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Question #42513

Difficulty: 8/10 | 🟢 Free

The Question

A large number of non-interacting electrons (mass $m$ , charge $-e$ ) are constrained to move in separate, identical circular non-conducting rings of radius $r$ lying in the $xy$ -plane. Initially, all electrons revolve counterclockwise, and their speeds follow the standard Maxwell-Boltzmann distribution of speeds for a 3D ideal gas at temperature $T$ . A uniform time-varying magnetic field $\vec{B}(t) = B(t) \hat{k}$ is applied, increasing slowly from $0$ to $B_0$ . Which of the following expressions represents the magnitude of the final average magnetic dipole moment per electron? (Here $k_B$ is the Boltzmann constant)

Options

A.
$\frac{er}{2}\sqrt{\frac{8k_BT}{\pi m}} + \frac{e^2 r^2 B_0}{4m}$
B.
$\frac{er}{2}\sqrt{\frac{3k_BT}{m}} + \frac{e^2 r^2 B_0}{4m}$
C.
$\frac{er}{2}\sqrt{\frac{8k_BT}{\pi m}} - \frac{e^2 r^2 B_0}{4m}$
D.
$\frac{er}{2}\sqrt{\frac{3k_BT}{m}}$

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

RMS vs Average vs Most Probable Speed|Magnetic Dipole Moment of Revolving Electron|Force on Current-Carrying Loop

Why This Question Is Tricky

Common Mistakes

This question tests two major conceptual traps in electromagnetism and thermodynamics.

First, students often indiscriminately use the RMS speed ( $v_{rms} = \sqrt{3k_BT/m}$ ) for any average property of a gas.

However, because the magnetic dipole moment $\mu = evr/2$ is linearly proportional to velocity, the correct statistical average requires the mean speed ( $\langle v \rangle = \sqrt{8k_BT/\pi m}$ ).

Second, it tests the application of Faraday's and Lenz's laws to a microscopic charge: a time-varying magnetic field induces a non-conservative electric field that does work on the electron, altering its orbital kinetic energy and consequently its magnetic moment (the classical basis of diamagnetism).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Kinetic Theory - Class 11 Part 2
Concept	RMS vs Average vs Most Probable Speed
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free