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Question #43437

Difficulty: 8/10 | 🟢 Free

The Question

A non-conducting solid sphere of mass $M$ and total charge $Q$ initially has a uniform mass and charge distribution. It is rotating with an angular velocity $\omega_0$ about a diametric axis and has an initial radius $R_0$ . Due to internal forces, the sphere expands symmetrically to a new radius $R_f = 2R_0$ . During this expansion, the material becomes non-uniform such that the new density is inversely proportional to the distance from the center, $\rho(r) = k/r$ (for $0 < r \le R_f$ ), but the local charge-to-mass ratio remains constant everywhere. Assuming no external torques act on the system, what is the final angular velocity $\omega_f$ and the final magnetic dipole moment $\mu_f$ of the sphere?

Options

A.
$\omega_f = \frac{3}{10}\omega_0, \quad \mu_f = \frac{1}{5} Q R_0^2 \omega_0$
B.
$\omega_f = \frac{3}{10}\omega_0, \quad \mu_f = \frac{6}{25} Q R_0^2 \omega_0$
C.
$\omega_f = \frac{1}{4}\omega_0, \quad \mu_f = \frac{1}{5} Q R_0^2 \omega_0$
D.
$\omega_f = \frac{1}{4}\omega_0, \quad \mu_f = \frac{1}{4} Q R_0^2 \omega_0$

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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

Gyromagnetic Ratio Invariance|Magnetic Moment of Orbiting Electron|Angular Velocity vs Linear Velocity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize the invariance of the gyromagnetic ratio ( $\gamma = Q/2M$ ) for a system where the local charge-to-mass ratio is constant, even as the mass distribution changes dynamically.

The primary trap lies in calculating the final magnetic moment by attempting to integrate the new non-uniform charge distribution or by incorrectly applying the uniform sphere formula ( $\mu = \frac{1}{5}QR^2\omega$ ) to the final state, which leads to Option B.

By realizing that angular momentum $L$ is conserved and $\mu/L$ is a constant, the final magnetic moment can be found instantly from the initial state, bypassing complex integration.

The calculation of $\omega_f$ serves as a necessary mechanical hurdle to verify they can handle variable mass distributions, contrasting with the elegant electromagnetism shortcut.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Gyromagnetic Ratio Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free