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Question #43274

Difficulty: 8/10 | 🟢 Free

The Question

A short magnetic dipole of magnetic moment $M_1$ is fixed at the origin of a coordinate system. Another short magnetic dipole of magnetic moment $M_2$ and mass $m$ is placed on the axis of the first dipole at a distance $r_0$ from it. Both dipoles are aligned in the same direction along the axis. If the second dipole is released from rest, what is its speed $v$ when it reaches a distance of $\frac{r_0}{2}$ from the first dipole? (Assume no other forces act on the system and $\mu_0$ is the permeability of free space).

Options

A.
$\sqrt{\frac{7\mu_0 M_1 M_2}{\pi m r_0^3}}$
B.
$\sqrt{\frac{3\mu_0 M_1 M_2}{2\pi m r_0^3}}$
C.
$\sqrt{\frac{7\mu_0 M_1 M_2}{2\pi m r_0^3}}$
D.
$\sqrt{\frac{15\mu_0 M_1 M_2}{\pi m r_0^3}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Magnetic Field on Axial vs Equatorial Line|Mechanical Energy Conservation|Non-Uniform Acceleration Constraint

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from constant-force kinematics to energy conservation in non-uniform fields.

The primary trap is the misapplication of Newton's equations of motion for constant acceleration ( $v^2 = u^2 + 2as$ ).

Students often calculate the initial magnetic force $F = \frac{3\mu_0 M_1 M_2}{2\pi r_0^4}$ , incorrectly assume it remains constant, and apply $v^2 = 2a(r_0/2)$ , leading to Option B.

The correct methodology requires recognizing that the force varies as $r^{-4}$ , necessitating either direct integration of work ( $\int F dr$ ) or the application of mechanical energy conservation using the dipole-dipole potential energy $U \propto r^{-3}$ .

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