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Question #43043

Difficulty: 8/10 | 🟢 Free

The Question

A point charge $Q$ is fixed at the origin of a cylindrical coordinate system $(r, \theta, z)$ . A magnetic field exists in the region, given by $\mathbf{B} = B_0 \hat{k}$ for $r < a$ , and $\mathbf{B} = \frac{B_0 a}{r} \hat{k}$ for $r \ge a$ , where $B_0$ and $a$ are positive constants. A particle of mass $m$ and charge $q$ ( $q > 0$ ) is released from rest at the position $(r=a, \theta=0, z=0)$ . Assuming the particle's motion is naturally confined to the $z=0$ plane, what is the minimum value of the charge $Q$ such that the particle can escape to infinity?

Options

A.
$\frac{2\pi\epsilon_0 q B_0^2 a^3}{m}$
B.
$\frac{\pi\epsilon_0 q B_0^2 a^3}{m}$
C.
$\frac{4\pi\epsilon_0 q B_0^2 a^3}{m}$
D.
$\frac{\pi\epsilon_0 q B_0^2 a^3}{2m}$

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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|Coulombic Central Force|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between conservative electric fields and non-conservative effective potentials induced by spatially varying magnetic fields.

Students often fall into the trap of blindly applying the Work-Energy Theorem, assuming that because the magnetic force does zero work, all initial electric potential energy can be converted into radial kinetic energy.

They fail to recognize that the spatial gradient of the magnetic field exerts an azimuthal force, changing the particle's mechanical angular momentum.

By applying Busch's Theorem (conservation of generalized angular momentum), one finds that the particle is forced to acquire a residual azimuthal velocity at infinity.

This 'locks up' a portion of the initial energy as transverse kinetic energy, creating an effective potential barrier that the central Coulombic force must overcome for the particle to escape.

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