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Question #43446

Difficulty: 8/10 | 🟢 Free

The Question

A small ideal bar magnet (modeled as a magnetic dipole) of magnetic moment $\vec{m} = m_0 \hat{k}$ is placed at the origin. It is enclosed in a spherical vacuum cavity of radius $R$ centered at the origin. The region outside the cavity ( $r > R$ ) is filled with an infinitely extended linear magnetic medium of relative permeability $\mu_r$ . Determine the uniform magnetic field $\vec{B}_{ind}$ induced inside the cavity due to the magnetization of the surrounding medium.

Options

A.
$\frac{\mu_0 m_0 (\mu_r - 1)}{2\pi R^3 (2\mu_r + 1)} \hat{k}$
B.
$\frac{\mu_0 m_0 (\mu_r - 1)}{2\pi R^3 (\mu_r + 2)} \hat{k}$
C.
$\frac{\mu_0 m_0 (\mu_r - 1)}{4\pi R^3 (2\mu_r + 1)} \hat{k}$
D.
$-\frac{\mu_0 m_0 (\mu_r - 1)}{2\pi R^3 (2\mu_r + 1)} \hat{k}$

Check Your Answer

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

Bar Magnet as Equivalent Solenoid|Cavity Field Shielding|Magnetic Field Line Behavior in Matter

Why This Question Is Tricky

Common Mistakes

This problem tests the rigorous application of magnetic boundary conditions and the distinction between the $\vec{B}$ and $\vec{H}$ fields in the presence of magnetizable media.

A common trap is conflating the boundary conditions, incorrectly assuming the tangential component of $\vec{B}$ is continuous or the normal component of $\vec{H}$ is continuous, which leads to incorrect shielding factors (Option B).

Another major conceptual trap is the sign of the induced field (Option D): students often intuitively assume the surrounding medium 'shields' the dipole by creating an opposing field at the center (Lenz's law misapplication).

However, for a paramagnetic/ferromagnetic medium ( $\mu_r > 1$ ), the return flux is drawn into the medium, effectively creating a bound surface current that enhances the field at the dipole's location in the $+\hat{k}$ direction.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Bar Magnet as Equivalent Solenoid
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free