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Question #43395

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical region of space of radius $R$ has its axis aligned with the $z$ -axis. Within this region ( $r \leq R$ ), there exists a steady, uniform conduction current density $\vec{J}_c = J_0 \hat{k}$ and a spatially uniform, time-varying electric field $\vec{E} = \left( \frac{2 J_0 t}{\epsilon_0} \right) \hat{k}$ , where $t$ is time, $J_0$ is a positive constant, and $\epsilon_0$ is the vacuum permittivity. Both fields are zero outside the cylinder ( $r > R$ ). Consider a closed circular loop $C$ of radius $r = R/2$ centered on the $z$ -axis and lying in the $xy$ -plane. What is the magnitude of the line integral $\oint_C \vec{B} \cdot d\vec{l}$ around this loop?

Options

A.
$\frac{3}{4} \mu_0 \pi R^2 J_0$
B.
$\frac{1}{4} \mu_0 \pi R^2 J_0$
C.
$\frac{1}{2} \mu_0 \pi R^2 J_0$
D.
$0$

Check Your Answer

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

Ampere’s Circuital Law Symmetry|Conservative vs Non-conservative|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of the Maxwell-Ampere law and the non-conservative nature of induced magnetic fields.

A common trap for JEE aspirants is to either assume $\oint \vec{B} \cdot d\vec{l} = 0$ by falsely equating magnetic field circulation with conservative electrostatic potential, or to rote-apply the static Ampere's Law ( $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{encl}$ ) while completely ignoring the displacement current generated by the time-varying electric field.

By superimposing a steady conduction current and a dynamic electric field, the problem forces students to independently evaluate and sum both the real charge flow and the Maxwell correction term to find the true circulation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Ampere’s Circuital Law Symmetry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free