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Question #42962

Difficulty: 8/10 | 🟢 Free

The Question

A large circular wire loop of radius $R$ lies in the $xy$ -plane, centered at the origin, and carries a steady current $I$ in the counter-clockwise direction (when viewed from the positive $z$ -axis). A small hard ferromagnetic sphere is fixed at the origin. It possesses a residual magnetic dipole moment $\vec{M}_r = \frac{\pi I R^2}{\sqrt{2}} \hat{i}$ . A small test loop of area $A$ carrying a steady current $i$ is placed at the point $(0, 0, R)$ . The test loop is free to rotate in 3D space. Assuming the test loop is small enough that the magnetic field over its area is uniform, what is the unit normal vector $\hat{n}$ of the test loop (indicating the direction of its magnetic moment) when it is in stable equilibrium?

Options

A.
$\frac{-\hat{i} + \hat{k}}{\sqrt{2}}$
B.
$\frac{\hat{i} + \hat{k}}{\sqrt{2}}$
C.
$\frac{2\hat{i} + \hat{k}}{\sqrt{5}}$
D.
$\frac{-2\hat{i} + \hat{k}}{\sqrt{5}}$

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

Soft vs Hard Magnetic Materials|Force on Current-Carrying Loop|Vector Decomposition Ambiguity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply the principle of superposition for magnetic fields and identify the correct stable equilibrium state of a magnetic dipole.

The primary trap lies in the vector decomposition of the residual magnetic field.

Students frequently misapply the magnetic dipole field formulas, either by confusing the axial and equatorial cases or by neglecting the crucial negative sign in the equatorial field equation ( $\vec{B}_{eq} = -\frac{\mu_0}{4\pi}\frac{\vec{M}}{r^3}$ ).

If a student incorrectly assumes the point $(0,0,R)$ is an axial point, they will use the $+2\vec{M}$ formula, leading to Option C or D.

If they correctly identify it as an equatorial point but forget the anti-parallel nature of the field, they will select Option B.

The question rigorously enforces dimensional and geometric consistency while probing a genuine conceptual boundary condition.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Magnetism and Matter - Class 12
Concept	Soft vs Hard Magnetic Materials
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free