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Question #43378

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The Question

A small bead of mass $m$ and charge $+q$ is threaded on a smooth, rigid, non-conducting circular ring of radius $a$ . The ring is fixed in the horizontal $xy$ -plane, centered at the origin. A time-varying, non-uniform magnetic field exists in the region, given by $\vec{B} = B_0 t \left( \frac{r}{a} \right) \hat{k}$ for $r \le a$ , where $r$ is the radial distance from the origin, $t$ is time, and $B_0$ is a positive constant. The bead is released from rest at $t = 0$ . Assuming gravity is negligible, find the magnitude of the radial normal force exerted by the ring on the bead at time $t$ .

Options

A.
$\frac{2 q^2 B_0^2 a t^2}{9 m}$
B.
$\frac{q^2 B_0^2 a t^2}{9 m}$
C.
$\frac{4 q^2 B_0^2 a t^2}{9 m}$
D.
$\frac{q^2 B_0^2 a t^2}{4 m}$

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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

Lorentz Force Vector Direction|Conservative vs Non-conservative|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Faraday's Law of Induction and the Lorentz force, specifically targeting the distinction between conservative and non-conservative fields in a dynamic setup.

A common trap is assuming a uniform magnetic field when calculating flux, leading to an incorrect induced electric field (Option D).

Furthermore, students often misapply Newton's Second Law in the radial direction by either ignoring the velocity-dependent magnetic Lorentz force entirely (treating the induced E-field as the only active force, Option B) or incorrectly adding the magnetic force and centripetal requirement (Option C).

The problem forces the realization that the non-conservative electric field does work to increase kinetic energy, while the magnetic field, though doing no work, dominates the radial dynamics and dictates the normal force.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Lorentz Force Vector Direction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free