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Question #43249

Difficulty: 8/10 | 🟢 Free

The Question

A solid sphere of radius $R$ has a uniform bare volume charge density $\rho_0$ . The sphere is composed of a theoretical non-uniform dielectric material whose relative permittivity varies with the radial distance $r$ from the center as $\epsilon_r(r) = \alpha r$ , where $\alpha$ is a positive constant. The sphere is rotated with a constant angular velocity $\omega$ about one of its diameters. Assuming that the bound polarization charges rotate along with the sphere, what is the total magnetic dipole moment of the rotating sphere?

Options

A.
$\frac{2\pi \rho_0 \omega R^4}{9} \left( 2R - \frac{1}{\alpha} \right)$
B.
$\frac{2\pi \rho_0 \omega R^4}{9 \alpha}$
C.
$\frac{4\pi \rho_0 \omega R^5}{15}$
D.
$\frac{\pi \rho_0 \omega R^4}{3 \alpha}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Magnetic Moment of Orbiting Electron|Conservative Field Path Independence|Non-Uniform Dielectric Filling

Why This Question Is Tricky

Common Mistakes

This question tests the profound misconception that the effective charge density in a non-uniform dielectric is simply $\rho_{bare} / \epsilon_r$ .

Students often blindly apply this without realizing that $\rho_{eff} = \nabla \cdot (\epsilon_0 \mathbf{E})$ , which couples to the spatial gradient of the permittivity.

Furthermore, candidates frequently forget that a spatially varying polarization field creates a surface bound charge at the boundary of the dielectric ( $\sigma_b = \mathbf{P} \cdot \hat{n}$ ), which also rotates and contributes to the total magnetic moment.

By forcing an integral formulation over both the modified volume charge and the surface bound charge, the problem heavily penalizes superficial application of standard formulas and tests deep understanding of Gauss's law in dielectrics.

Quick Facts About This Question

Question Information

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