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Question #42960

Difficulty: 8/10 | 🟢 Free

The Question

A magnetic core of mass $m$ (in $\text{kg}$ ) and volume $V$ (in $\text{m}^3$ ) is attached to a horizontal spring of force constant $k$ (in $\text{N/m}$ ) and oscillates on a frictionless surface. During its motion, the core passes through the equilibrium position where the external magnetic field reverses its direction spatially. This causes the core material to undergo exactly one full major hysteresis loop per mechanical cycle. The hysteresis loop of the core material is approximated as a rectangle with coercive force $H_c$ (in $\text{A/m}$ ) and remanent magnetic field $B_r$ (in $\text{T}$ ). Assume the energy dissipated per cycle is small compared to the total mechanical energy, and the core is always driven to saturation. If the initial amplitude of oscillation is $A_0$ (in $\text{m}$ ), which of the following expressions correctly represents the amplitude $A(t)$ (in $\text{m}$ ) as a function of time $t$ (in $\text{s}$ ) before the core comes to rest?

Options

A.
$A_0 \exp\left(-\frac{2 V H_c B_r}{\pi A_0^2 \sqrt{mk}} t\right)$
B.
$\sqrt{A_0^2 - \frac{4 V H_c B_r}{\pi \sqrt{mk}} t}$
C.
$A_0 - \frac{2 V H_c B_r}{\pi A_0 \sqrt{mk}} t$
D.
$\sqrt{A_0^2 - \frac{2 V H_c B_r}{\pi \sqrt{mk}} t}$

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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

Soft vs Hard Magnetic Materials|Hysteresis Loop Area|Damped Oscillation Equation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between viscous damping and constant-energy-loss damping (Coulomb-like damping).

The primary trap lies in the misapplication of the standard damped harmonic oscillator model.

Students are heavily conditioned to assume that any damping leads to an exponential decay of amplitude ( $A \propto e^{-\gamma t}$ ).

However, hysteresis loss per cycle is determined by the area of the $B-H$ loop, which remains constant as long as the core is driven to saturation, regardless of the oscillation amplitude.

A constant energy loss per cycle means the total energy decays linearly with time.

Since energy is proportional to the square of the amplitude, the amplitude must decay as a square root function of time.

Option A traps students who force-fit an equivalent viscous damping coefficient, while Option D traps those who miscalculate the peak-to-peak area of the rectangular hysteresis loop by missing a factor of 2.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Soft vs Hard Magnetic Materials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free