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Question #42666

Difficulty: 8/10 | 🟢 Free

The Question

A smooth, rigid, insulating circular ring of radius $a$ is fixed in the $xy$ -plane, with its equation given by $(x-a)^2 + y^2 = a^2$ . A point charge $+Q$ is fixed at the point $(-a, 0)$ . A uniform but time-varying magnetic field $\vec{B} = \alpha t \hat{k}$ exists in a cylindrical region $(x-a)^2 + y^2 \le R^2$ (where $R > 2a$ ), where $\alpha$ is a positive constant and $t$ is time. A small bead of mass $m$ and charge $+q$ is released from rest at the origin $(0,0)$ . Due to the induced electric field, the bead begins to slide along the upper semicircle ( $y \ge 0$ ) of the ring. What is the kinetic energy of the bead when it reaches the point $(2a, 0)$ for the first time? (Ignore gravity)

Options

A.
$\frac{qQ}{6\pi\epsilon_0 a} + \frac{q \alpha \pi a^2}{2}$
B.
$\frac{qQ}{6\pi\epsilon_0 a}$
C.
$\frac{qQ}{6\pi\epsilon_0 a} + q \alpha \pi a^2$
D.
$\frac{qQ}{4\pi\epsilon_0 a} + \frac{q \alpha \pi a^2}{2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Equipotential Surface Geometry|Trajectory of Charged Particle|Induced Electric Fields|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between conservative and non-conservative electric fields.

A common trap is to assume that the total work done on a charged particle is simply $q\Delta V$ , relying solely on the static potential difference and ignoring the path-dependent work done by the induced electric field.

Because the induced electric field forms closed loops ( $\oint \vec{E} \cdot d\vec{l} \neq 0$ ), it cannot be represented by a scalar potential.

Success requires applying the generalized Work-Energy Theorem, calculating the electrostatic work via potential difference, and elegantly determining the induced work using Faraday's Law and Stokes' theorem over a cleverly chosen closed loop where the return path contributes zero work.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Equipotential Surface Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free