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Question #43251

Difficulty: 8/10 | 🟢 Free

The Question

A classical, non-relativistic electron plasma of uniform number density $n$ is confined in a region permeated by a uniform external magnetic field $\vec{B} = B_0 \hat{k}$ . The plasma is initially in thermal equilibrium at temperature $T_0$ with an isotropic Maxwell-Boltzmann velocity distribution. Ignoring boundary effects, the bulk electrons undergo cyclotron motion, creating microscopic current loops. The plasma cools by radiating heat to its surroundings (maintained at a constant temperature $T_s = T_0 / 3$ ) according to Newton's law of cooling, with a cooling constant $\alpha$ . Which of the following expressions correctly represents the magnitude of the effective bulk magnetization $M(t)$ of the plasma at time $t$ ? (Assume the electron mass is $m$ , charge magnitude is $e$ , and Boltzmann constant is $k_B$ .)

Options

A.
$\frac{n k_B T_0}{B_0} \left( \frac{1}{3} + \frac{2}{3} e^{-\alpha t} \right)$
B.
$\frac{3 n k_B T_0}{2 B_0} \left( \frac{1}{3} + \frac{2}{3} e^{-\alpha t} \right)$
C.
$\frac{n k_B T_0}{2 B_0} \left( \frac{1}{3} + \frac{2}{3} e^{-\alpha t} \right)$
D.
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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

Magnetic Moment of Orbiting Electron|Maxwell-Boltzmann Distribution|Current Density Vector

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of microscopic electrodynamics, statistical mechanics, and macroscopic thermodynamics.

The primary trap lies in the dimensionality of the Maxwell-Boltzmann velocity distribution.

The magnetic moment of a cyclotron orbit depends strictly on the perpendicular velocity components ( $v_x$ and $v_y$ ).

Students who incorrectly apply the 3D average kinetic energy ( $\frac{3}{2} k_B T$ ) will calculate an erroneous magnetic moment of $\frac{3 n k_B T}{2 B_0}$ .

Additionally, students might fall into the 'Impulsive Averaging' trap, assuming that an isotropic velocity distribution implies a zero net magnetic moment, failing to realize that the external magnetic field breaks the symmetry of the orbits (all bulk orbits produce a magnetic moment anti-parallel to the field).

Finally, the non-linear decay introduced by Newton's Law of Cooling tests the correct formulation of the differential equation with a non-zero surrounding temperature.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Moment of Orbiting Electron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free