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Question #43283

Difficulty: 8/10 | 🟢 Free

The Question

A long straight wire carries a time-varying current $I(t) = k t^2$ , where $k$ is a positive constant. A stationary rectangular frame of height $h$ is placed coplanar with the wire. The inner and outer edges of the frame are at distances $R$ and $4R$ from the wire, respectively. The frame is completely filled with a composite magnetic core consisting of two distinct regions:

Region 1 ( $R \le r \le 2R$ ): A soft magnetic material with a non-linear magnetization response given by $\vec{M} = \gamma |\vec{H}| \vec{H}$ , where $\gamma$ is a positive constant and $\vec{H}$ is the auxiliary magnetic field.

Region 2 ( $2R \le r \le 4R$ ): A hard magnetic material with a frozen-in, permanent azimuthal magnetization $\vec{M} = M_0 \hat{\phi}$ (where $\hat{\phi}$ is the azimuthal unit vector). The hard material exhibits no additional magnetization due to the external field.

Assuming edge effects are negligible, what is the magnitude of the induced electromotive force (EMF) in the rectangular frame at time $t = t_0$ ?

Options

A.
$\frac{\mu_0 h k t_0}{\pi} \left[ 2 \ln 2 + \frac{\gamma k t_0^2}{2 \pi R} \right]$
B.
$\frac{\mu_0 h k t_0}{\pi} \left[ 2 \ln 2 + \frac{\gamma k t_0^2}{2 \pi R} + \frac{\pi R M_0}{k t_0^2} \right]$
C.
$\frac{\mu_0 h k t_0}{\pi} \left[ \frac{\gamma k t_0^2}{2 \pi R} \right]$
D.
$\frac{\mu_0 h k t_0}{\pi} \left[ 2 \ln 2 + \frac{\gamma k t_0^2}{\pi R} \right]$

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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

Soft vs Hard Magnetic Materials|Faraday’s Law Vector Form|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the application of Faraday's Law in non-homogeneous, non-linear magnetic media.

A common student pitfall is treating the permanent magnetization of a hard magnetic material as a dynamic contributor to the induced EMF, failing to recognize that frozen-in flux ( $d\Phi_{hard}/dt = 0$ ) does not generate an electric field.

Furthermore, students often incorrectly apply the shortcut $\Phi = BA$ or assume a linear relative permeability $\mu_r$ .

By introducing a spatially varying, non-linear susceptibility $\vec{M} \propto |\vec{H}|\vec{H}$ , the problem forces rigorous integration of the magnetic flux density field $\vec{B}(r,t)$ over the composite core before time differentiation, penalizing rote formula application.

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