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Question #43250

Difficulty: 8/10 | 🟢 Free

The Question

In a semi-classical model, an electron cloud of total charge $-e$ is treated as a continuous fluid flowing steadily in a fixed circular ring of radius $R$ . Due to an external perturbation, the speed of the fluid elements varies with angular position $\theta$ as $v(\theta) = v_0 \left(1 + \frac{3}{5} \cos \theta\right)$ . The unperturbed speed $v_0$ corresponds to a uniform circular Keplerian orbit of radius $R$ and period $T_0$ . Find the magnitude of the orbital magnetic moment of this electron cloud.

Options

A.
$\frac{4\pi e R^2}{5 T_0}$
B.
$\frac{\pi e R^2}{T_0}$
C.
$\frac{8\pi e R^2}{5 T_0}$
D.
$\frac{2\pi e R^2}{5 T_0}$

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

Magnetic Moment of Orbiting Electron|Kepler’s Third Law (Time Period)|Equation of Continuity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from discrete point-particle kinematics to continuum mechanics (hydrodynamic analogy) in electromagnetism.

The common trap is to misapply the point-particle magnetic moment formula $\mu = \frac{1}{2} e v_{avg} R$ or assume a uniform linear charge density $\lambda = e / (2\pi R)$ , which leads to Option B.

However, for a steady fluid flow with varying velocity, the Equation of Continuity dictates that the current $I = \lambda(\theta) v(\theta)$ must be strictly constant.

Consequently, the charge density $\lambda(\theta)$ must inversely follow the velocity profile.

By forcing the student to integrate the varying density to find the true constant current, the problem exposes the misconception of treating time-averaged continuum flux as a simple spatial average of velocity.

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