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Question #77224

Difficulty: 5/10 | 🟢 Free

The Question

A uniform time-varying magnetic field $\vec{B} = \beta t \hat{k}$ (where $\beta > 0$ is a constant) exists in a cylindrical region of radius $R$ centered at the origin. A smooth, rigid, insulating wire is bent into the shape of a parabola $y = h - \frac{x^2}{4h}$ and fixed in the $x-y$ plane, such that the entire wire lies within the magnetic field region ( $2h < R$ ). Gravity acts in the $-y$ direction with acceleration $g$ . A bead of mass $m$ and charge $+q$ is projected from the x-intercept $(2h, 0)$ along the wire towards the vertex $(0, h)$ . What is the minimum initial kinetic energy $K_0$ required for the bead to reach the vertex?

Options

A.
$mgh + \frac{4}{3} q\beta h^2$
B.
$mgh + \frac{2}{3} q\beta h^2$
C.
$mgh + q\beta h^2$
D.
$mgh$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Maximum Height Trap|Potential Gradient Direction|Faraday’s Law Vector Form

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 handle non-conservative induced electric fields and path-dependent work.

The primary trap is assuming that because magnetic forces do no work, the time-varying magnetic field has no effect on the energy required (leading to Option D).

A secondary trap is incorrectly assuming the induced electric field is conservative, which would lead students to calculate the work done along a simpler straight-line path instead of the actual parabolic path, yielding an incorrect coefficient for the work term (Option C).

By forcing the path integral $y dx - x dy$ along a specific curve, the question rigorously evaluates the application of Faraday's Law in a mechanical Work-Energy context, breaking the standard symmetry of projectile motion.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Maximum Height Trap
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free