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Question #43027

Difficulty: 8/10 | 🟢 Free

The Question

A solid sphere of radius $R$ is centered at the origin. It has a non-uniform volume charge density given by $\rho(r, \theta) = \rho_0 \left( \frac{r}{R} \right) \sin\theta$ , where $\rho_0$ is a positive constant, $r$ is the radial distance from the origin, and $\theta$ is the polar angle from the $z$ -axis. The sphere rotates with a constant angular velocity $\boldsymbol{\omega} = \omega \hat{k}$ about the $z$ -axis. Find the magnetic dipole moment $\boldsymbol{\mu}$ of the sphere and the net magnetic flux $\Phi_B$ through the entire closed surface of the sphere.

Options

A.
$\boldsymbol{\mu} = \frac{\pi^2 \rho_0 \omega R^5}{16} \hat{k}, \Phi_B = 0$
B.
$\boldsymbol{\mu} = \frac{\pi^2 \rho_0 \omega R^5}{20} \hat{k}, \Phi_B = 0$
C.
$\boldsymbol{\mu} = \frac{\pi^2 \rho_0 \omega R^5}{16} \hat{k}, \Phi_B = \mu_0 \rho_0 \omega R^4$
D.
$\boldsymbol{\mu} = \frac{\pi^2 \rho_0 \omega R^5}{12} \hat{k}, \Phi_B = 0$

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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 Dipole Moment Orientation|Gauss Law Applicability|Continuous Charge Distribution

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to derive the magnetic dipole moment from first principles for a non-uniform charge distribution, bypassing standard formulas.

A common trap is calculating the total charge $Q$ and incorrectly applying the uniform sphere formula $\mu = Q\omega R^2 / 5$ , which fails here because the radial and angular weighting of the current loops is altered by the $\rho(r, \theta)$ dependence.

Furthermore, the question tests the conceptual application of Gauss's Law for Magnetism; students often get bogged down attempting to integrate the complex, non-uniform magnetic field over the boundary, forgetting that $\nabla \cdot \mathbf{B} = 0$ universally dictates that the net magnetic flux through any closed surface is exactly zero.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Dipole Moment Orientation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free