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Question #43047

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The Question

An electron (rest mass $m_0$ , charge $-e$ ) is injected through a microscopic gap into the vacuum core of a toroidal solenoid. The toroid has $N$ turns carrying a steady current $I$ . The electron enters at a radial distance $R_1$ from the toroid's central axis of symmetry, with an initial velocity $\vec{v} = v_0 \hat{z}$ (parallel to the central axis), where $v_0$ is a relativistic speed. Inside the toroid, the electron experiences the azimuthal magnetic field $\vec{B}$ . Assuming the toroid's outer radius is sufficiently large so the electron does not collide with the outer wall, which of the following expressions gives the exact maximum radial distance $\rho_{max}$ reached by the electron? (Let $\gamma = \frac{1}{\sqrt{1 - v_0^2/c^2}}$ and assume the magnetic field inside the core follows Ampere's Law exactly).

Options

A.
$R_1 \exp\left( \frac{4 \pi \gamma m_0 v_0}{\mu_0 N I e} \right)$
B.
$R_1 \left( 1 + \frac{4 \pi \gamma m_0 v_0}{\mu_0 N I e} \right)$
C.
$R_1 \exp\left( \frac{4 \pi m_0 v_0}{\mu_0 N I e} \right)$
D.
$R_1 \exp\left( \frac{2 \pi \gamma m_0 v_0}{\mu_0 N I e} \right)$

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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

Toroidal Solenoid Boundary Conditions|Magnetic Dipole Moment of Revolving Electron|Dot Product Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of relativistic dynamics and non-uniform magnetic fields, specifically targeting the common misconception that a toroid's internal field can be approximated as uniform for particle trajectories.

Students often incorrectly assume a constant magnetic field $B = \mu_0 N I / (2\pi R_1)$ , leading them to calculate a simple circular trajectory and select the first-order Taylor expansion trap (Option B).

Furthermore, the problem requires recognizing that while the magnetic force alters the velocity vector, it does no work; thus, the relativistic Lorentz factor $\gamma$ remains strictly constant.

Students who forget the relativistic mass variation will choose Option C, while those who incorrectly set the turning point condition to $\dot{z} = 0$ instead of $\dot{z} = -v_0$ will fall for Option D.

The correct approach demands integrating the coupled differential equations in cylindrical coordinates.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Toroidal Solenoid Boundary Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free