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Question #43290

Difficulty: 8/10 | 🟢 Free

The Question

A thin bar magnet of length $2L$ and magnetic moment $M$ is cut transversely into two equal halves. One of the halves is placed along the $x$ -axis with its geometric center at the origin. A small magnetic compass needle of magnetic moment $m_0$ is placed on the $y$ -axis at a distance $y = L/2$ from the origin. The needle is free to rotate in the $xy$ -plane. What is the minimum work required to rotate the needle from its stable equilibrium position to its unstable equilibrium position? (Assume the pole model of magnetism, where the poles are located exactly at the ends of the magnet).

Options

A.
$\frac{\mu_0}{4\pi} \frac{2\sqrt{2} m_0 M}{L^3}$
B.
$\frac{\mu_0}{4\pi} \frac{8 m_0 M}{L^3}$
C.
$\frac{\mu_0}{4\pi} \frac{4\sqrt{2} m_0 M}{L^3}$
D.
$\frac{\mu_0}{4\pi} \frac{16 m_0 M}{L^3}$

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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

Effective Moment of Cut Magnets|Dipole Potential Angular Dependence|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question targets the 'Dipole Approximation Fallacy'.

Students frequently memorize the far-field equatorial dipole formula $B = \frac{\mu_0}{4\pi} \frac{M'}{r^3}$ and blindly apply it.

Here, the observation distance $r = L/2$ is comparable to the cut magnet's length $L$ , strictly invalidating the $r \gg L$ assumption.

Furthermore, students often miscalculate the new pole strength or magnetic moment after the cut.

By forcing the use of Coulomb's Law in vector form (superposition of discrete monopole fields) rather than the dipole approximation, the problem acts as a lethal trap for those relying solely on memorized far-field formulas.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Effective Moment of Cut Magnets
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free