Question

Consider a circular parallel plate capacitor of radius \( R \) with separation \( d \) between the plates (\( d \ll R \)). The plates are placed symmetrically about the origin. If a sinusoidal voltage \( V = V_0 \sin(\omega t) \) is applied between the plates, which of the following statement(s) is (are) true?

• A. The maximum value of the Poynting vector at \( r = R \) is \( \frac{V_0^2 \epsilon_0}{4 \pi R^2} \).
• B. The average energy per cycle flowing out of the capacitor is \( \frac{V_0^2}{4} \).
• C. The magnetic field inside the capacitor is constant.
• D. The magnetic field lines inside the capacitor are circular with the current flowing in the \( r \)-direction.

Hint:

In a capacitor with a time-varying voltage, the current produces circular magnetic fields inside the plates, and the Poynting vector describes the power flow.

Answer 1

The Correct Option is A, B, D

Step 1: Understanding the capacitor's behavior.
The time-varying electric field inside the capacitor generates a magnetic field in the surrounding region. The Poynting vector, which represents the power flow, reaches its maximum value at \( r = R \) and is given by \( \frac{V_0^2 \epsilon_0}{4 \pi R^2} \). Additionally, the average energy per cycle flowing out of the capacitor is \( \frac{V_0^2}{4} \). The current inside the capacitor creates circular magnetic field lines in the \( r \)-direction.
Step 2: Conclusion.
Thus, the correct answers are options (A), (B), and (D).