Before You Begin

Question #43646

Difficulty: 8/10 | 🟢 Free

The Question

An electron of mass $m$ and charge $-e$ is constrained to move in a fixed, non-conducting circular ring of radius $R$ in the $xy$ -plane. Initially, no magnetic field is present, and the electron is in a stationary state with an initial kinetic angular momentum $\vec{L}_0 = -n\hbar \hat{k}$ (where $n > 0$ is an integer and $\hat{k}$ is the unit vector along the $z$ -axis). A uniform, time-varying magnetic field $\vec{B}(t) = B(t) \hat{k}$ is then applied, increasing slowly from $B(0) = 0$ to a final constant value $B_0$ . Assuming the electron remains in the ring, which of the following expressions represents the exact work done by the induced electric field on the electron during this process? (Ignore self-inductance and electromagnetic radiation)

Options

A.
$-\frac{n e \hbar B_0}{2m} + \frac{e^2 R^2 B_0^2}{8m}$
B.
$\frac{n e \hbar B_0}{2m} + \frac{e^2 R^2 B_0^2}{8m}$
C.
$-\frac{n e \hbar B_0}{2m}$
D.
$0$

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

Energy Level Scaling|Magnetic Moment of Orbiting Electron|Faraday's Law of Induction

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between static and time-varying magnetic fields, specifically targeting the misconception that 'magnetic fields do no work'.

While the Lorentz force from a static magnetic field is always perpendicular to velocity and does zero work, a time-varying magnetic field generates a non-conservative induced electric field (Faraday's Law) which does perform work on the electron.

Students who blindly apply the static rule will incorrectly choose 0.

Furthermore, students who use a first-order approximation $W \approx -\vec{\mu} \cdot \vec{B}$ or assume constant angular velocity will miss the quadratic term, failing to account for the continuous change in the electron's magnetic moment as it accelerates.

This scenario bridges electromagnetic induction and semiclassical atomic physics, demonstrating how the Larmor diamagnetic term arises from the work-energy theorem.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Energy Level Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free