Before You Begin

Question #43301

Difficulty: 8/10 | 🟢 Free

The Question

A flexible conducting wire is anchored at two fixed points $(0, a)$ and $(0, -a)$ in the $xy$ -plane, where $a$ is a positive constant in meters. The wire is pulled in the $+x$ direction such that its shape at any time $t \ge 0$ (in seconds) is a parabola given by the equation $x(y, t) = \frac{v_0 t}{a^2} (a^2 - y^2)$ , where $v_0$ is a positive constant in $\text{m/s}$ . A non-uniform, time-independent magnetic field $\vec{B} = B_0 x^2 \hat{k}$ exists in the region, where $B_0$ is a positive constant in $\text{T/m}^2$ . The fixed points are connected by a straight stationary wire along the $y$ -axis to form a closed loop. What is the magnitude of the induced EMF in the loop at time $t$ ?

Options

A.
$\frac{32}{35} B_0 v_0^3 t^2 a \text{ V}$
B.
$\frac{16}{15} B_0 v_0^3 t^2 a \text{ V}$
C.
$\frac{16}{25} B_0 v_0^3 t^2 a \text{ V}$
D.
$\frac{32}{105} B_0 v_0^3 t^2 a \text{ V}$

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

Lenz’s Law Energy Conservation|Motional EMF for Arbitrary Shapes

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to calculate the induced EMF in a non-rigid, deforming loop within a non-uniform magnetic field.

The primary trap lies in misapplying the motional EMF formula or using an averaged magnetic field.

Students often incorrectly assume the entire wire translates with the peak velocity $v_0$ (leading to option B), or they evaluate the magnetic field at the centroid of the area and multiply by the total area (leading to option C).

By forcing the integration of a non-linear field ( $B \propto x^2$ ) over a time-dependent parabolic boundary, the problem demands rigorous application of Faraday's Law ( $\Phi = \int \vec{B} \cdot d\vec{A}$ ) or the exact convective line integral $\oint (\vec{v} \times \vec{B}) \cdot d\vec{l}$ , penalizing any geometric shortcuts.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Induction - Class 12
Concept	Lenz’s Law Energy Conservation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free