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Question #43017

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The Question

An infinite current slab of thickness $2d$ (spanning $-d \le x \le d$ ) carries a uniform volume current density $\vec{J} = J_0 \hat{k}$ . A conducting wheel of radius $R$ ( $R < d$ ) is placed in the $xz$ -plane, centered at the origin. The wheel consists of $N$ ( $N \ge 3$ ) thin radial conducting spokes, each of resistance $r_0$ , connected at the center to a perfectly conducting axle and at the edge to a perfectly conducting rim. The wheel is forced to rotate about the $y$ -axis with a constant angular velocity $\vec{\omega} = \omega \hat{j}$ . Find the external mechanical power $P_{ext}$ required to maintain this rotation. (Ignore self-inductance and friction).

Options

A.
$\frac{N \mu_0^2 J_0^2 \omega^2 R^6}{18 r_0}$
B.
$\frac{N \mu_0^2 J_0^2 \omega^2 R^6}{8 r_0}$
C.
$\frac{N \mu_0^2 J_0^2 \omega^2 R^6}{9 r_0}$
D.
$\frac{N \mu_0^2 J_0^2 \omega^2 R^6}{32 r_0}$

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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

Ampere’s Circuital Law Symmetry|Work-Energy Theorem for Rotation|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the coupling of Ampere's Circuital Law with motional EMF in a spatially varying magnetic field, and the application of the Work-Energy Theorem to equate mechanical driving power to Lorentz drag (Joule heating).

The primary trap lies in assuming a uniform magnetic field or misapplying the standard $\frac{1}{2}B\omega R^2$ formula.

Students must recognize that $B$ varies linearly with $r$ along the spoke ( $B \propto r \cos\theta$ ), requiring an $\int r^2 dr$ integration which yields a factor of $1/3$ .

A secondary trap is incorrectly summing the power across the spokes by ignoring the $\cos^2\theta$ geometric averaging over the wheel, which contributes an additional factor of $1/2$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Ampere’s Circuital Law Symmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free