When an AC source is connected to an ideal capacitor, show that the average power supplied by the source over a complete cycle is zero
The instantaneous power supplied to the capacitor is
P = im cos(ωt)vm sin(ωt)
P = imvm cos(ωt)sin(ωt)
Therefore, the average power is
`(:P:) = (:(i_mv_m)/2sin(2omegat):)=(i_mv_m)/2(:sin(2omegat):)`
Now, the average of sin(2ωt) over the cycle is zero.
∴ ⟨P⟩ = 0