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Question #43864

Difficulty: 8/10 | 🟢 Free

The Question

An infinitely long line charge with uniform linear charge density $\lambda > 0$ lies along the $z$ -axis, producing a radial electric field $\vec{E} = \frac{\lambda}{2\pi\epsilon_0 r} \hat{r}$ . A uniform magnetic field $\vec{B} = B_0 \hat{k}$ exists throughout the region. An observer is stationed in a non-inertial reference frame rotating with a constant angular velocity $\vec{\omega} = -\frac{qB_0}{2m} \hat{k}$ relative to the laboratory frame. The observer releases a particle of mass $m$ and positive charge $q$ from rest at a radial distance $r = R$ from the $z$ -axis. Assuming $\lambda$ is sufficiently large such that the particle moves outward, what is the kinetic energy of the particle, as measured by this rotating observer, when it reaches a radial distance $r = 2R$ ?

Options

A.
$\frac{q\lambda \ln 2}{2\pi\epsilon_0} - \frac{3 q^2 B_0^2 R^2}{8m}$
B.
$\frac{q\lambda \ln 2}{2\pi\epsilon_0} + \frac{3 q^2 B_0^2 R^2}{8m}$
C.
$\frac{q\lambda \ln 2}{2\pi\epsilon_0} - \frac{q^2 B_0^2 R^2}{8m}$
D.
$\frac{q\lambda \ln 2}{2\pi\epsilon_0}$

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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

Electric Field Symmetry|Frame of Reference Dependency|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Work-Energy Theorem in a non-inertial rotating reference frame, specifically targeting the 'False Equilibrium' trap in crossed electric and magnetic fields.

Students often incorrectly assume that fictitious forces (like the centrifugal force) are the only additions to the effective potential, neglecting the cross term $q(\vec{\omega} \times \vec{r}) \times \vec{B}$ arising from the transformation of the Lorentz force.

By choosing the Larmor frequency, the velocity-dependent forces cancel, leaving a modified conservative radial field.

Failing to account for the magnetic force acting on the frame's rotational velocity leads to a sign error in the effective radial spring constant, testing the student's rigorous adherence to transformation laws over intuitive static symmetry.

Quick Facts About This Question

Question Information

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