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Question #43383

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and positive charge $q$ is projected radially inwards from a distance $R$ towards a fixed positive charge $Q$ located at the origin. The initial speed of the particle is $v_0 = \sqrt{\frac{3qQ}{2\pi\epsilon_0 m R}}$ . A uniform magnetic field of magnitude $B$ exists in the entire region, directed perpendicular to the plane of motion. Due to the combined effect of the electric and magnetic fields, the distance of closest approach of the particle to the origin is observed to be $R/2$ . Find the magnitude of the magnetic field $B$ . (Assume gravity is negligible)

Options

A.
$\frac{4}{3R} \sqrt{\frac{m Q}{\pi \epsilon_0 q R}}$
B.
$\frac{4}{R} \sqrt{\frac{m Q}{\pi \epsilon_0 q R}}$
C.
$\frac{2\sqrt{2}}{3R} \sqrt{\frac{m Q}{\pi \epsilon_0 q R}}$
D.
$\frac{2}{3R} \sqrt{\frac{m Q}{\pi \epsilon_0 q R}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Trajectory of Charged Particle|Distance of Closest Approach|Impact Parameter Geometry

Why This Question Is Tricky

Common Mistakes

This problem tests the breakdown of the conservation of angular momentum in central force problems when a crossed magnetic field is introduced.

Students often incorrectly assume that because the target charge $Q$ exerts a central force, the angular momentum of the incoming particle remains zero (since it is projected radially).

However, the velocity-dependent Lorentz force has a non-zero azimuthal component that generates an external torque, coupling the radial distance to the angular momentum ( $dL = -qBr dr$ ).

Failing to account for this induced rotational kinetic energy at the distance of closest approach leads to incorrect energy balance equations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Trajectory of Charged Particle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free