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Question #43153

Difficulty: 8/10 | 🟢 Free

The Question

An electron of mass $m$ and charge $-e$ is initially moving in a circular orbit of radius $r$ with angular velocity $\omega_0$ in the $xy$ -plane. At $t=0$ , a uniform, time-varying magnetic field $\vec{B}(t) = B_0 \sin(\Omega t) \hat{k}$ is applied perpendicular to the orbital plane. The field increases from zero to $B_0$ over the time interval $0 \le t \le \frac{\pi}{2\Omega}$ . Assuming the orbital radius $r$ remains constant due to a strong central restoring force, what is the average power delivered to the electron by the induced electric field during this time interval?

Options

A.
$\frac{\Omega e r^2 B_0}{\pi} \omega_0$
B.
$\frac{\Omega e r^2 B_0}{\pi} \left( \omega_0 + \frac{e B_0}{4m} \right)$
C.
$\frac{\Omega e r^2 B_0}{\pi} \left( \omega_0 + \frac{e B_0}{2m} \right)$
D.
$\frac{\Omega e r^2 B_0}{2\pi} \left( \omega_0 + \frac{e B_0}{2m} \right)$

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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

Average Power over Cycle|Magnetic Moment of Orbiting Electron|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between the quasi-static potential energy of a magnetic dipole and the dynamic work done by an induced electric field (Faraday's Law).

A common trap is treating the electron's magnetic moment as a static vector $\vec{\mu}$ and calculating the work done simply as $\Delta U = -\vec{\mu} \cdot \vec{B}$ , which yields only the first term (leading to Option A).

However, the changing magnetic field induces an EMF that alters the electron's orbital angular momentum (Larmor precession).

Students must apply the Work-Energy Theorem to account for the quadratic dependence of kinetic energy on the changing angular velocity, revealing the additional Larmor precession energy term.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Average Power over Cycle
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free