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Question #42202

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The Question

A ferromagnetic isotropic material with Young's modulus $Y$ and Poisson's ratio $\sigma$ is placed in a uniform external magnetic field, reaching a saturation magnetization $M_s$ . The magneto-elastic coupling contributes an energy density $U_{me} = -\frac{1}{2} \mu_0 \gamma M_s^2 \theta^2$ , where $\theta$ is the volumetric strain and $\gamma$ is a positive dimensionless constant. If the material is subjected to a small external hydrostatic pressure $P$ , what is the magnitude of the equilibrium volumetric strain $|\theta|$ , and what condition must $\sigma$ satisfy for the material to remain mechanically stable under this magnetic saturation? (Assume the intrinsic thermodynamic limit $\sigma < 1/2$ holds).

Options

A.
$|\theta| = \frac{3P(1-2\sigma)}{Y - 3\mu_0 \gamma M_s^2 (1-2\sigma)}$ , \quad $\frac{1}{2} - \frac{Y}{6\mu_0 \gamma M_s^2} < \sigma < \frac{1}{2}$
B.
$|\theta| = \frac{3P(1-2\sigma)}{Y + 3\mu_0 \gamma M_s^2 (1-2\sigma)}$ , \quad $-1 < \sigma < \frac{1}{2}$
C.
$|\theta| = \frac{P(1+\sigma)}{3Y - \mu_0 \gamma M_s^2 (1+\sigma)}$ , \quad $\frac{1}{2} - \frac{Y}{3\mu_0 \gamma M_s^2} < \sigma < \frac{1}{2}$
D.
$|\theta| = \frac{3P(1-2\sigma)}{Y - \mu_0 \gamma M_s^2 (1-2\sigma)}$ , \quad $\frac{1}{2} - \frac{Y}{2\mu_0 \gamma M_s^2} < \sigma < \frac{1}{2}$

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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

Common Mistakes

This question tests the understanding of how magneto-elastic coupling modifies the effective macroscopic elastic constants of a material.

Students often incorrectly assume that the Bulk Modulus $K$ remains constant under magnetic saturation.

The trap lies in failing to recognize that the magnetic energy density, which depends on the volumetric strain, acts as a negative pressure term (Joule magnetostriction effect).

This modifies the effective Bulk Modulus.

Furthermore, it links this modified modulus to the theoretical limits of Poisson's ratio, requiring students to establish a new lower bound for $\sigma$ to prevent volumetric instability ( $K_{eff} \le 0$ ) under magnetic saturation.

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