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Question #42950

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The Question

An infinitely long cylindrical hard magnetic core of radius $R$ (in meters) has a frozen-in, non-uniform radial magnetization given by $\vec{M} = M_0 \left(\frac{r}{R}\right) \hat{r}$ , where $r$ is the radial distance from the cylinder's central axis, $\hat{r}$ is the radial unit vector, and $M_0$ is a positive constant (in $\text{A/m}$ ). A straight, thin wire carrying a steady free current $I$ (in Amperes) is placed along the central axis of the cylinder. Which of the following represents the magnitude of the magnetic induction $\vec{B}$ (in Teslas) at a radial distance $r = \frac{R}{2}$ inside the core? (Assume vacuum permeability is $\mu_0$ in $\text{T}\cdot\text{m/A}$ )

Options

A.
$\frac{\mu_0 I}{\pi R}$
B.
$\sqrt{\left(\frac{\mu_0 I}{\pi R}\right)^2 + \left(\frac{\mu_0 M_0}{2}\right)^2}$
C.
$\frac{\mu_0 I}{\pi R} + \frac{\mu_0 M_0}{2}$
D.
$\sqrt{\left(\frac{\mu_0 I}{\pi R}\right)^2 + (\mu_0 M_0)^2}$

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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

Superposition of Magnetic Fields|Soft vs Hard Magnetic Materials|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental distinction between the magnetic induction ( $\vec{B}$ ) and the magnetic field intensity ( $\vec{H}$ ) in the presence of hard magnetic materials.

A common trap is for students to misapply the superposition principle by assuming the total $\vec{B}$ -field is simply the vector sum of the field due to the free current and $\mu_0 \vec{M}$ .

This incorrectly yields a non-zero radial $\vec{B}$ -field component (leading to Option B).

By enforcing Gauss's Law for Magnetism ( $\nabla \cdot \vec{B} = 0$ ), students must recognize that a purely radial, frozen-in magnetization generates a corresponding demagnetizing $\vec{H}$ -field ( $H_r = -M_r$ ) such that the radial $\vec{B}$ -field remains strictly zero.

The problem eliminates candidates who rely on rote formulas without checking Maxwell's divergence conditions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Superposition of Magnetic Fields
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free