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Question #42062

Difficulty: 8/10 | 🟢 Free

The Question

A rigid semi-circular wire of radius $R$ is kept in the $xy$ -plane. At time $t=t_0$ , its ends are located at $(R, 0)$ and $(-R, 0)$ with the wire forming the curve $y = \sqrt{R^2 - x^2}$ . The wire is moving with a constant velocity $\vec{v} = v_0 \hat{i}$ . A magnetic field $\vec{B} = \alpha t x \hat{k}$ exists in the region, where $\alpha$ is a positive constant. What is the magnitude of the motional EMF induced across the wire at $t = t_0$ ?

Options

A.
$\frac{\pi}{2} \alpha v_0 t_0 R^2$
B.
$0$
C.
$2 \alpha v_0 t_0 R^2$
D.
$\pi \alpha v_0 t_0 R^2$

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|Vector Decomposition Ambiguity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to calculate motional EMF in a non-uniform magnetic field with a spatial gradient.

A common trap is assuming the effective length formula ( $B l_{eff} v$ ) applies universally.

Students often incorrectly use the straight-line distance between the endpoints ( $2R$ ) and conclude the EMF is zero since the integral of $x$ along the x-axis is zero.

However, because the magnetic field has a spatial gradient, the motional electric field $\vec{v} \times \vec{B}$ is non-conservative ( $\nabla \times (\vec{v} \times \vec{B}) \neq 0$ ), making the induced EMF strictly path-dependent.

The student must abandon the algebraic shortcut and perform the rigorous line integral along the actual arbitrary shape of the wire.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Vector Decomposition Ambiguity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free