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Question #43665

Difficulty: 8/10 | 🟢 Free

The Question

Two small particles, $P_1$ and $P_2$ , having masses $m$ and $2m$ , and charges $+q$ and $+2q$ respectively, are connected by an ideal massless spring of spring constant $k$ and natural length zero. The system is placed in a uniform magnetic field $\vec{B} = B \hat{k}$ . The particles are projected such that they undergo steady circular motion in the $xy$ -plane about their common center of mass, which remains at rest. If the particles rotate in the counter-clockwise direction, what is the steady angular frequency $\omega$ of their relative motion?

Options

A.
$\frac{\sqrt{q^2 B^2 + 6mk} - qB}{2m}$
B.
$\frac{\sqrt{q^2 B^2 + 6mk} + qB}{2m}$
C.
$\frac{\sqrt{q^2 B^2 + 4mk} - qB}{2m}$
D.
$\frac{\sqrt{9q^2 B^2 + 24mk} - 3qB}{4m}$

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

Reduced Mass Effect|Trajectory of Charged Particle|Conservative Field Path Independence|Vector Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Lorentz force in a two-body system.

A common trap is to incorrectly apply the magnetic force to the relative motion by either using the rest mass $m$ instead of the reduced mass $\mu$ , or by failing to derive the effective charge $q_{eff} = \mu (q/m)$ for the relative frame.

Furthermore, students often miscalculate the direction of the cross product $\vec{v}_{rel} \times \vec{B}$ , failing to recognize that it generates an outward 'centrifugal-like' force for counter-clockwise rotation, leading to a sign error in the radial differential equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Reduced Mass Effect
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free