AC through a capacitor

On dc a capacitor passes current only while it charges; once full, the current stops. On ac it is charged, discharged and charged the other way every cycle, so current keeps flowing in the leads.

With v = q/C = vm sin ωt, the current i = dq/dt = ωCvm cos ωt = im sin(ωt + π/2), where im = ωCvm.

So 1/ωC acts like a resistance. It is the capacitive reactance XC = 1/ωC, in ohm, and im = vm/XC.

XC falls as the frequency or the capacitance rises: a capacitor blocks dc (zero frequency) and passes high frequencies easily.

The current leads the voltage by π/2: it reaches each maximum a quarter period before the voltage.

As with the inductor, the average power into a pure capacitor over a cycle is zero.

Example 7.3: a lamp in series with a capacitor stays dark on dc but glows on ac. Making C smaller raises XC and dims the lamp on ac.

Example 7.4: 15.0 μF on 220 V, 50 Hz has XC = 212 Ω, rms current 1.04 A and peak current 1.47 A, leading the voltage by π/2. At double the frequency XC halves and the current doubles.

Example 7.5: a bulb in series with an inductor dims when an iron rod is pushed into the coil. The rod raises L and so XL, and more of the supply voltage appears across the coil.