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Question #90935

Difficulty: 5/10 | 🟢 Free

The Question

A rigid, frictionless, insulating circular ring of radius $R$ is fixed in the $xy$ -plane, centered at the origin. A small excited ion of mass $m$ and charge $+q$ is initially at rest on the ring. A uniform magnetic field $\vec{B} = \beta t \hat{k}$ is applied throughout the region for $t \ge 0$ , where $\beta$ is a positive constant. At $t = \tau$ , the ion emits a photon of energy $E$ tangentially along the ring in the direction opposite to its instantaneous velocity vector. Assuming the ion remains on the ring, what is the net work done by the induced electric field on the ion from $t = 0$ to $t = 2\tau$ ? (Here $c$ is the speed of light in vacuum).

Options

A.
$\frac{q^2 \beta^2 R^2 \tau^2}{2m} + \frac{q \beta R \tau E}{2mc}$
B.
$\frac{q^2 \beta^2 R^2 \tau^2}{2m} + \frac{q \beta R \tau E}{mc}$
C.
$\frac{q^2 \beta^2 R^2 \tau^2}{m} + \frac{q \beta R \tau E}{mc}$
D.
$\frac{q^2 \beta^2 R^2 \tau^2}{2m}$

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

Photon Neutrality|Lorentz Force Vector Direction|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the synthesis of Faraday's Law, Lorentz force, and photon momentum.

A common trap is assuming that because photons are neutral, their emission does not affect the work done by the macroscopic electromagnetic field (leading to Option D).

Students must recognize that the photon recoil increases the ion's velocity, which in turn increases the rate at which the induced electric field does work on the ion ( $P = \vec{F} \cdot \vec{v}$ ).

Furthermore, students often misapply the Work-Energy Theorem by equating the work done by the field to the total change in kinetic energy, forgetting that the photon emission itself converts internal energy into kinetic energy.

Direct integration of power correctly accounts for the cross-term between the field-induced velocity and the recoil velocity.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Photon Neutrality
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free