \mu_0^2 M^2} \sqrt{J \left(C + \frac{\mu_0 M I_0}{2\pi d}\right)}$
B.
$\frac{4 a^2 R}{\pi \mu_0^2 M^2} \sqrt{J C}$
C.
$\frac{2 a^2 R}{\pi \mu_0^2 M^2} \sqrt{J \left(C + \frac{\mu_0 M I_0}{2\pi d}\right)}$
D.
$\frac{16 a^2 R}{\pi \mu_0^2 M^2} \sqrt{J \left(C + \frac{\mu_0 M I_0}{2\pi d}\right)}$

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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

Torsional Oscillations|Dipole in Non-Uniform Field|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of mechanical and electromagnetic restoring torques alongside non-linear eddy current damping.

A common trap is neglecting the static magnetic field's contribution to the effective torsional constant, assuming only the fiber provides the restoring force.

Furthermore, students often struggle to calculate the magnetic flux of a dipole through a loop, either attempting a divergent surface integral or incorrectly applying the point-dipole field formula.

The elegant solution requires invoking the reciprocity theorem (mutual inductance) to find the flux, and subsequently applying Faraday's law to model the system as a damped harmonic oscillator.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Torsional Oscillations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free