Before You Begin

Question #42267

Difficulty: 8/10 | 🟢 Free

The Question

A thin non-conducting rod of length $2L$ lies along the $x$ -axis, centered at the origin. It carries a non-uniform linear charge density $\lambda(x) = \lambda_0 \frac{|x|}{L}$ , where $\lambda_0$ is a positive constant. An external agent moves a point charge $q$ from point $A(0, L, 0)$ to point $B\left(0, 0, \frac{3}{4}L\right)$ along a complex curved path $\Gamma$ parameterized by $\vec{r}(t) = L\sin(\pi t)\hat{i} + L\cos^2\left(\frac{\pi t}{2}\right)\hat{j} + \frac{3}{4}L\sin^2\left(\frac{\pi t}{2}\right)\hat{k}$ for $t \in [0,1]$ . What is the work done by the electric field on the charge $q$ during this motion?

Options

A.
$\frac{q \lambda_0}{4\pi\epsilon_0} (2\sqrt{2} - 3)$
B.
$\frac{q \lambda_0}{4\pi\epsilon_0} (3 - 2\sqrt{2})$
C.
$\frac{q \lambda_0}{2\pi\epsilon_0} (\sqrt{2} - 1)$
D.
$\frac{q \lambda_0}{4\pi\epsilon_0} (2\sqrt{2} - 1)$

Check Your Answer

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

Potential vs Field Relationship|Conservative Field Path Independence|Continuous Charge Distribution

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to recognize path independence in a conservative vector field, specifically avoiding the brute-force calculation of a complex line integral.

By providing a non-uniform charge distribution and an intimidating 3D parametric path that leaves the symmetry plane, students are baited into attempting to find the electric field vector at arbitrary points in space—a mathematically prohibitive task.

The correct approach requires identifying that the electrostatic force is conservative, meaning work depends only on the scalar potential difference between the initial and final points, both of which conveniently lie on the yz-plane where the potential integral is easily solvable.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Gravitation - Class 11 Part 1
Concept	Potential vs Field Relationship
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free