Consider a cylindrical wire in which electrons drifting at speed v produce a current I.
In time duration Dt, each free electron in the wire would have drifted forward a distance of vDt. This means that the charge carriers in the shaded volume would have passed through cross sectional area A of the wire.
This cylindrical space has a volume of
If there are n electrons per unit volume, the number of electrons in this volume is
Since each electron carries a charge of , the amount of charge in that volume is
Since current is charge per unit time, the current is
In general, the current could also be carried by ions (besides electrons). To include currents carried by charged particles that carry charge q each, the formula is rewritten as
Charge Carrier Density
One insight we can gain from this formula is this. For the same current, the drift velocity is higher for an insulator than a conductor, since an insulator has much lower mobile charge density (aka concentration) than a conductor. This means a stronger electric field and thus potential difference is required to produce the same current. It is harder to push a current through an insulator because those fewer available charge carriers have to be pushed harder. This is the primary reason for the higher resistance in insulators.
Drift Velocity is Sloooow
Also, if you plug in typical numbers into the formula, you would realize that the drift velocity in all practical situations is extremely slow. While individual electrons are buzzing about randomly in all directions at hyper supersonic speed of the order of 106 m s-1, they drift collectively at a snail pace of the order of 10-4 m s-1. So it takes hours for an electron to trudge its way from a battery to a light bulb. Yet a light bulb lights up the moment the switch is closed! This is because we don’t have to wait for electrons from the battery to arrive at the light bulb. There is plenty of electrons already in the filament of the light bulb. The moment the switch is closed, the electric field is set up throughout the circuit immediately (in fact, this signal propagates at the speed of light), and the electrons throughout the circuit start drifting. The bulb lights up the moment the electrons in the filament starts drifting.