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hi
i don't think charge carrier density is zero ! either it connected or not to a battery.of course in "conductor".
it can increase when the conductor connected to a battery with ionized electron and left valance band to conductive band.
in conductor there are a lot of charge carrier in conductive band so when it connect to a battery it cues to generate an electricity field that force the carrier charge to move in specific direction , electron: opposite of electricity field direction and hole :straight.
but in "electromagnetic field theory fundamentals" by hirizogulu book.they are solving that zero volume current density for conduction current density. please explain
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