Transport equation.
In a conductor although millions of electrons pass a given point every second although the actual speed of the electrons is very slow. This seems a bit non-intuitive but it is true. If you imagine a great big wide corridor absolutely full of students and the corridor is so wide that there are 5 million people in each row and they are packed tightly together. Even if they were moving very slowly large numbers could pass a given point in a short time. Electrons are like that in a conductor. It is hard to imagine how many there are. If zillions was a word there would be zillions of them.
OK now think of a wire. It is long and thin and circular in cross sectional area. Its area let us call A. It’s length let us call l (that’s a lower case L) so its volume is A times l = Al (easy so far? Of course it is)
OK so now lets consider that some of the electrons in that conductor are bound to the lattice structure and some are free to move. How many are free to move? Who knows let’s call it n. That sounds good n free electrons in a volume Al. So the number of free electrons per unit volume is given by nAl. Easy.
Now lets think about charge. Each of these free electrons carries a charge of 1.6 x 19-19 Coulombs and we call that little e so the total charge on these free change carriers is nAle (the number of free charge carriers per unit volume x the charge on each charge carrier) We call that total charge Q so we now have Q = nAle
Still with me? If not go back and digest that bit again. Have a cup of tea and think about it.
Ok so you’re confident to move on I see.
Now you should know that the current in a conductor is given by the formula … a wait a minute I need to insert an equation here … hold on …
Ah .. that looks good … now where were we? Oh yes almost there now.
Now one of the very first formulas you learn in physics is speed = distance over time and if you look at that last equation you should see the length of the wire in there … remember little L and it is divided by time. So
This, my good friends, is the transport equation. If you know the current, the number of charge carriers the area of the conductor and the charge on the charge carrier then you can calculate the drift velocity.
The formula can be generalised from dealing only with electrons to dealing with any charge if you replace the e with a q to represent any charge.
I = nAvq
Here ends the transport equation. Hopefully you followed all that. Make sure you know and understand it. You do not need to derive it, simply identify its component parts and use it.
Mr S