AC Source Connected to a Capacitor

A capacitor connected to an ac source. The instantaneous charge on the capacitor equals the instantaneous potential difference across it multiplied by the capacitance.

q = C V_m cos ωt

Since I= dq/dt,

I = –ωCV_m sin ωt

Unlike a resistor, the current I and potential difference V for a capacitor are not in phase. 

The first peak of the current-time plot occurs one quarter of a cycle before the first peak in the potential difference-time plot. Hence, the capacitor current leads capacitor potential difference by one quarter of a period. One quarter of a period corresponds to a phase difference of π/2 or 90°. The potential difference lags the current by 90°.

The quantity 1/ωC is called the capacitive reactance, and is denoted by the symbol X_C.

X_C = 1/ωC = 1/2πvC

Capacitive reactance is a measure of the extent to which the capacitor limits the ac current in the circuit. It depends on capacitance and the frequency of the generator. The capacitive reactance decreases with increase in frequency and capacitance. Resistance and capacitive reactance are similar in the sense that both measure limitations to ac current.

But unlike resistance, capacitive reactance depends on the frequency of the ac.

I_rms = V_rms/X_C

The instantaneous power delivered to the capacitor is the product of the instantaneous capacitor current and the potential difference.

P = VI

P = –ωCV² sin ωt cos ωt

P = –½ωCV² sin 2ωt

The sign of P determines the direction of energy flow with time. When P is positive, energy is stored in the capacitor. When P is negative, energy is released by the capacitor. The electric energy stored in the capacitor during a charging cycle is completely recovered when the capacitor is discharged. On an average, there is no energy stored or lost in the capacitor in a cycle.