16.1 Exercise 8

A potential difference $V \left(t\right) = V_{0} \text{sin} \omega t$ is maintained across a parallel-plate capacitor with capacitance C consisting of two circular parallel plates. A thin wire with resistance R connects the centers of the two plates, allowing charge to leak between plates while they are charging.
(a) Obtain expressions for the leakage current $I_{\text{res}} \left(t\right)$ in the thin wire. Use these results to obtain an expression for the current $I_{\text{real}} \left(t\right)$ in the wires connected to the capacitor.
(b) Find the displacement current in the space between the plates from the changing electric field between the plates.
(c) Compare $I_{\text{real}} \left(t\right)$ with the sum of the displacement current $I_{\text{d}} \left(t\right)$ and resistor current $I_{\text{res}} \left(t\right)$ between the plates, and explain why the relationship you observe would be expected.