Before You Begin

Question #43088

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate vacuum capacitor consists of two circular plates of radius $R$ separated by a distance $d$ . It is connected to an alternating voltage source $V(t) = V_0 \sin(\omega t)$ . Let $X_C$ be the capacitive reactance of the capacitor and $\mu_0$ be the permeability of free space. Neglecting edge effects, which of the following expressions represents the azimuthal magnetic field $B(t)$ at a radial distance $r = R/2$ from the central axis of the capacitor, in the region between the plates?

Options

A.
$\frac{\mu_0 V_0}{4 \pi R X_C} \cos(\omega t)$
B.
$\frac{\mu_0 V_0}{\pi R X_C} \cos(\omega t)$
C.
$\frac{\mu_0 V_0}{4 \pi R X_C} \sin(\omega t)$
D.
$\frac{\mu_0 V_0}{2 \pi R X_C} \cos(\omega t)$

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

Maxwell's Correction to Ampere's Law|Capacitive Reactance Dependency|Complex Phasor Analysis

Why This Question Is Tricky

Common Mistakes

This question tests the conceptual bridge between macroscopic AC circuit parameters ( $X_C$ ) and microscopic Maxwell's equations.

Students often memorize $X_C = 1/\omega C$ and treat the capacitor merely as a circuit element, ignoring the displacement current ( $I_d$ ) as a physical source of magnetic fields.

The primary trap (Option B) catches students who incorrectly apply Ampere's Law for a long straight wire, assuming all current is concentrated at the central axis rather than uniformly distributed as a changing electric flux.

Option C traps those who fail to account for the $\pi/2$ phase shift introduced by the time derivative of the electric field in Maxwell's correction.

Option D catches the misapplication of the area proportionality for enclosed flux, using a linear ratio instead of a quadratic one.

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