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Question #43255

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor consists of two circular plates of radius $a$ separated by a distance $d$ . The space between the plates is completely filled with a lossy dielectric material of uniform permittivity $\epsilon$ and uniform conductivity $\sigma$ . An alternating voltage $V(t) = V_0 \sin(\omega t)$ is applied across the plates. At a specific angular frequency $\omega_0$ , the amplitude of the conduction current density in the dielectric equals the amplitude of the displacement current density. What is the amplitude of the magnetic field $B_0$ at a radial distance $r$ ( $r < a$ ) from the central axis of the plates when the applied voltage has an angular frequency $\omega = \sqrt{3}\omega_0$ ?

Options

A.
$\frac{\mu_0 r \sigma V_0}{d}$
B.
$\frac{\sqrt{3} \mu_0 r \sigma V_0}{2 d}$
C.
$\frac{(1+\sqrt{3}) \mu_0 r \sigma V_0}{2 d}$
D.
$\frac{\mu_0 r \sigma V_0}{\sqrt{2} d}$

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

Ampere’s Law Displacement Current|Capacitive Reactance Dependency|Frequency Sensitivity of Z

Why This Question Is Tricky

Common Mistakes

This question tests the application of the Ampere-Maxwell Law in a lossy dielectric medium, bridging microscopic field theory with macroscopic AC circuit behavior.

A common trap is treating the capacitor as purely reactive (ignoring conduction current entirely, leading to Option B) or algebraically adding the amplitudes of conduction and displacement currents (leading to Option C).

Students must recognize that conduction current (in phase with the electric field) and displacement current (proportional to the time derivative of the electric field) are orthogonal in phase space.

Consequently, their amplitudes must be added in quadrature (phasor addition) to find the total enclosed effective current driving the induced magnetic field.

Quick Facts About This Question

Question Information

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