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Question #43417

Difficulty: 8/10 | 🟢 Free

The Question

A charged particle of mass $m$ and charge $+q$ moves in a cylindrically symmetric magnetic bottle. Near the $z$ -axis, the magnetic field is approximately given by $\vec{B} = -\frac{B_0 \alpha z}{2} x \hat{i} - \frac{B_0 \alpha z}{2} y \hat{j} + B_0\left(1 + \frac{\alpha}{2} z^2\right) \hat{k}$ , where $B_0$ and $\alpha$ are positive constants. At $z=0$ , the particle is injected near the origin with a velocity $\vec{v} = v_0 \sin\theta \hat{i} + v_0 \cos\theta \hat{k}$ (where $0 < \theta < \pi/2$ ). The particle spirals with its guiding center along the $z$ -axis. Assuming the radius of the helical path is small enough that the orbital magnetic moment of the particle remains an adiabatic invariant, what is the maximum $z$ -coordinate $z_{max}$ reached by the particle before it is reflected back?

Options

A.
$\sqrt{\frac{2}{\alpha}} \cot\theta$
B.
$\sqrt{\frac{2}{\alpha}} \cos\theta$
C.
$\sqrt{\frac{1}{\alpha}} \cot\theta$
D.
$\sqrt{\frac{2}{\alpha}} \tan\theta$

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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|Quantization of Angular Momentum

Why This Question Is Tricky

Common Mistakes

The core trap in this problem lies in the misapplication of the Lorentz force in a non-uniform magnetic field.

Students often assume that because the primary magnetic field is along the z-axis, the longitudinal velocity must remain constant (ignoring the radial components of the B-field).

This leads to the false conclusion that the particle never reflects.

In reality, the radial magnetic field components interact with the azimuthal velocity to produce a retarding longitudinal force, which decelerates the particle along the z-axis.

By invoking the adiabatic invariant (magnetic moment) alongside the conservation of kinetic energy (since magnetic forces do no work), the conversion of longitudinal kinetic energy into transverse kinetic energy is elegantly captured, revealing the 'magnetic mirroring' effect without requiring direct integration of the complex coupled differential equations.

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