6.04 Resistance and Resistivity
Let’s consider a mechanical example. Lets assume that we have a wooden board, something like this, and a bunch of nails embedded on it. Lets assume that we place a bunch of little BBs on this plate. When the plate is flat like this, parallel to the floor, then every point along this plate will have the same gravitational potential gravitational energy. If these BBs are at rest, initially, then the system will be in equilibrium and they will always be in rest position. If we try to determine, for example, a net flow of these BBs across a hypothetical line through this wooden board we are not able to determine any flow through this line.
Now, if we take this board and give it a kind of inclination from one end, in other words just lift it from one end of the board. Make it an incline plane something like this. In this case, we’ll generate a potential energy difference between the two ends of this wooden board. In other words, the left hand side will have certain height differences with respect to the ground in comparing to the right hand side.
So under this condition we naturally can guess that these BBs are going to roll down from high gravitational potential energy region to low gravitational potential energy region and then they will accelerate under the influence of the gravitational force component along the inclined plane. So if we just consider one of the BBs at the top we will see that it will be under the influence of the gravitational force. If we recall these forces, we are going to have mg in downward direction and we are going to have its component, which is parallel to the incline plane, this one. This force will cause the BB to move down along this inclined plane. The perpendicular compound components of the gravitational force with respect to the inclined plane will cancel with the normal force.
Under the influence of this force then, as this BB rolls down, it is going to make collisions with these nails. At each collision we can say that it will stop for a instant of time and then restart again. As it rolls down, instead of following a straight line of path, it is going to follow in a way a zig zig path due to these collisions. So at each collision we can infer that it is stopping for an instant of time and then restarting again and rolling down a zig zig path, something like this. Of course all the other BBs, also, will follow a simple type of motion.
If we match this case to the electrical case, a conducting medium, if we just consider a metal medium like a copper wire, originally the wire is filled with free electrons like these BBs. Of course in the case of mechanical example they were already addressed, but in the medium of copper its free electrons are moving in the random velocities proportional to the temperature of the medium in every possible direction. When we apply, of course, if I just re-draw my copper wire over here, something like this. So all these electrons inside of this medium, free electrons are moving in their random velocities. But since there is no potential difference between any two points in this medium we cannot detect a net electric current from point a to point b.
But if we take the same wire and connect its ends to a terminals of a power supply, a battery let us say, in that case, we simply generate a potential difference between the ends of that conducting medium. If the potential difference between the terminals of that power supply — this battery’s in V, volts — as soon as we turn the switch on, we will generate the same potential difference between the ends of this copper wire. Therefore, as soon as we turn the switch on, the induced, or set up an electric field which is going to be pointing from positive to negative end of this medium, and this electric field then immediately generates coulomb force on each one of these free electrons.
Let’s assume that, although we know that they are negatively charged electrons, but let’s assume that these are positively charged particles. So if we just follow the motion or track the motion of one of them, there is a positive, charged particle which feels the coulomb force in the same direction of the electric field, they will start to move. As they move they will make collisions with the other free electrons or the other charges in the medium or the atoms of the medium. Therefore, instead of following a straight line of path it will follow a zig zig path due to these collisions, which is, in a way, similar to the BB that will follow a path as it rolls down along this inclined plane with these embedded nails.
Here, the nails represents the atoms of the medium or the other charges in the medium and in this case also, as the electrons or the charge carriers are drifting, while making collisions with the other ones, that they stop at each collision for an instant of time and then start again. So they acquire these average speed which we have introduced that speed as drift speed.
The interesting point over here, is when we do this experiment with different materials, like a piece of copper wire and a piece of glass, for example, then the amount of current that we’re going to be drawing from, the same potential difference will change. As a matter fact, we’ll see that the more the collisions that these particles are making, the lesser the current is going to be flowing through this medium. In other words, if we have many, many more nails over here and then we try to determine the net flow of these BBs on this hypothetical line once we introduce this inclination, then we are going to determine lesser numbers are flowing through this line for a given time interval than if we have a case where we have a lesser number of nails embedded in the wooden plate.
So the amount of current, therefore, flowing through the medium for a given potential difference will depend on the characteristics of that medium. Obviously the number of collisions that the charge carriers are making as they drift from a high potential region to a low potential region will be directly dependent or determine the amount of current that we are going to end up throughout that medium. In order to take that quantity into account associated with the current we will introduce the concept of resistance.
We are going to denote resistance by capital R. And the resistance is defined as the potential difference between two points divided by the current flowing through those points. So in symbolic form then, R is equal to V over i. If we look at the units of resistance, the units of potential difference in SI units system is in volts and the unit of current is in amperes. Therefore volt per ampere is the unit of resistance in SI units system and we have a special name for this ratio. It is called “ohm” and we are going to use capital Greek letter, Ω, in order to represent unit ohm.
Now, sometimes instead of dealing with a particular object or with a particular component in the electrical circuit, again, as in the case of current density, we might be interested with the properties of the medium. In that case, in order to define or take the collisions that the charge carriers are making as they drift in that medium, we will introduce the concept of resistivity, which we are going to denote this by ρ, and here you have to be careful not to confuse with the volume charge density because, well, we have used the same symbol for that quantity.
The resistivity is defined as the ratio of electric field in the medium to the current density of the medium. So ρ is equal to E over J. And if we look at the units of resistivity in SI unit system, since the electric field is volt per meter, or newtons per meter, and the current density is current per unit area, which is amps per meter squared, that will be equal to volt per meter times meter squared per amp. One of these meters will cancel. We have just introduced that the volt per amp is nothing but the unit of resistance, which is ohms, then the unit of resistivity becomes equal to ohm meter in SI unit system.
Now, let’s also introduce the symbol that we’ll use for the resistance in electric circuits by recalling the other symbols that we have used so far. This symbol represented the power supply, which we also call it as “electromotive force” or power supply, and we have used this symbol in order to represent electric switch, a two-way switch. We used straight lines to represent wires in the electric circuit. And we used this symbol to represent a capacitor and now we have introduced the concept of resistance and we are going to represent a resistor with a resistance of R with this symbol.
Now if we, therefore, recall that resistance, this is directly associated with the number of collisions that the charge carriers are making as they drift from high potential region to low potential region as the ratio of the potential difference between two points to the amount of current passing through those points, and the resistivity as the electrical field in the medium divided by the current density of the medium at that region. These two quantities, therefore, in a way, related to one another and we will see the relationship between them in a moment.
Now, naturally the resistivity is also associated with the number of collisions that the charge carriers are making in the medium as they drift from one point to another point. The inverse of this quantity, we’ll denote that by σ, which is called conductivity, and that is equal to 1 over ρ. Therefore it will have the units of 1 over ohm meter is associated with the fact that how easy that the charges can move or can be conducted in a way in the medium of interest. So the greater the resistivity means the harder the charges can move around in that region of the medium and the greater the conductivity means how easy, actually, it can move in that region from point a to point b.