S.T. the voltage across a pure capacitor lags the current by `90^(@)` or `(pi)/(2)` rad.

Explanation

Let the applied voltage across the capacitor be `v_(c)=v=v_(m)sinomegat`.
By definition, `i=(dq)/(dt)`
where, `q=Cv_(m)sinomegat`
i.e., `i=(d)/(dt)(Cv_(m)sinomegat)`
i.e., `i=(v_(m))/((1)/(Comega))cosomegat`
`i=i_(m)sin(omegat+pi//2)`
where `X_(C)=(1)/(Comega)and(v_(m))/(X_(c))=i_(m)`
`X_(C)` is the capacitive reactance which is the a.c. resistance offered by a capacitor.
Note : `X_(C)=(1)/(Comega)=(1)/(2pifC)`. Hence `X_(c)prop(1)/(f)andX_(c)prop(1)/(C)andbarp=0`