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Question #43278

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical ferromagnetic core of length $L$ and cross-sectional area $A$ is rigidly clamped between two immovable walls. The core is subjected to an alternating magnetic field of frequency $f$ . The area of the hysteresis loop (B-H curve) for the material is $W_h$ (in $\text{J m}^{-3}$ ). The material has density $\rho$ , specific heat capacity $c$ , Young's modulus $Y$ , and coefficient of linear thermal expansion $\alpha$ . Assuming all hysteresis loss is converted into internal heat, no heat is lost to the surroundings, and eddy current losses are negligible, what is the time $t$ required for the compressive thermal stress in the core to reach a magnitude of $\sigma_0$ ?

Options

A.
$\frac{\sigma_0 \rho c}{Y \alpha W_h f}$
B.
$\frac{3 \sigma_0 \rho c}{Y \alpha W_h f}$
C.
$\frac{\sigma_0 \rho c}{Y \alpha W_h f L}$
D.
$\frac{\sigma_0 c}{Y \alpha W_h f}$

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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

Hysteresis Loop Area|Thermal Stress in Constrained Rods|Thermodynamics Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple microscopic volumetric energy dissipation (magnetic hysteresis) with macroscopic 1D mechanical constraints (thermal stress).

A primary trap lies in the dimensional transition: because hysteresis loss is defined per unit volume, students frequently and incorrectly apply the volumetric thermal expansion coefficient ( $3\alpha$ ) instead of the linear coefficient ( $\alpha$ ) when calculating the constrained stress.

Additionally, students often fail to realize that the geometric dimensions (length and area) perfectly cancel out during the calorimetry phase, leading them to select dimensionally inconsistent options that retain $L$ or $A$ .

The problem demands a rigorous, step-by-step application of energy conservation, calorimetry, and Hooke's Law without conflating 3D heat generation with 1D mechanical restriction.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Hysteresis Loop Area
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free