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Question #84060

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The Question

A long isotropic ferromagnetic rod of intrinsic Young's modulus $Y$ and Poisson's ratio $\nu$ is placed in a uniform external magnetic field $H_0$ parallel to its axis. Due to the magnetostriction effect, the magnetic susceptibility $\chi$ of the material depends on the volumetric strain $\theta = \frac{\Delta V}{V_0}$ as $\chi(\theta) = \chi_0 (1 + \lambda \theta^2)$ , where $\chi_0$ and $\lambda$ are positive dimensionless constants. The system is maintained at a constant temperature. Which of the following expressions correctly represents the effective Bulk Modulus $K_{eff}$ of the rod for small volumetric strains, and the critical external magnetic field $H_c$ at which the rod becomes mechanically unstable?

Options

A.
$K_{eff} = \frac{Y}{3(1-2\nu)} - \mu_0 \chi_0 \lambda H_0^2 \quad ; \quad H_c = \sqrt{\frac{Y}{3\mu_0 \chi_0 \lambda (1-2\nu)}}$
B.
$K_{eff} = \frac{Y}{3(1-2\nu)} + \mu_0 \chi_0 \lambda H_0^2 \quad ; \quad \text{No instability occurs}$
C.
$K_{eff} = \frac{Y}{1-2\nu} - \mu_0 \chi_0 \lambda H_0^2 \quad ; \quad H_c = \sqrt{\frac{Y}{\mu_0 \chi_0 \lambda (1-2\nu)}}$
D.
$K_{eff} = \frac{Y}{3(1-2\nu)} - \frac{1}{2}\mu_0 \chi_0 \lambda H_0^2 \quad ; \quad H_c = \sqrt{\frac{2Y}{3\mu_0 \chi_0 \lambda (1-2\nu)}}$

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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

Poisson's Ratio Limits|Magnetic Permeability and Susceptibility

Why This Question Is Tricky

Examiner Levers

Visualization Stress

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

Difficulty Boost : +4

Common Mistakes

This question tests the thermodynamic coupling between elasticity and magnetism (magnetostriction).

A common trap is using the positive magnetic energy density $u_m = +\frac{1}{2}\mu H^2$ instead of the magnetic Gibbs free energy $g_m = -\frac{1}{2}\mu H^2$ for a system at constant external field $H_0$ .

Using the positive sign leads to the false conclusion that the magnetic field stiffens the material (Option B).

Furthermore, it tests the fundamental relation between Young's modulus, Poisson's ratio, and Bulk Modulus, requiring students to recognize how the intrinsic resistance to volume change is counteracted by the magnetic pressure, ultimately leading to mechanical instability.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Poisson's Ratio Limits
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free