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Question #42952

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical bar magnet of length $L$ and radius $R$ ( $R \ll L$ ) has a non-uniform magnetization. It is modeled as a tightly wound solenoid with an azimuthal effective surface current density $K(x) = c x^2$ , where $x$ is the distance from the center of the magnet along its axis ( $-L/2 \le x \le L/2$ ) and $c$ is a positive constant. The total magnetic moment of the magnet is $M$ . The magnetic field at a point on the axis at a distance $r$ ( $r \gg L$ ) from the center can be expressed as $B = \frac{\mu_0 M}{2\pi r^3} \left( 1 + \beta \frac{L^2}{r^2} + \mathcal{O}\left(\frac{L^4}{r^4}\right) \right)$ Find the value of the dimensionless constant $\beta$ .

Options

A.
$\frac{9}{10}$
B.
$\frac{1}{2}$
C.
$\frac{9}{20}$
D.
$\frac{3}{5}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Magnetic Field on Axial vs Equatorial Line|Bar Magnet as Equivalent Solenoid|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

Students often memorize the finite-length correction factor for a uniform bar magnet (which yields $\beta = 1/2$ , equivalent to the field of two magnetic monopoles at the ends).

This question introduces a non-uniform magnetization modeled by a surface current density $K(x) \propto x^2$ .

The trap tests whether the student blindly applies the standard uniform dipole correction or correctly applies the Superposition Principle and Biot-Savart Law.

By forcing the integration of non-uniform current loops and a rigorous binomial expansion of $(1-x/r)^{-3}$ , the 'point-dipole' safety net is removed, exposing misconceptions about how spatial distribution affects the multipole expansion.

Quick Facts About This Question

Question Information

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