Source across Capacitor

Source across Capacitor\(i\left( t \right)=C\frac{dV}{dt}=C\frac{d}{dt}~\left( {{V}_{m}}~sin\omega t \right)\).

\(=\omega C{{V}_{m}}\cos \omega t=~\frac{~{{V}_{m}}}{\frac{1}{\omega C}}\text{sin}\left( \omega t+{{90}^{0}} \right)\).

\(=\frac{{{V}_{m}}}{{{X}_{c}}}\sin \left( \omega t+{{90}^{0}} \right)\).

Where, \({{X}_{C}}=\frac{1}{\omega C}\) and is known as capacitive reactance

i (t) = Im sin (ωt + 90⁰)

Instantaneous power

p (t) = v(t) i(t)

= Vm sinωt Im cos ωt

\(P\left( t \right)=\frac{{{V}_{m}}{{I}_{m}}}{2}\sin 2\omega t\).

Average power

\({{P}_{avg}}=\frac{1}{2\pi }~\underset{0}{\overset{2\pi }{\mathop \int }}\,p\left( t \right)~d\omega t=0~\).

Now vector diagrams are drawn below:Vector DiagramHence in a capacitor current leads to voltage by 90° (or voltage lags to currents by 90°).