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Question #42980

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly conducting rigid wire is bent into the shape of a parabola $y = \frac{x^2}{2L}$ and is bounded between $x = -L$ and $x = L$ . The wire is placed in a time-independent, non-uniform magnetic field given by $\vec{B} = \beta x \hat{k}$ , where $\beta$ is a positive constant. At a specific instant $t = t_0$ , the wire is positioned such that its vertex is at the origin and its axis of symmetry is aligned with the positive y-axis. At this exact instant, the wire is undergoing a combined motion: its vertex translates with a velocity $\vec{v}_{v} = \gamma L t_0 \hat{i}$ , and the entire wire rotates about its vertex with an angular velocity $\vec{\omega} = \gamma t_0 \hat{k}$ , where $\gamma$ is a positive constant. Determine the magnitude of the instantaneous motional electromotive force (EMF) induced across the ends of the wire at $t = t_0$ .

Options

A.
$\frac{1}{5} \beta \gamma t_0 L^3$
B.
$\frac{2}{3} \beta \gamma t_0 L^3$
C.
$\frac{13}{15} \beta \gamma t_0 L^3$
D.
$0$

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

Motional EMF for Arbitrary Shapes|Non-Uniform Acceleration Constraint|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Faraday's Law and Lorentz force for motional EMF in a non-uniform field, specifically targeting the trap of using simplified macroscopic formulas like $\varepsilon = B_{avg} v_{avg} L_{eff}$ .

Students often incorrectly assume that for a translating and rotating rigid body, they can decouple the motions or use the velocity of the center of mass.

By making the magnetic field position-dependent ( $\vec{B} = \beta x \hat{k}$ ) and the velocity field spatially varying due to rotation ( $\vec{v} = \vec{v}_{v} + \vec{\omega} \times \vec{r}$ ), the problem forces the use of the fundamental line integral $\int (\vec{v} \times \vec{B}) \cdot d\vec{l}$ .

Students who ignore the rotational velocity component will arrive at the plausible distractor $\frac{2}{3} \beta \gamma t_0 L^3$ , while those who rely on symmetry arguments might incorrectly guess zero.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Motional EMF for Arbitrary Shapes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free