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Question #42081

Difficulty: 8/10 | 🟢 Free

The Question

An infinitely long straight wire carrying a uniform positive linear charge density $\lambda$ is fixed along the horizontal bottom edge of a smooth non-conducting inclined plane of inclination $\theta = 30^\circ$ . A particle of mass $m$ and positive charge $q$ is projected from a point on the incline at a distance $r_0$ from the wire. The initial velocity of the particle is $v_0$ , directed at an angle $\alpha = 60^\circ$ to the wire, along the inclined plane. If the system parameters satisfy the relation $\frac{q\lambda}{2\pi\epsilon_0} = m g r_0$ (where $g$ is the acceleration due to gravity), what is the speed of the particle when its distance from the wire becomes $2r_0$ ?

Options

A.
$\sqrt{v_0^2 + g r_0 (2 \ln 2 - 1)}$
B.
$\sqrt{v_0^2 + g r_0}$
C.
v_0
D.
$\sqrt{v_0^2 + 2 g r_0 \ln 2}$

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

Projectile on Inclined Plane|Trajectory of Charged Particle|Gauss Law Applicability

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize that standard projectile motion equations (constant acceleration) fail when a particle moves in a spatially varying electric field.

The lethal trap occurs when students calculate the initial electric force ( $F = mg$ ) and incorrectly assume it remains constant, leading to a net work of $mgr_0 - mgr_0/2 = mgr_0/2$ and an incorrect final speed of $\sqrt{v_0^2 + gr_0}$ (Option B).

Furthermore, the initial projection angle $\alpha = 60^\circ$ acts as a secondary trap, tempting students to unnecessarily resolve velocity components and potentially mishandle the 2D kinematics.

By forcing the integration of the coordinate-dependent electric force $F(r) = qE(r)$ derived from Gauss's Law alongside the constant gravitational component, the problem penalizes rote application of kinematic formulas and rewards rigorous use of the Work-Energy Theorem.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Projectile on Inclined Plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free