So far we have considered the cases that the electric charges were either at rest or they were moving at very low velocities relative to one another. We called this part of the electrical interactions as “electrostatic”. Now, we’re going to start to consider the cases that the charges are in motion, which will bring us to the concept of electric current.
Let’s consider a copper wire placed in the shape of a rectangle loop. We know that copper is a metal and therefore it has abundance of free electrons, which are free to move under the influence of any force, and it is a good electrical conductor because as these free electrons move from one point to another point, they carry the charge with themselves because they’re negatively charged fundamental particles.
So if we look at such a medium and let’s say these little blue dots represent the free electrons in this medium. Of course these free electrons are moving at the random velocities inside of this region. In a way, their speed is proportional to the temperature of the medium. Let’s assume that we’d like to determine the number of charges or amount of charge passing through the cross-sectional area of this wire at a specific point.
Let’s say we’ll look through this cross sectional area and try to figure out if there is a net movement of charges from one side of the surface to the other side. If we do this experiment for a specific time interval, we will record 0. In other words, no net amount of charge is going to be passing through the surface either to the right or left. The reason for that is even though these free electrons are moving at their random velocities in this medium, for a given time interval the number moving to the right will cancel the number moving to the left, and we will just not determine a specific net amount of charge passing through the surface per unit of time.
Now let’s take the same wire, and in this case, cut it at a certain point and place a power supply, a battery, between the ends of that wire. In other words, now, let’s take our same wire and cut it at this point. Of course I’m drawing the thickness of this wire in an exaggerated way so we can get the feeling of what is taking place and connect these ends to the terminals of our, let’s say, battery. Let’s assume that this terminal is the positive one and this terminal is the negative one. Here we have again our free electrons that they’re moving at their random velocities in this medium.
Well, as soon as we turn the switch on and connect this battery to the ends of this wire, we’re going to generate certain potential difference between these two ends of the wire. Because we know that the potential associated with the positive charge so the end connected to the positive terminal of the power supply will have a greater potential than the one which is connected to the negative terminal of the battery because the potential associated with negative point charge is smaller than the positive charge. One of them is positive, the other one is negative.
Therefore, if the power supply is generating the volts of potential difference, you can take our volts meter and read the potential difference between the ends of this wire. We will read the same voltage, in other words, same potential difference.
Furthermore, doing this, in other words, connecting these two ends to the terminals of the power supply, therefore generating the potential difference, we will immediately set up an electrical field through this wire pointing from positive terminal towards the negative terminal because we know that the field lines emerge from the positive and enters into the negative one. So we’re going to introduce a net electric field through this wire, which is going to be pointing from positive end towards the negative end.
Now, we have an external electric field, induced into this medium, and then we have these charged particles. We know that then, from Coulomb’s Law, that this external electric field, is going to exert a net force on these particles. Therefore, they’re going to start to oscillate or move under the influence of this force. Now, we’re talking about negative charges, which is the case for a metal medium, these free electrons. That force is going to be equal to minus q times E. As you recall, force is equal to charge times the electric field. If the charge is positive, that force is in the same direction, with the direction of the electric field. If the charge is negative, then that force is going to be in the opposite direction to the direction of the electric field.
So, if these were therefore free electrons, then each one of these electrons will be under the influence of the net force, which is going to be acting in opposite direction to the direction of the electric field. That will be the Coulomb force acting on these electrons. If you call the electron charge magnitude, e, as e, and we know that it is 1.6 times 10-19 Coulombs in SI unit system. Therefore, the force is going to be equal to –e times the electric field external electric field. And due to the sign of the charge it will be in opposite direction of the external electric field. Under the influence of this force therefore, these free electrons will move in a way from negative to positive terminal direction.
In this case, if we do the same experiment that we did for the previous case, now we’re going to be able to determine net amount of charge movement through the cross sectional area of this wire for a given time interval. As a matter of fact, if we just consider the magnitude of these charges, and if we say that Δq is the amount of charge passing through cross-sectional area of the wire in time interval of Δt seconds, we will see that this amount of charge that passing through the cross-sectional area of this wire is going to be directly proportional to the amount of time that we’re waiting or we’re observing.
In other words, Δq is going to be directly proportional to the time interval. The longer that we wait and count the amount of charge passing through this surface will be greater. If we do this experiment over and over again, we can find a proportionality constant in order to express this proportionality in the form of an equation. It turns out to be that that Δq is going to be equal to some proportionality constant, let’s call that i times Δt. Here, i is the proportionality constant and if you leave i alone in this equation, that’s going to be equal to Δq over Δt. In other words, the amount of charge passing through the cross sectional area of this wire per unit time.
We have a special name for this proportionality constant. It is called electric current. We can express this relationship in a form such that by taking the limit of Δt as it goes to 0 of Δq over Δt. In other words, by making the Δt interval smaller and smaller, then of course the amount of charge passing through that surface during that interval will get smaller and smaller. If you recall from calculus, this is nothing but the derivative of charge with respect to time. In other words, change of charge with respect to time.
That is the definition of electric current: the amount of charge passing through a surface per unit time and that surface, in general, the cross sectional area of the conducting medium. If you look at the units of this quantity, since charge in SI unit system is in Coulombs, and the unit of time is in seconds. So the Coulombs per second is the unit of electric current. And we have a special name for this ratio. It is called Ampere.
If we look at the definition of electric current, we see that the charge is a scalar quantity. In other words, it doesn’t have any directional properties. Time is a scalar quantity. It also does not have any directional properties. Whenever divide any scalar by another scalar, we always end up with a scalar. Therefore, current is a scalar quantity. In other words, it does not have any directional properties.
Having said that, we have a direction associated with the flow of charges. As a matter of fact, when we look at our example over here, since for a given metal medium like copper wire that we considered over here, the charge carriers are negatively charged free electrons because the atoms of the medium, they’re much much heavier than the electrons. As a result of that, we can assume that they almost remain at their fixed positions. But the free electrons with much, much less mass, that they move around on the [inaudible 16:44] of Coulomb force. We see that the direction of flow of these electrons is actually, in a way, in the opposite direction to the electric field, and it is therefore from the negative end of this wire toward the positive end.
But when the electromagnetic theory was originally being developed, it was assumed that the charge carriers were the positive charges. Therefore, the theory was established relative to the motion of the positive charges. And at that time it was established and assumed that the direction of flow of electric current is from positive towards the negative terminal of the power supply, which generates the electrical potential energy to the circuit. So because of that reason, the conventional direction of flow of current has been chosen as from positive to negative rather than negative to positive, which is the actual case for the metals, because the charge carriers in those cases are negatively charged free electrons. So we have to keep that in mind.
But then as conventional direction of flow of current will always choose the direction as pointing from positive to negative end. Let’s make a note of that. Conventional flow direction of current is from positive terminal to negative terminal, despite the fact that the actual direction of flow of charges or current for the metals is just the opposite since the charge carriers are the negatively charged free electrons in metal mediums. So current itself is a scalar quantity, but we have a flow direction associated with the electric current.
I can give you a simple example relating to these facts. If you consider a piece of wire, let’s say something like this, and it branches out in to two branches like this at this point, now, if current i is flowing from left to right, whenever it comes to this point some fraction of that current will go through this first branch, let’s say as i1, and the remaining fraction will go through this branch as i2. Of course from conservational charge principle, this i entering into this point, which we will call that point as “junction point”, will be equal to the sum of the amount of current passing coming out from that point. In other words, i will be equal to i1 plus i2.
If this were a vector quantity, as you know, to be able to verify that, we had to introduce a coordinate system and resolve these into their components and so on and so forth. But in this case that is completely irrelevant because instead of this shape of wire if we had a wire of something like this for example. Again, we have our junction point over here, and if the current is coming in to this point as i, again, some fraction will go through the first one as i1 and the remaining fraction will go through the second one as i2. In both cases, therefore, i will be equal to i1 plus i2. And as well as here, i is equal to i1 plus i2. Irrelevant of the flow of direction of these currents. Of course it will be determined by the shape of the wire as the charges flow through that medium.
So you should always remember that whenever you’re dealing with current, you’re dealing with a scalar quantity. It does not have a directional property. But on the other hand it does have a flow direction, which is always from positive terminal towards the negative terminal.