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Question #90440

Difficulty: 5/10 | 🟢 Free

The Question

An infinitely long straight wire lies along the $x$ -axis of a Cartesian coordinate system and carries a constant current $I$ in the $+x$ -direction. A thin, straight conducting rod of length $L$ is pivoted at a point $P$ with coordinates $(0, a, 0)$ , where $a > 0$ . The rod rotates in the $xy$ -plane with a constant angular velocity $\omega$ about the pivot $P$ . Find the magnitude of the induced electromotive force (EMF) across the ends of the rod at the exact instant when the rod is aligned parallel to the $y$ -axis (i.e., occupying the region from $y = a$ to $y = a + L$ ). Let $\mu_0$ be the permeability of free space.

Options

A.
$\frac{\mu_0 I \omega}{2\pi} \left[ L - a \ln\left(1 + \frac{L}{a}\right) \right]$
B.
$\frac{\mu_0 I \omega L}{2\pi}$
C.
$\frac{\mu_0 I \omega L^2}{4\pi a}$
D.
$\frac{\mu_0 I \omega}{2\pi} \left[ L + a \ln\left(1 + \frac{L}{a}\right) \right]$

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

Motional EMF in Rotating Rods|Superposition Principle|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to calculate motional EMF in a non-uniform magnetic field while correctly identifying the velocity profile of a rotating body.

A common trap is to misapply the standard formula $\epsilon = \frac{1}{2}B\omega L^2$ , which is only valid for a uniform magnetic field, leading students to incorrectly evaluate $B$ at the pivot or the center of the rod.

Another critical trap is failing to shift the reference frame for velocity; students often incorrectly use $v = \omega y$ instead of $v = \omega(y-a)$ , which erroneously simplifies the integral and leads to the distractor $\frac{\mu_0 I \omega L}{2\pi}$ .

Mastery requires rigorous application of $d\epsilon = (\vec{v} \times \vec{B}) \cdot d\vec{l}$ with position-dependent functions for both velocity and magnetic field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Motional EMF in Rotating Rods
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free