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Question #42308

Difficulty: 8/10 | 🟢 Free

The Question

An infinite non-conducting sheet lies in the $xy$ -plane with a variable surface charge density $\sigma(x) = \alpha x$ , where $\alpha$ is a positive constant. In the region $z > 0$ , the electrostatic field $\vec{E}$ is conservative, and its $z$ -component is given by $E_z = \frac{\sigma(x)}{2\epsilon_0}$ . It is also given that the $x$ -component of the electric field vanishes on the sheet ( $E_x = 0$ at $z=0$ ) and the $y$ -component is zero everywhere. A particle of charge $q$ is moved from the origin $(0,0,0)$ to the point $(a, 0, b)$ along the path $x = a\left(\frac{z}{b}\right)^2$ . What is the work done by the electric field on the particle?

Options

A.
$\frac{q \alpha a b}{2\epsilon_0}$
B.
$\frac{q \alpha a b}{3\epsilon_0}$
C.
$\frac{q \alpha a b}{6\epsilon_0}$
D.
$\frac{q \alpha a b}{\epsilon_0}$

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

Variable Force Integration|Field due to Infinite Sheet|Superposition of Potentials

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental property of electrostatic fields being conservative ( $\nabla \times \vec{E} = 0$ ).

Students often mistakenly assume that the electric field from the sheet only has a $z$ -component $E_z = \sigma(x)/2\epsilon_0$ and ignore the $x$ -component required to prevent the field from curling.

If a student falls into this trap and only integrates the non-conservative force component $F_z = q E_z$ along the specified curved path $x = a(z/b)^2$ , they will incorrectly obtain $\frac{q \alpha a b}{6\epsilon_0}$ .

Recognizing that a spatially varying charge density necessitates a transverse field component is crucial for applying the work-energy theorem correctly in electrostatic fields.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Variable Force Integration
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free