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Question #43432

Difficulty: 8/10 | 🟢 Free

The Question

A series AC circuit consists of a resistor with resistance $R$ (in $\Omega$ ) and a parallel-plate capacitor with capacitance $C$ (in $\text{F}$ ). The capacitor plates are circular, each having a radius $a$ (in $\text{m}$ ). The circuit is driven by an ideal alternating voltage source $V(t) = V_0 \sin(\omega t)$ , where $V_0$ is the peak voltage (in $\text{V}$ ) and $\omega$ is the angular frequency (in $\text{rad s}^{-1}$ ). Neglecting any edge effects (fringing fields) of the capacitor, what is the magnitude of the magnetic field $B$ (in $\text{T}$ ) at a radial distance $r = a/2$ from the central axis of the capacitor plates, at the exact instant when the applied source voltage reaches its maximum positive value $V_0$ ?

Options

A.
$\frac{\mu_0 V_0 R \omega^2 C^2}{4 \pi a (1 + \omega^2 R^2 C^2)}$
B.
$0$
C.
$\frac{\mu_0 V_0 \omega C}{4 \pi a \sqrt{1 + \omega^2 R^2 C^2}}$
D.
$\frac{\mu_0 V_0 R \omega^2 C^2}{\pi a (1 + \omega^2 R^2 C^2)}$

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

Ampere’s Law Displacement Current|Phasor Rotation Convention|Phase Relationship

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize AC phasor analysis with the Ampere-Maxwell law, specifically targeting the misconception that displacement current is driven directly by the time-derivative of the source voltage.

Students falling into this trap will erroneously calculate $I_d = C(dV/dt)$ using $V(t) = V_0 \sin(\omega t)$ , leading them to conclude that the displacement current (and thus the magnetic field) is zero when the source voltage is at its maximum (Option B).

Furthermore, students who ignore the phase shift entirely might assume current is maximum when voltage is maximum (Option C), or they may fail to account for the spatial distribution of the displacement current density, missing the area ratio factor (Option D).

The correct approach requires recognizing that the displacement current is continuous with the conduction current, which is phase-shifted from the source voltage due to the RC impedance, and then correctly applying the Ampere-Maxwell law to the enclosed Amperian loop.

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