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Question #43400

Difficulty: 8/10 | 🟢 Free

The Question

A straight wire of length $2R$ is placed along the $z$ -axis with its center at the origin. The wire carries a uniform linear charge density $\lambda$ and a steady current $I$ . A point charge $q$ is moved slowly from a point $(2R, 0, 0)$ to a point $(R, 0, 0)$ by an external agent. Let $W$ be the work done by the external agent against the electrostatic force. Let $B$ be the magnitude of the magnetic field produced by the wire at the point $(R, 0, 0)$ . If $c$ is the speed of light in vacuum, which of the following expressions gives the ratio $\frac{W}{q B}$ ?

Options

A.
$\frac{\sqrt{2} \lambda R c^2}{I} \ln \left( \frac{2(\sqrt{2} + 1)}{\sqrt{5} + 1} \right)$
B.
$\frac{\sqrt{2} \lambda R c^2}{I} \ln(2)$
C.
$\frac{\lambda R c^2}{I} \ln \left( \frac{2(\sqrt{2} + 1)}{\sqrt{5} + 1} \right)$
D.
$\frac{\lambda R c^2}{I} \ln(2)$

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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

Infinite Wire vs. Finite Wire|Potential of Concentric Shells|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between the fields of finite and infinite sources, a common pitfall in JEE Advanced.

Students frequently memorize the logarithmic potential difference $\Delta V = \frac{\lambda}{2\pi\epsilon_0}\ln(r_2/r_1)$ for infinite cylindrical symmetries and blindly apply it to finite wires.

Simultaneously, they might correctly use the finite wire formula for the magnetic field, leading to a mismatched ratio (Option B).

Alternatively, they might approximate both as infinite (Option D).

The problem forces rigorous integration for the electrostatic potential of a finite segment, breaking the false assumption of infinite cylindrical symmetry, while combining it with the Biot-Savart law for a localized magnetic field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Infinite Wire vs. Finite Wire
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free