Before You Begin

Question #42377

Difficulty: 8/10 | 🟢 Free

The Question

A thin insulating disk of radius $R$ lies in the $xy$ -plane with its center at the origin. It has a non-uniform mass surface density $\sigma_m(r) = \sigma_0 \left(1 - \frac{r}{R}\right)$ and a non-uniform charge surface density $\sigma_q(r) = \sigma_1 \frac{r}{R}$ , where $r$ is the radial distance from the center. The disk is rotated with a constant angular velocity $\omega$ about a chord that lies in the $xy$ -plane at a perpendicular distance of $R/2$ from the center. Find the ratio of the magnitude of the magnetic dipole moment to the angular momentum of the disk, both measured about this axis of rotation.

Options

A.
$\frac{11 \sigma_1}{8 \sigma_0}$
B.
$\frac{\sigma_1}{\sigma_0}$
C.
$\frac{11 \sigma_1}{4 \sigma_0}$
D.
$\frac{2 \sigma_1}{\sigma_0}$

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

Parallel and Perpendicular Axes|Magnetic Moment of Orbiting Electron|Integration

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume that the gyromagnetic ratio $\mu/L = q/2m$ holds universally, failing to realize it requires a uniform charge-to-mass ratio.

This question traps students who blindly apply the standard formula or fail to express the differential magnetic moment as $d\mu = \frac{1}{2}\omega r_{\perp}^2 dq$ .

It rigorously tests the synthesis of the Perpendicular and Parallel Axis Theorems applied to both mass and charge distributions, forcing the realization that the 'effective' gyromagnetic ratio for non-uniform, non-proportional densities is a ratio of their respective displaced moments of inertia.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Parallel and Perpendicular Axes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free