Capacitative circuits

When a capacitor is connected to a voltage supply the result is quite different for an a.c. supply from that for d.c.
In the circuit in Figure 1(a) no current flows and the lamp does not light. But if the supply is a.c., as in Figure 1(b), the lamp lights showing that some current must be flowing through it. This can be explained as follows.

When a capacitor is connected to an a.c. supply the plates of the capacitor are continually charging and discharging, and so an alternating current flows in the connecting wires. Current does not actually flow through the capacitor itself.

This can be shown mathematically as follows.

Let the voltage applied to the capacitance C be v = v_osin (ωt)

The charge q on the capacitor plates is given by q = Cv = Cv_osin(ωt)
But the current i = dq/dt = <ω Cv_ocos(ωt) = ω Cv_osin (πt + ω/2)

The current and voltage shown in Figure 7 are not in phase; in fact, the current is 90^o ahead of the voltage, as shown by the vector diagram.

Initially there is no voltage across the capacitor. As soon as the voltage begins to increase a large current flows. This current falls as the capacitor charges up, i.e as the p.d across it increases, and is zero when the capacitor is fully charged.