Driving in my car for five hours today, I had plenty of time to think about velocity. There’s not much you can do about it, speeding doesn’t take much time off the trip but it does add the risk of an expensive ticket. We single classical particles don’t have a lot of options for getting from place to place very quickly.
The situation is a little more interesting with flows of current. If you want to fill a large bucket with water from a garden hose at one cubic foot per second, the hose is going to have to be turned on very high. Water will be flying out of the end of the hose at some preposterous speed. On the other hand, the Mississippi River can deliver more than half a million cubic feet of water per second with the water flowing at a walking pace. The power of the current doesn’t just depend on the speed of the flow.
Electricity is like that as well. 1 ampere of current is a fairly typical value for the flow through an incandescent light bulb. That much current corresponds to a flow of 6 million trillion electrons per second. It’s a lot, but electrons are very tiny. To find out how fast the electrons are moving, we need the following formula:
Where I’m using the slightly non-standard notation that I is the current in electrons per second, rho is the density of electrons in the wire, and A is the cross-sectional area of the wire.
But we don’t know what the density – how many conduction electrons per cubic meter – there are in the wire. It’s actually a fairly delicate experiment to find out, but it has been done. The density of conduction electrons in something like copper is huge. It’s on the order of 8e28 per cubic meter. Plugging that into our equation with some typical wire values gives a drift velocity of a fraction of a millimeter per second. The electrons in your wires are moving pretty slowly.
Then how does a light turn on so quickly when you hit the switch? Like the water in a garden hose, the wire is already filled with electrons. The pressure causes them to all start moving, and the ones already at the bulb being to be pushed through, causing light. It’s along the same lines as how there’s a delay between turning on the hose and the water coming out only if the hose starts off empty. If there was already water in it, the flow begins almost immediately.