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Question #82466

Difficulty: 8/10 | 🟢 Free

The Question

An isolated solid metallic sphere of mass $M$ , initial radius $R_0$ , and specific heat capacity $c$ is rotating with an initial angular velocity $\omega_0$ about the $z$ -axis in a vacuum. A total charge $Q$ is uniformly distributed over its surface. The sphere is immersed in a weak uniform magnetic field $B_0 \hat{k}$ . Initially at a temperature $T_0$ , the sphere cools by radiating thermal energy into the $0\text{ K}$ surroundings. The surface has an emissivity $\epsilon$ . As it cools, the sphere undergoes isotropic thermal contraction with a coefficient of linear expansion $\alpha$ . Assuming the charge distribution remains uniform, no charge is lost, and thermal radiation carries away negligible angular momentum, what is the magnitude of the rate of change of its magnetic dipole moment, $\left| \frac{d\vec{\mu}}{dt} \right|$ , at $t=0$ ? (Given: Stefan-Boltzmann constant is $\sigma$ )

Options

A.
$0$
B.
$\frac{8\pi \epsilon \sigma \alpha Q \omega_0 R_0^4 T_0^4}{3 M c}$
C.
$\frac{4\pi \epsilon \sigma \alpha Q \omega_0 R_0^4 T_0^4}{M c}$
D.
$\frac{2\pi \epsilon \sigma \alpha Q \omega_0 R_0^4 T_0^4}{3 M c}$

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

Emissivity and Absorptivity|Magnetic Dipole Moment of Revolving Electron|Integration

Why This Question Is Tricky

Examiner Levers

Extraneous Data

Provides unnecessary variables (Red Herrings) to test conceptual focus.

Difficulty Boost : +3

Visualization Stress

Requires mental rotation or 3D mapping (Optics/Geometry/Vectors).

Difficulty Boost : +4

Common Mistakes

This question tests the student's ability to recognize invariant physical quantities amidst complex, coupled dynamic processes.

The trap is set by presenting a highly involved scenario: a sphere cooling via Stefan-Boltzmann radiation, contracting thermally, and spinning up due to conservation of angular momentum.

Students will typically attempt to calculate the rate of change of temperature, radius, and angular velocity, often making errors in the chain rule or neglecting the work done by thermal contraction against centrifugal forces.

The 'lethal collision' of concepts is bypassed entirely if the student recognizes the gyromagnetic ratio: for a uniformly expanding/contracting body where charge and mass distributions scale identically, the magnetic dipole moment is strictly proportional to the angular momentum.

Since no external torques act on the system, angular momentum is conserved, making the magnetic dipole moment a dynamical constant.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Emissivity and Absorptivity
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free