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Question #42956

Difficulty: 8/10 | 🟢 Free

The Question

Region 1 ( $z > 0$ ) is vacuum with magnetic permeability $\mu_0$ . Region 2 ( $z < 0$ ) is filled with a linear, isotropic magnetic material having a magnetic susceptibility $\chi_m = 3$ . A uniform magnetic field exists in both regions. In Region 1, the magnetic field is given by $\mathbf{B}_1 = B_0 (\hat{i} + \sqrt{3}\hat{k})$ , where $B_0$ is a positive constant in Tesla. Assuming there are no free surface currents on the interface at $z = 0$ , what is the ratio of the magnetic energy density in Region 2 to the magnetic energy density in Region 1, $\frac{u_2}{u_1}$ ?

Options

A.
$\frac{19}{16}$
B.
$\frac{1}{4}$
C.
4
D.
$\frac{49}{16}$

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

Magnetic Permeability and Susceptibility|Flux and Surface Geometry|Boundary Conditions

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of magnetic boundary conditions at the interface of two media with different permeabilities.

The primary trap is the incorrect assumption that the magnetic field vector $\mathbf{B}$ is entirely continuous across the boundary, or conflating the continuity of the normal component of $\mathbf{B}$ with the tangential component.

Students often misapply Ampere's Law, failing to realize that the tangential component of the auxiliary field $\mathbf{H}$ is continuous (in the absence of free surface currents), which forces a jump discontinuity in the tangential component of $\mathbf{B}$ proportional to the relative permeability.

The options specifically target these false continuity assumptions, such as assuming $\mathbf{B}$ is continuous (yielding 1/4) or assuming $\mathbf{H}$ is continuous (yielding 4).

Quick Facts About This Question

Question Information

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