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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?

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If by “enclosed” you mean totally enclosed, yes, it’s zero always, isolated or not.

Formally, draw an (arbitrary) surface totally inside the conductor that encloses wherever you’re interested in. The field everywhere on this surface is zero (it’s a conductor) so the E field flux through it is also zero, so by Gaus’s law the enclosed charge is zero.

Operationally, charge will move around inside the conductor until this is so. I.e. if you place an extra charge Q in a void, it will attract -Q to neutralize.

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