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Question #43040

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and charge $+q$ is released from rest at the origin of a Cartesian coordinate system. The region $x > 0$ contains a uniform electric field $\vec{E} = E_0 \hat{i}$ and a non-uniform magnetic field $\vec{B} = B_0 x \hat{k}$ , where $E_0$ and $B_0$ are positive constants. Which of the following represents the maximum x-coordinate ( $x_{max}$ ) reached by the particle?

Options

A.
$\left( \frac{8 m E_0}{q B_0^2} \right)^{1/3}$
B.
$\left( \frac{4 m E_0}{q B_0^2} \right)^{1/3}$
C.
$\left( \frac{2 m E_0}{q B_0^2} \right)^{1/3}$
D.
$\left( \frac{m E_0}{q B_0^2} \right)^{1/3}$

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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

Moving Charge in Non-Uniform B|Potential Gradient Direction|Vector Integration

Why This Question Is Tricky

Common Mistakes

The trap tests the student's ability to handle a spatially varying magnetic field and its coupling with the Work-Energy Theorem.

Students often incorrectly assume that because the magnetic field does no work, it does not affect the energy balance at all, leading them to write $q E_0 x = \frac{1}{2} m v_x^2$ and failing to account for the kinetic energy diverted into the y-direction.

Furthermore, those who do realize $v_y$ is non-zero often misapply the Lorentz force integration, forgetting to use the chain rule ( $v_x dt = dx$ ) to find the exact dependence of $v_y$ on $x$ .

This question rigorously demands the synthesis of non-uniform path integration with the scalar energy theorem.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Moving Charge in Non-Uniform B
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free