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Question #42967

Difficulty: 8/10 | 🟢 Free

The Question

A perfectly rigid, macroscopic insulating ring of mass $M$ and radius $R$ is suspended by a torsionless thread in a region of zero initial magnetic field. The ring contains $N$ identical, non-interacting atoms. Each atom is modeled as a fixed nucleus of charge $+e$ and a single electron of mass $m_e$ and charge $-e$ in a spherically symmetric orbit with mean square radius $\langle r^2 \rangle = a^2$ (where $a \ll R$ ). A uniform magnetic field $\vec{B} = B_0 \hat{k}$ is slowly turned on. What is the net angular momentum $\Delta \vec{L}_{\text{total}}$ acquired by the entire ring system (lattice + electrons) due to the induced electric field?

Options

A.
$0$
B.
$-N \frac{e a^2}{3} B_0 \hat{k}$
C.
$+N \frac{e a^2}{3} B_0 \hat{k}$
D.
$+N \frac{e R^2}{2} B_0 \hat{k}$

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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

Diamagnetic Property of Materials|Gyromagnetic Ratio Invariance|Integration

Why This Question Is Tricky

Common Mistakes

This question tests a deep misconception regarding the conservation of angular momentum and induced electric fields in diamagnetic materials.

Students often incorrectly assume that because a macroscopic object is electrically neutral, the induced electric field from a changing magnetic field exerts zero net torque on it, leading them to choose 0 (Option A).

However, they fail to account for the microscopic structure of the atoms.

The finite size of the electron orbits creates a non-zero quadrupole moment.

The induced electric field varies over the size of the atom, exerting a net external torque that exactly accounts for the internal realignment of the orbital magnetic moments (Larmor precession).

By integrating this torque, one finds that the system acquires a net angular momentum from the external field, which perfectly explains why the macroscopic lattice of a diamagnet does not rotate to conserve angular momentum.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Diamagnetic Property of Materials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free