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Question #42915

Difficulty: 8/10 | 🟢 Free

The Question

An infinitely long non-conducting cylinder of radius $R$ has a time-varying uniform volume charge density $\rho(t) = \beta t$ , where $\beta$ is a positive constant. The cylinder rotates about its longitudinal axis (the $z$ -axis) with a constant angular velocity $\omega \hat{k}$ . At time $t = t_0$ , a particle of charge $q$ and mass $m$ is projected radially inward from a distance $r = R/2$ from the axis with a speed $v_0$ . If the instantaneous azimuthal force (in the $\hat{\phi}$ direction) on the particle is zero at the moment of projection, find the value of $v_0$ .

Options

A.
$\frac{7R}{24t_0}$
B.
$\frac{R}{3t_0}$
C.
$\frac{7R}{32t_0}$
D.
$\frac{5R}{24t_0}$

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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

Gauss Law Applicability|Lorentz Force Vector Direction|Trajectory of Charged Particle

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of Maxwell's equations in a dynamic setup, specifically targeting the misconception that a uniform charge distribution only produces a conservative radial electric field.

Students often fail to recognize that the rotating time-varying charge constitutes a time-varying convection current, which generates a time-varying magnetic field.

This, in turn, induces a non-conservative azimuthal electric field via Faraday's Law.

A common trap is assuming the magnetic field is uniform across the cross-section when calculating the flux, which leads to an incorrect induced electric field (Option B).

Equating the electric force from this induced field with the magnetic Lorentz force ( $\vec{v} \times \vec{B}$ ) isolates the correct projection velocity.

Quick Facts About This Question

Question Information

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