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Question #42459

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $q > 0$ is projected from the origin $(0,0,0)$ with an initial velocity $\vec{v} = v_0 \hat{i}$ . A spatially varying magnetic field $\vec{B} = B_0 x \hat{k}$ exists in the region $x > 0$ , where $B_0$ is a positive constant. Neglecting gravity, what is the maximum $x$ -coordinate ( $x_{\text{max}}$ ) reached by the particle?

Options

A.
$\sqrt{\frac{2m v_0}{q B_0}}$
B.
$\sqrt{\frac{m v_0}{q B_0}}$
C.
$\sqrt{\frac{m v_0}{2q B_0}}$
D.
$2\sqrt{\frac{m v_0}{q B_0}}$

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

Impulse-Momentum Theorem|Trajectory of Charged Particle|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze motion in a spatially varying magnetic field, specifically targeting the trap of blindly applying the constant-field cyclotron radius formula $R = mv/qB$ .

If a student incorrectly assumes the maximum penetration depth is simply the local radius of curvature ( $x = mv_0/q(B_0 x)$ ), they will erroneously obtain $x = \sqrt{mv_0/qB_0}$ .

The correct approach requires recognizing that the impulse provided by the Lorentz force must be integrated over the spatial gradient, utilizing the relation $v_x dt = dx$ to find the change in momentum components, ultimately revealing a factor of $\sqrt{2}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Impulse-Momentum Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free