Before You Begin

Question #43114

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor consists of two circular plates of radius $R$ separated by a distance $d$ . The space between the plates is completely filled with a lossy dielectric medium of uniform permittivity $\epsilon$ and uniform conductivity $\sigma$ . An alternating voltage $V(t) = V_0 \sin(\omega t)$ is applied across the plates. Neglecting edge effects, what is the amplitude of the magnetic field $B_0$ at a radial distance $r$ ( $r < R$ ) from the central axis of the capacitor?

Options

A.
$\frac{\mu_0 r V_0}{2d} \sqrt{\sigma^2 + \epsilon^2 \omega^2}$
B.
$\frac{\mu_0 r V_0}{2d} \epsilon \omega$
C.
$\frac{\mu_0 r V_0}{2d} \sigma$
D.
$\frac{\mu_0 R^2 V_0}{2 r d} \sqrt{\sigma^2 + \epsilon^2 \omega^2}$

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

Medium Dependency|Maxwell's Correction to Ampere's Law|Dynamic Resistance

Why This Question Is Tricky

Common Mistakes

Students often incorrectly assume that the magnetic field inside a capacitor is generated solely by Maxwell's displacement current, treating the medium as a perfect vacuum or ideal dielectric ( $\sigma = 0$ ).

This question tests the complete Ampere-Maxwell law where the total current density $J_{tot} = J_c + J_d$ must be evaluated.

In a lossy semiconductor medium, the conduction current $J_c = \sigma E$ and displacement current $J_d = \epsilon \frac{\partial E}{\partial t}$ are $\pi/2$ out of phase.

Failing to add them as orthogonal phasors leads to ignoring the complex impedance of the medium, resulting in an incomplete expression for the induced magnetic field.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electromagnetic Waves - Class 12
Concept	Medium Dependency
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free