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Question #43273

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The Question

A small magnetic dipole of magnetic moment $m$ and moment of inertia $I$ is pivoted at the origin and is free to rotate in the $xy$ -plane. A uniform external magnetic field $\vec{B} = B_0 \hat{j}$ is present in the region. A stationary circular conducting ring of radius $R$ , electrical resistance $r$ , and negligible self-inductance is placed parallel to the $xz$ -plane, with its center fixed at $(0, d, 0)$ . The dipole is given a small initial angular displacement $\theta_0 \ll 1$ from its equilibrium position (where $\vec{m}$ is aligned with $\vec{B}$ ) and released. Assuming the energy dissipated per cycle is very small (so the amplitude decreases negligibly during a single cycle), what is the approximate mechanical energy dissipated in the ring during the first complete oscillation cycle?

Options

A.
$\frac{\pi \mu_0^2 R^4 m^2}{4 r (R^2+d^2)^3} \sqrt{\frac{m B_0}{I}} \theta_0^2$
B.
$\frac{\pi \mu_0^2 R^4 m^2}{16 r (R^2+d^2)^3} \sqrt{\frac{m B_0}{I}} \theta_0^4$
C.
$\frac{\pi \mu_0^2 R^4 m^2}{4 r (R^2+d^2)^3} \sqrt{\frac{m B_0}{I}} \theta_0^4$
D.
$\frac{\pi \mu_0^2 R^4 m^2}{8 r (R^2+d^2)^3} \sqrt{\frac{m B_0}{I}} \theta_0^2$

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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

Magnetic Field on Axial vs Equatorial Line|Dipole in Uniform Field|Eddy Currents Dissipation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize magnetostatics, electromagnetic induction, and non-linear dynamics, while setting a lethal trap for those who reflexively apply standard linear damping formulas.

In standard damped SHM, the damping force is proportional to velocity ( $F_d \propto \dot{\theta}$ ), leading to an energy dissipation per cycle proportional to the square of the amplitude ( $\theta_0^2$ ).

However, because the magnetic flux through the ring depends on $\cos\theta$ , the induced EMF is proportional to $\sin\theta \cdot \dot{\theta}$ .

For small angles, this introduces an extra spatial factor $\theta$ , making the power dissipation proportional to $\theta^2 \dot{\theta}^2$ .

Consequently, the energy lost per cycle scales with the fourth power of the amplitude ( $\theta_0^4$ ).

Students who blindly assume $\Delta E \propto \theta_0^2$ or incorrectly linearize the flux as $\Phi \propto \theta$ will fall into the trap.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Magnetic Field on Axial vs Equatorial Line
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free