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Question #42838

Difficulty: 8/10 | 🟢 Free

The Question

An infinitely long solid cylinder of radius $R$ has a uniform volume charge density $\rho$ . The axis of the cylinder coincides with the $z$ -axis. A uniform time-varying magnetic field $\vec{B} = \alpha t \hat{k}$ is present throughout the region, where $\alpha$ is a positive constant and $t$ is time. What is the line integral of the net electric force $W = \int_A^B q\vec{E} \cdot d\vec{l}$ on a point charge $q$ along a straight line path from point $A(R, 0, 0)$ to point $B(0, R, 0)$ at a given instant $t$ ?

Options

A.
0
B.
$-\frac{q \alpha R^2}{2}$
C.
$-\frac{\pi q \alpha R^2}{4}$
D.
$\frac{q R^2}{2} \left( \frac{\rho}{\epsilon_0} - \alpha \right)$

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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|Conservative Field Path Independence|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the limits of the conservative field assumption.

Students often default to calculating potential differences ( $\Delta V$ ) using Gauss's Law for symmetric charge distributions, concluding that work done between equidistant points is zero.

The trap lies in ignoring the Maxwell-Faraday equation ( $\nabla \times \vec{E} = -\partial \vec{B}/\partial t$ ).

The presence of a time-varying magnetic field generates a non-conservative induced electric field, making the line integral path-dependent.

By choosing a straight-line path instead of an arc, the problem forces students to either perform a complex vector line integral or elegantly apply Stokes' Theorem, distinguishing rote learners from those with a deep understanding of vector calculus in electromagnetism.

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