Example- Drift Speed
Alright. Let’s do an example related to the drift speed.
Let’s try to determine the drift speed of the conduction electrons, or the free electrons, in a copper wire. Let’s say “What is the drift speed of the conduction electrons in a copper wire, say, with radius r is equal to 900 micrometer when it has a uniform current i, which is equal to 17 milliamps?”
We will assume that each copper atom contributes one conduction electron to the current, and the current density is uniform across the wire’s cross section. So, let’s say assume that current density j is constant through the cross sectional area of the wire and a number of atoms is equal to number of conduction electrons. In other words, the free electrons. So, each atom contributes one free electron to the medium.
Alright. Now if we recall that we have defined the current density as the current per unit cross sectional area, and we have also shown that the drift speed is related to the current density as j divided by number density of the charge carriers times the charge of a single, magnitude of the charge of a single electron, or the charge of the charge carrier, which can also be expressed as i, the current, divided by n times a time e.
Here in the problem we’re given the current flowing through the copper wire, we know what the magnitude of the electron charge is, and that is equal to 1.6 times 10 to the -19 Coulombs, and since we know the radius of the wire, which is 900 micrometer, we can easily calculate the cross sectional area of this wire.
Therefore, the only quantity that we have to deal with over here, or determine, is the number density of charge carriers.
Number density of charge carriers is defined as n is equal to number of charge carriers per unit volume. Since we have assumed that the number of charge carriers, in this case, free electrons, or conduction electrons, equal to number of atoms, this quantity also becomes equal to number of atoms per unit volume.
Let’s try to obtain this quantity. In order to do this we will take advantage of Avagadro’s number, and as you recall, this number gives us the atoms per mole.
So, we can express n as atoms, number of atoms per mole, which is the Avagadro’s number, times moles per unit mass, and this quantity is nothing but the inverse of the molar mass of the object, in this case, copper, and times mass per unit volume, and that is nothing but the density of the object, or, in other words, the density of copper.
If we look at over here, the moles will cancel, and the masses will cancel, and as a result of this product we’re going to end up with atoms per unit volume, and that is nothing but the quantity that we’re trying to determine, and the number of atoms per unit volume, eventually, will be equal to the number of charge carriers, in this case, the number of conduction electrons, or free electrons per unit volume.
Well, atoms per mole is the Avagadro’s number. Let’s denote that by n sub a. N sub as is the Avagadro’s number, and that is equal to 6.02 times 10 to the 23 number of atoms per unit mole.
And let’s denote moles per unit mass, which is the inverse of the molecular, or molar mass, and let’s denote the molar mass with capital n and introduce subscript of cu for copper, and this quantity can be obtained from the periodic table, and that is equal to 63.54 times 10 to the -3 kg per mole.
And, moving on, the last term is nothing but the density, let’s denote that by [inaudible 09:08] sub cu for copper, and [inaudible 09:12] sub cu is equal to, again, from the periodic table, that is 8.96 times 10 to the 3 kilograms per meter cube.
Okay. Now from here if we calculate the number density of atoms per unit volume and hence the number of charge carriers per unit volume, that’s going to be equal to Avagadro’s number. Let me just write down these in symbolic form first, n sub a times 1 over M sub cu, times [inaudible 10:08] sub cu, and substituting these quantities over here, we will have 6.02 times 10 to the 23 number per mole times 8.96 times 10 to the 3 kilograms per meter cube, divided by the molar mass of copper, which is 63.54 times 10 to the -3 kilograms per mole.
And, again, if we look at the units over here, we see that this mole and that mole will cancel and the kilograms will cancel and we’re going to end up with the number per meter cubed is nothing but number of charge carriers per unit volume.
And if we do this calculation, n will turn out to be 8.49 times n to the 28 number per meter cubed.
Alright. Once we determine the number density, then we can easily determine the drift velocity. Drift velocity was equal to current divided by number density of charge carriers times the cross sectional area of the conductor, times the magnitude of the charge in a single carrier, and since the copper wire is in cylindrical form, we know the radius. We can easily write this in more explicit form as i over n times pir squared, times the magnitude of the electron charge.
Substituting the associated values for these quantities, the current is 17 times 10 to the -3 amps divided by 8.49 times 10 to the 28 number per meter cubed for the number density.
And if we do this calculation, pir square, we will end up with 2.54 times 10 to the -6 meters square, and finally, the magnitude of the electron charge is 1.6 times 10 to the -19 Coulombs.
If we do the calculation, then the drift velocity of these conduction electrons will turn out to be 4.9 times 10 to the -7 meters per second, which is also equal to 1.8 millimeters per hour.
Therefore, this is a very, very low speed. It’s actually slower than the speed of a snail, and by looking at this quantity, one can easily ask the question “How come we can immediately see the light emitted from the light bulb as soon as we turn the switch on in a typical electric circuit?”
The answer to that question can be explained in terms of a mechanical example. Let us assume that we have a hose, cylindrical hose, and let me draw it this way, it is connected at one side to the faucet, and it’s our experience that if the hose is originally empty, if there isn’t any water inside of the hose, when we turn the faucet on one side of the hose, we wait for a while until the water starts to flow from the other side of the hose.
On the other hand, if we fill the hose, initially with water, then, when we turn the faucet on on one side of the hose we will almost simultaneously observe that the water will flow on the one side of the hose.
As a matter of fact, the pressure wave that we generate by turning the water on on the faucet side of the hose will propagate throughout the medium of water at the speed of sound throughout this medium. As a result of that, we observe that the water is coming out from the other end almost instantaneously.
In a similar way, when we look at this copper wire, we see that it is filled with free electrons, like the hose, like the water fills the whole hose.
So, in this case, what it counts is not that this electron, which is located at one end of this wire, let’s say, just next to the switch, will start to move and drift in order to reach to the other end, the major quantity over here is that the order that each electron will get ot move, to start to drift, in a specific direction, and that order is set by the electric field, and the electric field is set throughout this medium at the speed of light.
As soon as we turn the switch on, if this end is connected to the positive terminal and the other end is connected to the negative terminal of a power supply, as soon as we turn the switch on we set the electric field throughout this medium at the speed of light.
And this electric field, therefore, immediately exerts Coulomb’s force on each one of these free electrons, and as a result of this force they all start to drift, since they are negatively charged particles in opposite direction to the direction of this electric field.
So, in other words, the first electron next to the switch, and the electron just next to the light bulb will start to move almost instantaneously, and as a result of that, once we turn the switch on, we will see the light emitted from the bulb almost instantaneously.
Now, the drift velocity of the electrons as they drift throughout a conducting medium, like in this case of copper wire, despite the fact that they’re very, very slow, like in this example, 1.8 millimeters per hour, in other words, it’s going to take hours, for example, to travel several meters of distance. Their random velocity is much, much faster than these. We know that these electrons move inside of this conducting medium at their random velocities, and that velocity at room temperature is about on the order of 10.6 meters per second. So they’re continuously moving around making collisions with one another and etc., but once we apply a potential difference between two points in that medium, then, still, as they move at their random velocities they start to drift at very, very low speeds like this.
Another mechanical way of explaining drift speed is, can be done by looking at a very crowded sidewalk from the top view. Let’s say, if we consider a sidewalk from the top view, something like this, and if we mark people that they’re going in upward direction, let’s say, with dots, that they’re moving upward, and the ones that they’re moving downward direction with cross signs, like this, when we look at it from the top view and try to determine net number of people passing through a specific direction, either up or down, along a hypothetical line across the sidewalk, for a given time in [inaudible 21:25] we’re not going to be able to detect a net movement of people because we have so many people that some of them are moving in downward directions, some of them are moving in upward directions. Eventually those quantities will cancel one another as an average and we will now determine a net movement of people through this line, either up or down.
And here we will represent these motion of the people up and down along this one dimensional system in a way like the random motion of electrons, free electrons, in a wire, in this copper wire, for example.
On the other hand, if we just take the same sidewalk, now looking from the front view, and give some inclination, in other words, lift the sidewalk from one side, like this, and in this case, again, some people are moving into the plane direction, let’s say those are the dots, and some people are moving outside of plane direction along this sidewalk, and when we look at the same sidewalk from the top view, this tilted sidewalk now, as the people move up and down directions, in both possible directions, we’re going to detect a net shift of these people from this high potential region towards the low potential region. In other words, we’re going to detect a drift speed, or drift velocity, pointing from high gravitational potential energy region, towards the low gravitational energy region.
And, again, the people are moving in these two possible directions, up and down, but as they do this, as they’re doing away their random motion along this one dimensional case, they will also drift from the high potential towards the low potential region.
It is this velocity, in a way, we can match to the drift velocity of the electrons, because introducing a potential gravitational energy difference is equivalent to introducing a potential difference between two points in a conducting medium, and it is this drift velocity in the mechanical system, is, in a way, matches to the case of the drift velocity of the electrons as they drift throughout, through the high potential region towards the low potential region.