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Question #43013

Difficulty: 8/10 | 🟢 Free

The Question

Two charged particles, $P_1$ and $P_2$ , are moving in the $xy$ -plane under the influence of a uniform magnetic field $\vec{B} = B_0 \hat{k}$ . Particle $P_1$ has mass $m$ , charge $q$ (where $q > 0$ ), and at $t=0$ is at the origin $(0,0)$ moving with velocity $\vec{v}_1 = v_0 \hat{i}$ . Particle $P_2$ has mass $2m$ , charge $2q$ , and at $t=0$ is also at the origin moving with velocity $\vec{v}_2 = v_0 \hat{j}$ . The two particles undergo a perfectly inelastic collision at the origin and merge into a single particle. What is the magnitude of the change in the total orbital magnetic dipole moment of the system due to this collision?

Options

A.
$\frac{2 m v_0^2}{3 B_0}$
B.
$\frac{m v_0^2}{3 B_0}$
C.
$\frac{5 m v_0^2}{6 B_0}$
D.
$\frac{3 m v_0^2}{2 B_0}$

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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

Cyclotron Frequency Invariance|Center of Mass (COM) Frame|Collision Impulses

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of Center of Mass frame energetics with electrodynamic invariants.

The primary trap is computational: students often attempt to solve this by finding the post-collision velocity vector, calculating the new radius of curvature, and then evaluating the new area and time period to find the final magnetic moment.

This brute-force method is highly prone to algebraic errors and consumes excessive time.

The conceptual hurdle is failing to recognize the invariant relationship $\mu = K/B$ for cyclotron motion.

Furthermore, students may not realize that the loss in total magnetic moment corresponds exactly to the loss in kinetic energy, which is most elegantly and rapidly found using the Center of Mass frame's internal kinetic energy formula ( $\frac{1}{2}\mu_{red}v_{rel}^2$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Cyclotron Frequency Invariance
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free