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Question #42559

Difficulty: 8/10 | 🟢 Free

The Question

A classical ideal gas consists of $N$ non-interacting atoms per unit volume. Each atom has a permanent magnetic dipole moment $\mu$ arising from a single orbiting electron. The gas is in thermal equilibrium at a temperature $T$ , and the average translational kinetic energy of a single atom is $K$ . A weak uniform magnetic field $B$ is applied in the $z$ -direction such that the magnetic interaction energy is much less than the thermal energy ( $\mu B \ll K$ ). Using classical Boltzmann statistics, which of the following expressions correctly represents the net macroscopic magnetization $M$ of the gas in the $z$ -direction?

Options

A.
$\frac{N \mu^2 B}{2 K}$
B.
$\frac{N \mu^2 B}{3 K}$
C.
$\frac{2 N \mu^2 B}{3 K}$
D.
$\frac{N \mu^2 B}{K}$

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

Equipartition Theorem|Magnetic Moment of Orbiting Electron|Gyromagnetic Ratio Invariance

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of classical statistical mechanics (Boltzmann distribution) with the equipartition theorem.

The primary trap lies in the spatial averaging of the magnetic dipoles and the definition of thermal energy.

Students often incorrectly assume the thermal energy disrupting alignment is simply $K$ or $k_B T$ without performing the proper 3D angular integration, leading to missing the $1/3$ factor inherent in Curie's Law.

Furthermore, they frequently conflate the average translational kinetic energy $K = \frac{3}{2} k_B T$ with the characteristic thermal energy scale $k_B T$ , leading to incorrect numerical prefactors like $1/3$ or $2/3$ in the final expression.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermodynamics - Class 11 Part 2
Concept	Equipartition Theorem
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free