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Question #42726

Difficulty: 8/10 | 🟢 Free

The Question

A thin insulating disk of uniform mass density, total mass $M$ , and radius $R$ lies in the $xy$ -plane. It is rotating with a constant angular velocity $\vec{\omega} = \omega \hat{k}$ about its central axis (the $z$ -axis). The disk has a non-uniform surface charge density given by $\sigma(r) = \sigma_0 \frac{r}{R}$ , where $r$ is the radial distance from the center and $\sigma_0$ is a positive constant. A uniform magnetic field $\vec{B} = B_0 \hat{i}$ is applied across the region. Assume the setup is in a gravity-free space and the disk is pivoted at its center. The magnetic torque causes the rotation axis of the disk to undergo gyroscopic precession. Assuming the precession angular velocity $\Omega_p$ is much smaller than $\omega$ , what is the magnitude of the precession angular velocity $\Omega_p$ ?

Options

A.
$\frac{2 \pi \sigma_0 B_0 R^2}{5 M}$
B.
$\frac{\pi \sigma_0 B_0 R^2}{3 M}$
C.
$\frac{\pi \sigma_0 B_0 R^2}{5 M}$
D.
$\frac{\pi \sigma_0 B_0 R^2}{2 M}$

Check Your Answer

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

Current Density Vector|Gyroscopic Stability|Lorentz Force Vector Direction|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the gyromagnetic ratio shortcut and the application of gyroscopic precession.

A common trap for students is to blindly apply the gyromagnetic ratio $\vec{\mu} = \frac{Q}{2M}\vec{L}$ , which is only valid when the charge-to-mass ratio is spatially uniform.

In this setup, the mass density is uniform, but the charge density varies linearly with the radius.

If a student calculates the total charge $Q = \frac{2}{3}\pi\sigma_0 R^2$ and uses the ratio, they will incorrectly arrive at a precession rate of $\frac{\pi \sigma_0 B_0 R^2}{3 M}$ .

The correct physical approach requires integrating the Lorentz force density (or the elemental magnetic moments) explicitly to find the true magnetic torque, and then equating it to the gyroscopic rate of change of angular momentum $\vec{\Omega}_p \times \vec{L}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Current Electricity - Class 12
Concept	Current Density Vector
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free