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Question #43447

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical magnetic rod of radius $R$ and length $L = \frac{4}{3}R$ is placed along the $z$ -axis with its flat ends at $z=0$ and $z=L$ . The material possesses a non-uniform permanent magnetization given by $\vec{M} = M_0 \left(1 - \frac{z}{L}\right) \hat{k}$ , where $M_0$ is a positive constant with units of $\text{A/m}$ , and $\hat{k}$ is the unit vector along the $z$ -axis. Assuming there are no external magnetic fields or free currents, what is the magnitude of the magnetic field $\vec{B}$ at the origin ( $z=0$ )?

Options

A.
$\frac{\mu_0 M_0}{4}$
B.
$\frac{\mu_0 M_0}{5}$
C.
$\frac{2\mu_0 M_0}{5}$
D.
$\frac{3\mu_0 M_0}{10}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Bar Magnet as Equivalent Solenoid|Superposition of Magnetic Fields|Dimensional Independence vs. Scalar/Vector Nature

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from a uniform magnetization model to a non-uniform distribution requiring the integration of bound currents.

A common lethal trap is misapplying the superposition principle by taking the spatial average of the magnetization ( $M_{avg} = M_0/2$ ) and plugging it into the standard formula for a finite uniform solenoid, which yields the incorrect option $\mu_0 M_0 / 5$ .

This flawed approach ignores the non-linear dependence of the Biot-Savart law on distance; current loops closer to the origin contribute significantly more to the magnetic field than those further away, rendering a simple arithmetic average of the source term physically invalid.

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