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Question #42965

Difficulty: 8/10 | 🟢 Free

The Question

A uniform cylindrical rod of an isotropic diamagnetic material is perfectly constrained between two rigid, immovable walls at its flat ends. The material has Young's modulus $Y$ , Poisson's ratio $\nu$ , and coefficient of linear thermal expansion $\alpha$ . The initial volume magnetic susceptibility of the rod at temperature $T$ is $\chi_0$ (where $\chi_0 < 0$ ). The temperature of the rod is increased by a small amount $\Delta T$ . According to a theoretical model, the mean square atomic radius $\langle r^2 \rangle$ of the atoms in the material scales with the macroscopic volume $V$ of the rod as $\langle r^2 \rangle \propto V^{2/3}$ . Find the change in the volume susceptibility $\Delta \chi$ of the rod. (Assume $\alpha \Delta T \ll 1$ ).

Options

A.
$-\frac{2}{3} \chi_0 \alpha \Delta T (1 + \nu)$
B.
$\chi_0 \alpha \Delta T (1 - 2\nu)$
C.
$-\chi_0 \alpha \Delta T$
D.
$\frac{4}{3} \chi_0 \alpha \Delta T (1 + \nu)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Diamagnetic Property of Materials|Thermal Stress in Constrained Rods|Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of thermal stress, elasticity, and microscopic magnetic properties.

A common trap is incorrectly calculating the volume change of the constrained rod by treating the uniaxial compressive stress as a hydrostatic pressure and applying the bulk modulus ( $\Delta V/V = \sigma_x/B$ ), which leads to Option B.

Another trap is ignoring the number density dependence ( $1/V$ ) in the volume susceptibility formula and only considering the orbital radius expansion, leading to Option D.

The correct approach requires recognizing that the rod expands laterally due to both thermal expansion and the Poisson effect from the longitudinal constraint, and that volume susceptibility depends on both atomic density and orbital radius.

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