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Question #43439

Difficulty: 8/10 | 🟢 Free

The Question

A solid sphere of mass $M$ and radius $R$ has a total charge $Q$ uniformly distributed over its surface. The sphere is spinning about its diameter with a constant, high angular speed $\omega_s$ . It is placed in a uniform magnetic field of magnitude $B$ directed along the positive z-axis. The sphere is pivoted at its center of mass such that it can rotate freely in 3D space. If the spin axis is initially tilted by a very small angle $\theta_0$ with respect to the z-axis, what is the time period of the subsequent motion of the spin axis? (Assume all quantities are in standard SI units and ignore any radiation losses).

Options

A.
$\frac{12\pi M}{5QB}$
B.
$\frac{4\pi M}{QB}$
C.
$2\pi \sqrt{\frac{6M}{5QB\omega_s}}$
D.
$2\pi \sqrt{\frac{2M}{QB\omega_s}}$

Check Your Answer

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

Gyromagnetic Ratio Invariance|Torque and Potential Energy in B-Field|Equilibrium Stability

Why This Question Is Tricky

Common Mistakes

This question tests the distinction between static equilibrium stability and dynamic stability (Larmor precession) for a rotating magnetic dipole.

A common lethal mistake is treating the system like a physical pendulum, where students incorrectly apply $\tau = I\alpha$ to the tilt angle, yielding a false simple harmonic oscillation period (Option C).

They fail to recognize that because the magnetic moment is coupled to a large spin angular momentum, the torque $\vec{\mu} \times \vec{B}$ induces a change in the direction of $\vec{L}$ (precession) rather than a change in the tilt angle.

Additionally, it tests the invariance of the gyromagnetic ratio, trapping students who blindly use $Q/2M$ without accounting for the differing spatial distributions of mass (solid) and charge (surface).

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