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Question #43149

Difficulty: 8/10 | 🟢 Free

The Question

A toroidal solenoid of rectangular cross-section has an inner radius $a$ , an outer radius $b$ , and consists of $N$ tightly wound turns carrying a steady current $I$ . A particle of mass $m$ and positive charge $q$ is projected from the inner edge ( $r=a$ ) radially outwards with an initial speed $v_0$ in the mid-plane of the toroid. Assume the axial height of the toroid is sufficiently large so the particle does not strike the top or bottom walls, and the magnetic field outside the toroid is zero. Which of the following expressions gives the minimum initial speed $v_0$ required for the particle to just reach the outer edge ( $r=b$ )?

Options

A.
$\frac{\mu_0 q N I}{2\pi m} \ln\left(\frac{b}{a}\right)$
B.
$\frac{\mu_0 q N I (b-a)}{\pi m (a+b)}$
C.
$\frac{\mu_0 q N I (b-a)}{2\pi m a}$
D.
$\frac{\mu_0 q N I}{2\pi m} \left(\frac{b-a}{b}\right)$

Check Your Answer

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

Transformer Flux Linkage|Moving Charge in Non-Uniform B|Toroidal Solenoid Boundary Conditions

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze the motion of a charged particle in a non-uniform magnetic field with a radial gradient ( $B \propto 1/r$ ).

The primary trap exploits the habit of assuming a uniform magnetic field (often evaluated at the mean radius or inner radius) to calculate the Lorentz force and subsequent trajectory.

By doing so, students incorrectly apply the formula for the radius of curvature of a circular path ( $R = mv/qB$ ) and fail to recognize that the varying magnetic field couples the radial and axial velocities logarithmically.

The correct approach requires integrating the equations of motion directly and applying the work-energy theorem (conservation of kinetic energy) to find the boundary condition where radial velocity vanishes.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Transformer Flux Linkage
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free