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Question #43089

Difficulty: 8/10 | 🟢 Free

The Question

A thin, infinite sheet lying in the $xy$ -plane ( $z=0$ ) carries a radial surface current density $\vec{K} = -\alpha r \hat{r}$ , where $\alpha$ is a positive constant, $r = \sqrt{x^2+y^2}$ is the radial distance from the origin, and $\hat{r}$ is the radial unit vector in the $xy$ -plane. This surface current causes a time-varying surface charge density $\sigma(t)$ to accumulate on the sheet. Assuming the sheet is initially uncharged at $t=0$ , what is the induced magnetic field vector $\vec{B}$ at a point $(x, y, z)$ in the region $z > 0$ ?

Options

A.
$\frac{\mu_0 \alpha}{2} (-y \hat{i} + x \hat{j})$
B.
$\frac{\mu_0 \alpha}{4} (-y \hat{i} + x \hat{j})$
C.
$\frac{\mu_0 \alpha}{2} (y \hat{i} - x \hat{j})$
D.
$\vec{0}$

Check Your Answer

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

Maxwell's Correction to Ampere's Law|Wave Propagation Direction|Vector Cross-Product

Why This Question Is Tricky

Common Mistakes

This problem tests the profound implications of the Maxwell-Ampere law and the continuity equation in 2D.

The primary trap (Option D) catches students who incorrectly assume that because the real current $\vec{K}$ is purely radial and confined to the $xy$ -plane, it produces no magnetic field, completely ignoring the displacement current generated by the accumulating surface charge.

A secondary, highly lethal trap (Option B) catches students who misapply the 2D divergence operator in cylindrical coordinates by simply taking the radial derivative $\frac{\partial K_r}{\partial r} = -\alpha$ instead of $\frac{1}{r}\frac{\partial}{\partial r}(r K_r) = -2\alpha$ , leading to a missing factor of 2 in the displacement current density.

The correct approach requires synthesizing the continuity equation, Gauss's law, and the integral form of Maxwell's correction to find the solenoidal induced magnetic field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Maxwell's Correction to Ampere's Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free