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Question #42702

Difficulty: 8/10 | 🟢 Free

The Question

A long cylindrical region of radius $R$ contains a time-varying uniform magnetic field $\vec{B} = \beta t \hat{k}$ , where $\beta$ is a positive constant and $t$ is time. A uniform line charge of linear charge density $\lambda$ is fixed along the $z$ -axis. A point charge $q$ is moved slowly by an external agent from point $P(a, 0, 0)$ to point $Q(b, \pi/2, 0)$ in cylindrical coordinates $(r, \theta, z)$ , where $a < b < R$ . The chosen path consists of a circular arc from $(a, 0, 0)$ to $(a, \pi/2, 0)$ in the $z=0$ plane, followed by a radial line from $(a, \pi/2, 0)$ to $(b, \pi/2, 0)$ . What is the total work done by the external agent? (Assume $\epsilon_0$ is the permittivity of free space).

Options

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

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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

Potential of Concentric Shells|Conservative vs Non-conservative|Variable Force Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of non-conservative fields and path dependence in the Work-Energy Theorem.

In pure electrostatics, work is a state function ( $\Delta V = -W/q$ ), making it path-independent.

However, a time-varying magnetic field generates an induced electric field that is non-conservative ( $\nabla \times \vec{E} \neq 0$ ).

Students commonly fall into the trap of either ignoring the induced field entirely (Option D) or assuming the work is still path-independent, calculating it for a different, seemingly equivalent path like radial-then-arc (which leads to the trap Option B).

This forces the realization that the line integral of the total electric field must be explicitly evaluated along the exact specified path.

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