The drift velocity does not depend on the length or the cross sectional area of the wire, when dealing with a macroscopic (ordinary, everyday life) wire. However, if the wire is, say, too short, e.g. comparable to the average distance a charge carrier travels before undergoing a collision, then it might begin to depend on the wire length, but for all practical intents and purposes a wire won't be that short.And we also know that,$v _{ d }=\frac{ I }{ eNA }$ where $I =$ current, $e =$ charge on electron, $N =$ no. of electrons,$A=$ area of cross-section.
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