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Question #42337

Difficulty: 8/10 | 🟢 Free

The Question

A rigid, closed conducting loop of arbitrary shape has mass $m$ , enclosed area $A$ , and electrical resistance $R$ . It is constrained to translate without rotation on a smooth horizontal $xy$ -plane. A non-uniform vertical magnetic field $\vec{B} = \alpha x \hat{k}$ exists in the region, where $\alpha$ is a positive constant. At $t=0$ , the loop is at rest and is subjected to an instantaneous mechanical impulse $J$ in the positive $x$ -direction. Neglecting self-inductance, what is the total distance $\Delta x$ traveled by the loop before it comes to rest?

Options

A.
$\frac{J R}{\alpha^2 A^2}$
B.
$\frac{J R}{2 \alpha^2 A^2}$
C.
$\frac{2 J R}{\alpha^2 A^2}$
D.
$\frac{J R}{\pi \alpha^2 A^2}$

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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

Collision Impulses|Motional EMF for Arbitrary Shapes|Resistor Phase Invariance|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of impulsive mechanics and electromagnetism in non-uniform fields.

The primary trap is the 'arbitrary shape' in a spatially varying magnetic field; students often abandon the problem, not realizing that flux and magnetic force can be rigorously evaluated using the centroid definition and Green's Theorem ( $\oint x dy = A$ ).

Furthermore, it addresses the misconception of treating the impulse as a continuous driver rather than an initial condition generator ( $v_0 = J/m$ ) for the subsequent Lenz's Law velocity decay governed by $m(dv/dt) = -kv$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Collision Impulses
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free