Question

An AC generator with output voltage \(V\) and frequency \(f\) is connected to the plates of an air-filled parallel plate capacitor of plate area \(A\) and plate separation \(d\). The maximum value of the displacement current is:

• A. \(\dfrac{2\pi \epsilon_0 f V A}{d}\)
• B. \(\dfrac{\pi \epsilon_0 f V A}{d}\)
• C. \(\dfrac{2\pi \epsilon_0 f A}{V d}\)
• D. \(\dfrac{2\pi \epsilon_0 V A}{f d}\)

Hint:

Use \(I_d = C\,dV/dt\) with \(C = \epsilon_0 A/d\) and peak of \(dV/dt = 2\pi f V\).

Answer 1

The Correct Option is A

Step 1: The displacement current between the capacitor plates equals the conduction current in the connecting wires and is given by
\[I_d = \epsilon_0 \frac{d\Phi_E}{dt}\]
where \(\Phi_E = E A\) is the electric flux.

Step 2: For a parallel plate capacitor the field is \(E = \dfrac{V(t)}{d}\), so the flux is \(\Phi_E = \dfrac{A\,V(t)}{d}\). Thus
\[I_d = \frac{\epsilon_0 A}{d}\frac{dV}{dt}\]
Step 3: Let the AC output be \(V(t) = V_0 \sin(\omega t)\) with peak value \(V\) (so \(V_0 = V\)) and \(\omega = 2\pi f\). Then
\[\frac{dV}{dt} = V\,\omega \cos(\omega t) = 2\pi f V \cos(\omega t)\]
Step 4: Substitute:
\[I_d = \frac{\epsilon_0 A}{d}\,(2\pi f V)\cos(\omega t)\]
The maximum occurs when \(\cos(\omega t) = 1\):
\[I_{d,\max} = \frac{2\pi \epsilon_0 f V A}{d}\]
This is option (A).
\[\boxed{I_{d,\max} = \dfrac{2\pi \epsilon_0 f V A}{d}}\]