6.03 Drift Speed
All right. We’re going to now introduce an important quantity associated with the electric current which is called “drift speed” or “drift velocity”. In order to obtain this quantity, let’s consider a piece of current-carrying wire. Let’s take a piece of wire and connect one end to the positive terminal and the other end to the negative terminal of a power supply. In doing that, we immediately generate a potential difference between the two ends of this wire and also immediately set up an electric field through the wire pointing from positive end towards the negative end.
Since we know that wire is a metal and it has an abundance of free electrons, therefore the charge carriers in this medium will be negatively charged electrons. Each one of them will have the fundamental charge of e. In other words, e magnitude represents the magnitude of electron charge. As soon as we set up the electric field by turning the switch on, this electric field will immediately exert coulomb force on each one of these electrons. That force, as you recall, will be equal to –e times E since the electron charge is –e. If you recall, the value of e was 1.6 times 10-19 coulombs.
Every electron under the influence of this force, therefore, will accelerate in opposite direction to direction of electric field. With this acceleration though, they are not going to gain a velocity which will continuously increase because they will make collisions with one another and as well as with the atoms of the medium. At each collision, they will lose some of their energy. We can visualize the situation such that they stop for an instant of time and then restart again. In other words, at every collision, they will stop for a moment and then they will start to move and gain a velocity until the next collision and then repeat the process.
As a result of that, they will require speed and this speed, as an average, will be constant throughout the medium. That is the speed what we call as “drift speed”. So since the force is acting in opposite direction to the electric field for a negatively charged carrier in this case, free electrons or the conduction electrons, therefore the force is to the left and so the drift velocity that they will gain as a result of this process is going to be pointing in opposite direction to the direction of the electric field.
Now, let’s concentrate along a specific length of this wire. Let’s call that length L such that along this length an electron enters into this region right after making its last collision, moves along this length and then makes the next collision at the end of this length. In other words, L is a distance such that as soon as one electrons enters by making its last collision, continues without making any other collision to the other end along this length and then makes the next collision. During this interval, the average velocity of the electron is vd.
Let’s say then L is the distance that the electron travels after making its last collision, let’s say in time Δt, until its next collision. Relative to those parameters then, vd, the drift speed becomes equal to distance traveled divided by the corresponding time, which is L over Δt. In terms of these quantities, we can represent this distance L as vd times Δt.
Here, let’s define another parameter. We will denote that by n. This is basically the number of charge carriers per unit volume and that is simply nothing but number density of the charge carriers.
Okay. What we’d like to do is now express the net amount of charge in this region of interest. In other words, along the segment of this wire, which has length L. Let’s also say that A represents the cross sectional area of this wire. So now we’re going to try to express the amount of charge inside of this region.
Okay. Since we know the number density of the charge carriers, in other words, number density of free electrons, or number of free electrons per unit volume, if we multiply that density by the volume of the region that we’re interested with, then we will get the number of charge carriers inside of this region. Since we know the charge of each carrier, which is e, or this much of coulombs, 1.6 times 10-19 coulombs, with a negative sign, therefore once we obtain the total number of carriers in this region, simply multiplying that quantity by the charge of a single carrier, then we’re going to be able to get that whole charge in that region.
Let’s say Δq represents the net charge inside of the region of interest. That charge is going to be equal to number density of the charge carriers, which is number of charges per unit volume, times the volume of the region of interest. Then, since this is going to give us the total number of charges in the region of interest, and if we multiply this whole quantity by the charge of a single carrier, then we will get that total charge.
Let’s express these quantities in explicit form. The number density of the charge carriers is n times the volume of the region of interest. If we go back to our diagram, the cross-sectional area is A, and the length is L, so the volume of this cylindrical region will be equal to A times L and the charge of a single carrier is e. I’m just using the magnitude of that charge. Of course, with the sign for the electrons, we have the negative sign over here, but our main interest is with the magnitude of the charge in that region of interest.
Since we express the L in terms of the drift speed as vd times Δt, we will substitute this here into this equation. Then Δq becomes equal to n times A times vd times Δt times e. Now let’s divide both sides by Δt. On the left hand side then we will have Δq over Δt is equal to n times A times vd. Once we divide the right hand side by Δt also that will cancel and we have just e over here. If we look at this expression, on the left-hand side we have change in charge with respect to time interval in which that is changing. In other words this is nothing but the definition of electric current, so i becomes equal to n times A times vd times the electron charge.
From here, if we solve for the drift speed, then we end up with an equation which is equal to current divided by number density of the charges times the cross sectional area of the wire times the electron charge magnitude. If we consider this ratio over here, which is current per unit area, and that is, by definition, nothing but the current density, we can also express the speed as, in terms of the current density, vd is equal to magnitude of the current density J divided by number density of the charges times the electron charge. Again, in terms of the current, that is equal to n times A times e. Therefore we obtain an expression for the drift velocity of the charge carriers in terms of the current density as well as the current.