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Question #43164

Difficulty: 8/10 | 🟢 Free

The Question

An AC voltage source $V(t) = V_0 \sin(\omega t)$ with internal resistance $R$ is connected across the input terminals of an infinite ladder network of capacitors. The ladder consists of series capacitors $C_1 = 2C$ and shunt capacitors $C_2 = C$ . The first element connected to the source is a series capacitor $C_1$ , whose plates are circular with radius $a$ . Given that $R = \frac{1}{\omega C}$ , what is the magnetic field $B$ induced by the displacement current at a radial distance $r = a/2$ from the central axis of the capacitor $C_1$ as a function of time? (Assume a uniform electric field between the plates and ignore edge effects).

Options

A.
$\frac{\sqrt{2} \mu_0 \omega C V_0}{8 \pi a} \sin\left(\omega t + \frac{\pi}{4}\right)$
B.
$\frac{\mu_0 \omega C V_0}{4 \pi a} \cos(\omega t)$
C.
$\frac{\sqrt{2} \mu_0 \omega C V_0}{4 \pi a} \sin\left(\omega t + \frac{\pi}{4}\right)$
D.
$\frac{\sqrt{2} \mu_0 \omega C V_0}{8 \pi a} \sin(\omega t)$

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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|Infinite Series Capacitance|Lenz’s Law Energy Conservation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize steady-state AC circuit theory with Maxwell's equations.

A common trap is treating the displacement current as a global parameter without evaluating the convergent equivalent capacitance of the infinite ladder ( $C_{eq} = C$ ).

Students often ignore the internal resistance's effect on the complex impedance, incorrectly assuming the circuit is purely capacitive, which leads to a missing phase shift (Trap B).

Furthermore, students frequently misapply the Ampere-Maxwell law by forgetting to scale the enclosed displacement current by the area ratio $r^2/a^2$ when calculating the magnetic field inside the capacitor plates (Trap C), or they completely drop the phase angle while retaining the magnitude (Trap D).

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