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Question #42796

Difficulty: 8/10 | 🟢 Free

The Question

A long cylindrical region of radius $R$ contains a spatially uniform magnetic field $\vec{B} = \alpha t \hat{k}$ , where $\alpha$ is a positive constant, $t$ is time, and $\hat{k}$ is the unit vector along the z-axis. An infinitely long straight wire carrying a uniform linear charge density $\lambda$ is fixed along the z-axis. A point charge $q$ is moved from point $A$ with cylindrical coordinates $(a, 0, 0)$ to point $B$ with coordinates $(b, \pi/2, 0)$ , where $a < b < R$ . The charge is first moved radially outward from $r=a$ to $r=b$ at $\theta=0$ , and then moved along a circular arc of constant radius $b$ from $\theta=0$ to $\theta=\pi/2$ . What is the total work done by the net electric field on the point charge $q$ during this entire process?

Options

A.
$\frac{q\lambda}{2\pi\epsilon_0} \ln\left(\frac{b}{a}\right) - \frac{\pi q \alpha b^2}{4}$
B.
$\frac{q\lambda}{2\pi\epsilon_0} \ln\left(\frac{b}{a}\right)$
C.
$\frac{q\lambda}{2\pi\epsilon_0} \ln\left(\frac{b}{a}\right) - \frac{\pi q \alpha a^2}{4}$
D.
$\frac{q\lambda}{2\pi\epsilon_0} \ln\left(\frac{b}{a}\right) + \frac{\pi q \alpha b^2}{4}$

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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

Potential of Concentric Shells|Work against Conservative Force|Induced Electric Fields

Why This Question Is Tricky

Common Mistakes

This question tests the critical distinction between conservative electrostatic fields and non-conservative induced electric fields.

Students commonly fall into the trap of treating all electric work as path-independent ( $W = -q\Delta V$ ), calculating only the electrostatic potential difference and completely ignoring the induced EMF (Option B).

By explicitly specifying a two-part path, the problem forces the evaluation of the line integral $\int \vec{E}_{ind} \cdot d\vec{l}$ , highlighting that work done by a time-varying magnetic field is strictly path-dependent.

Option C traps students who incorrectly use the initial radius 'a' for the arc, and Option D traps those who make a sign error applying Lenz's Law.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Potential of Concentric Shells
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free