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Question #42065

Difficulty: 8/10 | 🟢 Free

The Question

A rigid conducting rod $OP$ of length $L$ rotates with a constant angular velocity $\vec{\omega} = \omega \hat{k}$ about the origin $O$ in the $xy$ -plane. The region contains a static, non-uniform magnetic field given by: $\vec{B} = B_0 \left[ \sin\left(\frac{\pi x}{L}\right) \cosh\left(\frac{\pi y}{L}\right) \hat{i} - \cos\left(\frac{\pi x}{L}\right) \sinh\left(\frac{\pi y}{L}\right) \hat{j} + \left(\frac{x^2 + y^2}{L^2}\right) \hat{k} \right]$ An observer $O'$ moves with a constant velocity $\vec{v}_0 = \omega L \hat{i} + \omega L \hat{j}$ relative to the laboratory frame. What is the magnitude of the motional electromotive force (EMF) across the rod $OP$ as measured by the observer $O'$ ?

Options

A.
$\frac{1}{4} B_0 \omega L^2$
B.
$\frac{1}{3} B_0 \omega L^2$
C.
$\frac{1}{2} B_0 \omega L^2$
D.
$\frac{5}{4} B_0 \omega L^2$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Relative Velocity in 2D|Motional EMF in Rotating Rods|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests two lethal traps in electromagnetic induction.

First, it presents a highly complex, non-uniform magnetic field to bait students into tedious component-wise cross products.

The logical bypass requires recognizing that the scalar triple product $(\vec{v} \times \vec{B}) \cdot d\vec{l} = (d\vec{l} \times \vec{v}) \cdot \vec{B}$ strictly isolates the $z$ -component of the field, rendering the intimidating in-plane components irrelevant.

Second, it introduces a moving observer to trigger a false relative velocity frame transformation.

Students often incorrectly add a translational EMF term, failing to realize that the total Lorentz force per unit charge (and thus the measured EMF) is invariant under Galilean transformations due to the induced electric field $\vec{E}' = \vec{v}_0 \times \vec{B}$ in the moving frame.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Relative Velocity in 2D
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free