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Question #42908

Difficulty: 8/10 | 🟢 Free

The Question

A fixed circular loop of radius $R$ lies in the $yz$ -plane with its center at the origin, carrying a steady current $I$ in the counter-clockwise direction as seen from the $+x$ -axis. A very small current loop with a constant magnetic dipole moment $\vec{m} = m_0 \hat{j}$ is placed on the $x$ -axis at a distance $x = \sqrt{3}R$ from the origin. What is the magnetic force $\vec{F}$ acting on the small current loop?

Options

A.
\vec{0}
B.
$\frac{3\sqrt{3} \mu_0 I m_0}{128 R^2} \hat{j}$
C.
$-\frac{3\sqrt{3} \mu_0 I m_0}{64 R^2} \hat{i}$
D.
$-\frac{3\sqrt{3} \mu_0 I m_0}{128 R^2} \hat{j}$

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

Dipole in Non-Uniform Field|Torque on Current Loop|Force on Current-Carrying Loop

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of the Stern-Gerlach force $\vec{F} = \nabla(\vec{m} \cdot \vec{B})$ in a non-uniform magnetic field.

A common trap is assuming that because the magnetic field on the axis is purely in the $x$ -direction ( $\vec{B} = B_x \hat{i}$ ) and the dipole is in the $y$ -direction ( $\vec{m} = m_0 \hat{j}$ ), the dot product $\vec{m} \cdot \vec{B}$ is zero everywhere, leading to zero force.

Students fail to recognize that the force depends on the gradient of the field components off-axis.

By applying $\nabla \cdot \vec{B} = 0$ and cylindrical symmetry, one must deduce the transverse field gradient $\frac{\partial B_y}{\partial y}$ to find the correct transverse force.

Another trap is blindly applying $F = m \frac{dB}{dx}$ without considering the vector dot product, leading to an incorrect force along the $x$ -axis.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electric Charges and Fields - Class 12
Concept	Dipole in Non-Uniform Field
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free