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Question #43286

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The Question

Two infinitely long straight parallel wires, Wire 1 and Wire 2, carry equal currents $I$ in the $+y$ direction. Wire 1 is located at $x = -d, z = 0$ and Wire 2 is located at $x = d, z = 0$ . The $y-z$ plane ( $x=0$ ) separates two semi-infinite regions. The region $x < 0$ is filled with a linear magnetic material of relative permeability $\mu_r = 3$ . The region $x > 0$ is vacuum ( $\mu_r = 1$ ). A small magnetic dipole of magnetic moment $\vec{m} = m_0 \hat{k}$ is placed on the x-axis at $x = d/2$ . Find the magnetic force $\vec{F}$ experienced by the dipole.

Options

A.
$-\frac{22 \mu_0 m_0 I}{9 \pi d^2} \hat{i}$
B.
$-\frac{20 \mu_0 m_0 I}{9 \pi d^2} \hat{i}$
C.
$-\frac{24 \mu_0 m_0 I}{9 \pi d^2} \hat{i}$
D.
$-\frac{14 \mu_0 m_0 I}{9 \pi d^2} \hat{i}$

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

Magnetic Field Line Behavior in Matter|Superposition of Magnetic Fields|Equilibrium Stability|Dimensional Consistency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle magnetic fields across non-linear or discontinuous media boundaries.

A common misconception (Trap 1) is to apply the principle of superposition naively, treating the field as the simple sum of vacuum fields from the two wires, which yields an incorrect coefficient of 20.

Another trap (Trap 2) is simply scaling the field of the wire in the medium by its local permeability, yielding 24.

In reality, the boundary conditions for $\vec{B}$ and $\vec{H}$ at the interface require the use of image currents and transmitted currents.

The susceptibility of the medium effectively 'drags' the field lines, creating an effective current of $2I$ at $x=-d$ instead of $I$ , significantly altering the magnetic field gradient and the resulting force on the dipole.

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