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Question #43285

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The Question

A long straight wire carrying a steady current $I = 50 \text{ A}$ lies along the $z$ -axis. It is surrounded by a non-homogeneous paramagnetic cylindrical shell of inner radius $a = 0.1 \text{ m}$ and outer radius $b = 0.2 \text{ m}$ . The magnetic susceptibility of the shell varies radially as $\chi_m(r) = k r$ , where $k = 20 \text{ m}^{-1}$ and $r$ is the perpendicular distance from the $z$ -axis. Calculate the total magnetic flux $\Phi$ passing through a rectangular cross-section of the shell of length $L = 2.0 \text{ m}$ parallel to the $z$ -axis. (Assume the permeability of free space is $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$ ).

Options

A.
$20 (\ln 2 + 2) \text{ \mu Wb}$
B.
$80 \ln 2 \text{ \mu Wb}$
C.
$60 \ln 2 \text{ \mu Wb}$
D.
$100 \ln 2 \text{ \mu Wb}$

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 Field Line Behavior in Matter|Magnetic Permeability and Susceptibility|Continuous Charge Distribution|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply magnetic field relations locally in a non-homogeneous medium.

A common trap in JEE preparation is treating magnetic permeability $\mu$ as a macroscopic constant and pulling it out of the flux integral, often by calculating an 'average' susceptibility (e.g., $\chi_{avg} = 20 \times 0.15 = 3$ , leading to the incorrect $80 \ln 2 \text{ \mu Wb}$ ).

By defining $\chi_m$ as a function of $r$ , the problem forces the application of the differential form of the constitutive relation $\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$ before integrating over the area.

It highlights that the bound volume current density $\mathbf{J}_b = \nabla \times \mathbf{M}$ continuously modifies the local magnetic field, invalidating lumped-parameter approaches.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Magnetic Field Line Behavior in Matter
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free