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Question #43326

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 filled with a non-magnetic dielectric material whose relative permittivity varies with time as $\epsilon_r(t) = 1 + \alpha \sin(\omega t)$ , where $\alpha$ is a positive dimensionless constant ( $\alpha \ll 1$ ). An alternating voltage $V(t) = V_0 \cos(\omega t)$ is applied across the plates. Neglecting edge effects, what is the peak magnitude of the induced magnetic field at the edge of the capacitor plates (at a radial distance $R$ from the central axis) specifically for the component oscillating at angular frequency $2\omega$ ? (Let $c$ be the speed of light in vacuum)

Options

A.
$\frac{\alpha \omega R V_0}{2 d c^2}$
B.
$\frac{\alpha \omega R V_0}{4 d c^2}$
C.
$\frac{\alpha \omega R V_0}{d c^2}$
D.
$\frac{\alpha \omega R V_0}{8 d c^2}$

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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

Phasor Rotation Convention|Maxwell's Correction to Ampere's Law|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the fundamental definition of displacement current in a time-varying medium.

A common trap is for students to blindly apply the static-capacitor formula $I_d = C(t) \frac{dV}{dt}$ .

However, because the dielectric permittivity is time-dependent, the capacitance $C(t)$ is also a function of time.

The correct total displacement current requires the product rule: $I_d = \frac{d}{dt}(C(t)V(t)) = C(t)\frac{dV}{dt} + V(t)\frac{dC}{dt}$ .

Ignoring the $V(t)\frac{dC}{dt}$ term leads exactly to half the correct amplitude for the $2\omega$ sideband frequency, trapping students who rely on rote memorization rather than Maxwell's equations ( $J_d = \frac{\partial D}{\partial t}$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Phasor Rotation Convention
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free