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Question #43261

Difficulty: 8/10 | 🟢 Free

The Question

A rigid, non-conducting rod of length $L$ is pivoted at one end ( $x=0$ ) and is free to rotate in a horizontal plane. The rod has a non-uniform mass density $\lambda_m(x) = \lambda_0 x$ and a non-uniform charge density $\lambda_q(x) = \alpha x$ , where $x$ is the distance from the pivot. The total mass of the rod is $M$ and its total charge is $Q$ . Initially, the rod is at rest. A uniform magnetic field, perpendicular to the plane of rotation, is switched on and slowly increased from $0$ to a steady value $B$ . Assuming no friction, no radiation losses, and that all quantities are in SI units, find the final kinetic energy $K$ acquired by the rod and the motional electromotive force $\mathcal{E}$ developed across its ends in the steady state.

Options

A.
$K = \frac{Q^2 B^2 L^2}{16 M}, \quad \mathcal{E} = \frac{Q B^2 L^2}{4 M}$
B.
$K = \frac{Q^2 B^2 L^2}{24 M}, \quad \mathcal{E} = \frac{Q B^2 L^2}{4 M}$
C.
$K = \frac{Q^2 B^2 L^2}{16 M}, \quad \mathcal{E} = \frac{Q B^2 L^2}{3 M}$
D.
$K = \frac{Q^2 B^2 L^2}{24 M}, \quad \mathcal{E} = \frac{Q B^2 L^2}{3 M}$

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

Gyromagnetic Ratio Invariance|Motional EMF in Rotating Rods|Work-Energy Theorem for Rotation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple electromagnetic induction with rigid body dynamics in a non-uniform system.

The primary trap lies in the miscalculation of the moment of inertia and the motional EMF.

Students often reflexively use $I = \frac{1}{3}ML^2$ for a rod, ignoring the linear density gradient, which leads to an incorrect kinetic energy (Option B).

Furthermore, a common misconception in calculating motional EMF is using the velocity of the center of mass ( $v_{cm} = \omega x_{cm}$ ) instead of integrating $\int v B dx$ .

For this non-uniform rod, $x_{cm} = \frac{2}{3}L$ , which yields an incorrect EMF of $\frac{Q B^2 L^2}{3 M}$ if the integration principle is bypassed (Option C).

Option D catches students who make both conceptual errors.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Gyromagnetic Ratio Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free