6.02 Current Density
Alright, we have introduced the electric current as the amount of charge passing through a surface per unit time. Since both charge and the time are scalar quantities, we concluded that the current is a scalar quantity. In other words it doesn’t have any directional properties. On the other hand, we said that current has a flow direction, and we define the conventional flow direction of current as from positive terminal towards the negative terminal.
Now, when we are dealing with a specific conductor, then we talk about the amount current passing through that conductor’s cross-sectional area as i. On the other hand, sometimes we might want to take a view and focus our attention on the flow of charge carriers at a particular point within a conductor. In order to do this, we will introduce the concept of current density.
We’re going to denote current density by capital J, and it is simply defined as current per unit area. Or, simply J is equal to i over A. Therefore, we’re looking at a specific surface and let’s say the amount of current i is flowing from left to right through this surface. The surface area is always perpendicular to the surface, and this current divided by that area simply gives us what we call the current density. If we consider the units, in SI unit system, since current unit is in amperes in SI system, and the area is in meters squared, therefore, amp per meter squared is the unit of current density.
From here, we know that the area is a vectorial quantity, which is perpendicular to the surface of interest and it has the magnitude of the surface area of the region. And, although current is a scalar quantity, whenever we divide or multiply any vectorial quantity by a scalar, we end up with a new vector, therefore, current density is a vectorial quantity. In other words, it has a specific direction.
Well, of course from this expression we can easily see that it is in the same direction with the area vector. Now, if we solve i from this equation, that’s going to be equal to J times A, but since these are vectorial quantities, it is actually equal to J dot A because the orientation of the area of interest is important as we are trying to define the amount of current passing through that surface. In other words, if I choose a surface such that it is aligned horizontal or parallel to the current direction, then the amount of current flowing through that area is going to be equal to 0. In order to take that into account, therefore, we say that I is equal to J dot A.
Now, if the current density is not constant, in that case, we take and incremental area element with surface area of dA along the surface of interest and calculate the amount of current passing through that surface, and then, just go ahead and do a similar type of process for the next area, and so on so forth. Eventually, then add all those currents to one another to be able to get the total current passing through that area of interest. So, in that case i becomes equal to integral of J dot dA. And, obviously the orientation of the surface relative to the current flow is going to be, then, determined through this dot product. In other words, this is equal to the projection of area vector along the current density vector.
Now, let’s see which direction that this current density is pointing. Since we have two different types of charges, positive and negative, let’s consider two different conducting mediums. One of them, the charge carriers are positively charged particles, +q, and we apply a potential difference between the two ends of this medium by connecting those ends to the positive and negative terminals of a power supply, which generates, let’s say, V volts of potential difference. So, whenever we do this, we generate a potential difference between the right hand side and the left hand side of this conducting medium. As soon as we turn the switch on, in other words as soon as we connect these ends to the terminals of that power supply, say battery, we will induce electric field pointing from right to left, according to this configuration.
So, the electric field is going to be pointing to the left from positive to negative terminal. So, as soon as we set up the electric field, then this electric field is going to exert coulomb force on each one of these charges. And, that force is equal to q times E, from Coulomb’s law. And, since the charges are positively charged, then the direction of this force is going to be pointing in the same direction with the electric field.
Under influence of this force, the charges are going to accelerate in the same direction with the electric field. But, as they move along this conducting medium, they’re not going to get faster and faster continuously because, as they move, they’re going to make collisions with the other charges and also with the atoms of the medium. At each collision they will stop for an instant of time and then they will start to move again. We’re going to denote the average velocity that they will acquire between the collisions as vd. And, we will call this velocity as the drift velocity.
In this configuration, then, the drift velocity is going to be pointing in the same direction with the electric field, that is, to the left. The current density vectors of these charges as they move from high potential regions towards a low potential region is in the same direction with the electric field, therefore, vector J, current density vector, is going to be pointing to the left.
Now, let’s consider another conducting medium, and in this case the charge carriers are negative charges. This can be, for example, the case for the metals. We know that in metal mediums the charge carriers are negatively charged electrons. Again, we apply a potential difference between the ends of this medium by connecting one end to the positive terminal, the other end to the negative terminal of a battery, therefore, generating V volts of potential difference between the two ends. And, as soon as also turning the switch on, again, we setup an electric field which is pointing from positive towards a negative terminal.
Under the influence of this electric field, again, each one of these charges, –q‘s, negative charges, are going to be under the influence of coulomb force generated by this external electric field and if we draw these vectors again, electric field is pointing to the left, and the coulomb force on each charge is going to be –qE. And, therefore, the force is going to be pointing in opposite direction to the direction of external electric field. Therefore, the charges are going to move in the opposite direction to the electric field, they will accelerate in that direction.
But, as in the case of positive charges, they will not get faster and faster and faster continuously. They’re going to make collisions, and at each collision, they will restart again. So, they will acquire an average velocity between the collisions, and again, we will call that velocity, drift velocity. In this case that is going to be pointing in opposite direction to the direction of electric field.
But, in this case, also, the current density vector is going to be pointing to the left. So, in other words, no matter what type of charge carriers that we’re dealing with, no matter what the sign of the charge of the carriers, the current density vector always points in the direction of the electric field.
So, if we make the note of these facts, we say that first, positive charges drift in the direction of applied electric field, and the negative charges drift in opposite direction to the applied electric field. Having said that, the direction of current density, vector J, and also, let’s say the sense of the current, i, were always in the same direction. In other words, they were in such a way that the charge carriers were all positive.
Here, the conventional current direction is going to be pointing to the left, sense of current flow, in other words, from positive to negative. And, again, despite the fact that the actual flow direction of charges here, for the negative charge carriers, from negative to positive, we will still choose the current flow direction as in the same case of positive charge carriers. In other words, again, from positive to negative.
So, we make a note over here, by saying that the direction of the current density J and the sense of current flow by convention as if the carriers were positive. So, in other words, J, current density J is in the same direction with applied electric field.
Alright, then we can say that the general relationship between current density and current is such that, for a particular surface, which does not need to be a plane, that cuts across a conductor, current is the flux of the current density vector J over that surface. In other words, i is equal to integral of J dot dA. Here, again, dA is an elemental surface area, and the integral is taken over the surface in question.
We can look at what the current density vectors represent through a mechanical system if we consider a constricted pipe, for example, something like this. And, if the fluid is flowing through this pipe from this high-pressure region towards the low pressure region, the velocity vectors, or the streamlines are actually the velocity vectors of this fluid, which is flowing through this constricted tube. Therefore, the streamlines here, the streamlines are the velocity vectors of the fluid, making a laminar flow through a constricted pipe.
So, the density of these lines, in other words, the number of lines passing through per unit area, simply indicates that the velocity is hiding those regions in comparing to the ones that the number of field lines passing through that surface, or the number of stream lines, I should say, passing through a surface is less. So, if you take a similar type of situation and now consider a constricted conductor such that the current is flowing through this conductor, in this case, streamlines are actually the current density vectors, streamlines, these are current density vectors representing flow of charge through a constricted conductor.