Discuss power in AC circuit containing only capacitor.

Explanation

The instantaneous power supplied to the capacitor is,
`P_(C ) = IV = I_(m) cos ( omega t ) V_(m) sin ( omega t ) `
`= I_(m) V_(m) cos ( omegat ) sin ( omega t ) `
`= ( I_(m) V_(m))/( 2) . ( 2 cos omega t sin omega t ) `
`= ( I_(m) V_(m))/( 2) sin ( 2 omega t )`
Average power,
`P_(C ) = langle(I_(m) V_(m) sin 2 omega t )/(2) rangle = ( I_(m) V_(m))/( 2) lt sin e omega t gt `
Since ` sin ( 2 omega t) =0` over a complete cycle. This is explained by figure in detail.
`:.` Average power `P_(C ) = 0 `

0-1 the current i flows as shown and from the maximum at 0, reaches a zero value at 1. The plate A is charged to positive polarity while negative charge q builds up in B reaching a maximum at 1 until the current becomes zero. The voltage `V_(c ) = q//C ` is in phase with q and reaches maximum value at 1. Current and voltage are both positive. So, `P = V_(c ) I` is positive. Energy is absorbed from the source during this quarter cycle as the capacitor is charged.

1-2 the current I reverse its direction. The accumulated charge is depleted means, the capacitor is discharged during this quarter cycle. The voltage gets reduced but is still positive. The current is negative . Their product the power is negative. The energy absorbed during the `1//4` cycle 0-1 is returned during this quarter.

One complete cycle of voltage / current. Note that the current leads the voltage.

2-3 As I continues to flow from A to B, the capacitor is charged to reversed polarity means the plate B acquires positive and A acquires negative charge. Both the current and the voltage are negative . Their product p is positive. The capacitor absorbs energy during this `1//4` cycle.

3-4 the current I reverse its direction at 3 and flows from B to A.The accumalated charge is deplected and the magnitude of the voltage `V_(c )` reduced. `V_(c )` becomes zero at 4 when the capacitor is fully discharged. The power is negative. Energy absorbed during 2-3 is returned to the source. Net energy absorbed is zero.