# Electrostatics: Is conductor charge 0 only in isolated conductor?

I'm having a little bit of confusion in electrostatic analysis of conductors. Is the total charge Qenclosed in a conductor equal to zero in isolated conductors? If we are applying a voltage Vo between ground and a conductor this conductor will be at a potential Vo does that allow the presence of charges in the conductor now ?

Source of confusion : If in conductors Q=0 then why does $Q=\int_a^b \epsilon \vec E. d\vec S \neq 0$ in all cases?