Lesson Video: Resistance and Resistivity of Conductors Physics

Video Transcript

In this video, we will learn how the material that makes up a conductor determines its resistivity, mostly due to the structure of its atomic lattice. We will also learn how, in addition to the resistivity of the material, the physical dimensions of the conductor, like its length and cross-sectional area, also affect its resistance. We will first derive a formula that expresses the resistance in terms of the resistivity length and cross-sectional area and then explain the physics behind each of these quantities.

Let’s consider an empirical definition for resistance that will be very helpful later when we consider electrons moving around inside of the material. We start with some block of material and we want to measure its resistance. We start by attaching both ends to a voltage source, say a battery, so that there is now a potential difference across the block. The battery will have a fixed voltage. Let’s call it 𝑉. With the battery putting a potential difference across our material, we’ll now introduce an ammeter to measure the current. We’ll call the current 𝐼, and it is a value that we measure as a function of the voltage. We now define the resistance of our material to be the ratio of the known voltage 𝑉 to the measured current 𝐼.

For uniform materials, this ratio will have a fixed value regardless of the particular value of the voltage. It turns out this definition will actually be very helpful to us in understanding the physics in our later discussions. For now, though, it gives us an experimental way to measure the resistance. So let’s use it to determine which properties of a block of material contribute to that block’s resistance. We can actually determine all of the relevant factors for the time being with six very similar experiments. Our measurement setup will be almost identical to what we had before, but with the role of current and voltage reversed.

This time we’ll pass a fixed current 𝐼 through a block of material and measure the resulting potential drop across the block, 𝑉. The length of our block measured parallel to the direction of current will be lowercase 𝑙. And the cross-sectional area of the block, measured perpendicular to the direction of current, will be lowercase 𝑎. We’ll also consider two almost-identical setups, but where the block has different dimensions. In one setup, the block has twice the length but the same cross-sectional area as the original setup. And in the other setup, the block has the same length but twice the cross-sectional area. Finally, to make a total of six experiments, we’ll perform each of these measurements on two different materials, which we’ve called material one and material two.

For the first setup, which we can think of as our baseline, we’ll measure a resistance of 𝑅 one for material one and 𝑅 two for material two. The particular values of these resistances are not important. But what is important is that they are different. We’ll come back to this difference after we gather the data from the other two experimental setups. For the second setup where the block is twice as long, the measured resistance doubles for both materials. On the other hand, in the third setup where the cross-sectional area is doubled, we find that the resistance for both materials is halved. Because doubling the cross-sectional area leads to half the resistance regardless of the material, we know that the resistance of the block must be inversely proportional to the cross-sectional area.

Similarly, if doubling the length leads to double the resistance regardless of the material, the resistance of the block must be directly proportional to its length. Finally, because we measured two different baselines for the two materials but found the same functional dependence on the length and the cross-sectional area of the block, the resistance must also depend in some multiplicative way on the particular material we’re using. We’ll call this dependence the resistivity of the material, and we’ll represent it with the Greek letter 𝜌.

Because we have defined resistance as directly proportional to resistivity, objects with larger resistivities have larger resistances. Materials with very low intrinsic resistivity like gold and copper are conductors, while materials with very high intrinsic resistivity like glass or many plastics are insulators. Okay, let’s now combine these three dependences into a single formula. We have that the resistance of a block of material is equal to the resistivity of the material times the length of the block divided by the block’s cross-sectional area. This formula allows us to directly calculate resistance without having to resort to a current–voltage measurement.

We’re now going to investigate the physical basis for this formula. That is, we’re going to try and understand where resistivity comes from and why longer lengths are associated with larger resistances and larger cross-sectional areas are associated with smaller resistances. It is important to stress that in the coming discussion, we will be sticking almost exclusively to a classical description of electrons moving around inside of solids. Because a full picture requires some advanced quantum mechanics, there will be points in the coming discussion where we do not and indeed cannot give a fully satisfactory classical explanation. Anyway, let’s start with the resistivity to get a good idea of the kind of classical picture that we’ll be using.

Resistivity is a material’s intrinsic opposition to the flow of charge, that is, current. Typical units for resistivity are ohm meters so that when we multiply by a length measured in meters and divide by an area measured in meters squared, we are left with ohms, which is a unit of resistance. Values for resistivity at room temperature actually range over more than 30 orders of magnitude. Resistivities between 10 to the negative eighth and 10 to the negative sixth ohm meters are typical of many metals and other good conductors, while resistivities between 10 to the 10th and 10 to 25th ohm meters are typical of good insulators. Of course, there are many materials with resistivity in between these two ranges such as many semiconductors which form the basis of most computing technology.

All right now, let’s look at how charge might actually flow through a material to understand how the material can oppose that flow. We’ll model a typical solid as a lattice of atoms, that is, a regularly spaced array of nuclei surrounded by electrons. In this picture, we’ve represented the nuclei by larger red dots and the electrons by smaller blue dots. Most of the electrons are actually bound quite closely to the nuclei, but some of the electrons, the ones that we’ve drawn here, are free to move around inside of the lattice. In conductors, there’re typically many such free electrons, often several per atom like we’ve drawn in our picture. For semiconductors, specifically around room temperature, there are still some free electrons but far fewer than there are in a typical conductor.

In insulators around room temperature, there are almost no free electrons, perhaps only one for every several atoms. To understand why the number of free electrons make such a difference, recall our original experimental definition for resistance. Resistance is the constant ratio of the applied voltage across a block of material to the current measured through that material. But current itself is just the amount of charge flowing divided by the time it takes for that charge to flow. This means that more charge flowing in the same amount of time leads to a larger current. But a larger current with the same applied voltage means a smaller resistance.

Now, when left alone, the free electrons in the lattice are moving mostly in random directions. However, when we connect a voltage source across the lattice, the resulting electric field causes the electrons to, on average, start moving in the same direction. But a number of electrons, all moving in roughly the same direction, is precisely the net flow of charge with time that defines a current. And of course, the more free electrons there are in the lattice, the more electrons there are moving in the same general direction. And so we have more charge flowing in the same amount of time, in other words, a larger current for the same applied voltage and thus a smaller resistance.

So the resistivity of the material depends inversely on the free-electron density because materials with a higher free-electron density pack more free electrons into the same unit volume, allowing more charge to flow for the same applied voltage. The other factor that goes into determining the current for an applied voltage is the time that it takes for the charge to flow. Classically, the current is made up of electrons that leave the negative terminal of the battery, travel around the circuit, that is, across the lattice, and then reenter the positive terminal of the battery. So the time that we are interested in when calculating the current is the time that it takes a single electron to move across the entire lattice.

To understand how this depends on the particular structure of the atomic lattice, let’s track the motion of one electron as it moves. We’ll follow the motion of this particular electron as it moves across the lattice in the general direction opposite to the applied electric field. Instead of traversing the lattice in a straight line from left to right, the electron bounced around several times, colliding with the nuclei in the lattice. Because of these collisions, the electron traveled a longer distance and thus took a longer time to traverse the lattice than it would have if it had traveled in just a straight line. In fact, the higher the frequency of collisions, the longer it takes electrons to cross the lattice.

But the longer that it takes electrons to cross the lattice, the larger the 𝑡 is in our expression for the current. But that means that the current is smaller for the same applied voltage, and thus the resistance is larger. So the resistivity of the material depends directly on the collision frequency of its electrons because the higher the collision frequency, the more time it takes for electrons to move through the lattice and thus the larger the resistance. To get a feel for this idea, let’s consider two examples of atomic lattices where we would expect a larger collision frequency than the regular lattice that we’ve drawn.

In this lattice, we’ve represented the presence of several impurity atoms with larger red dots. These impurities could occur naturally or be added intentionally in the process of making an alloy. In any case, the presence of these impurities disrupts the regularity of the lattice, resulting in more electron collisions. The impurities caused the electron to take a much more roundabout path across the lattice, resulting in a much longer time to cross the lattice, hence lower current and higher resistance. The same sort of collisions can happen when the nuclei in a regular lattice are displaced significantly from their usual positions.

The displaced atoms then act similarly to the impurity atoms when scattering electrons. These displacements occur because the nuclei have enough energy to move from their usual positions and the nuclei have more energy at higher temperatures. So at higher temperatures, we would expect to see more of these displacements. This leads us directly into the last factor that we’ll consider when discussing resistivity, and that is that the resistivity of a material always depends on its temperature. But the effect is opposite for conductors compared with semiconductors and insulators.

In addition to more displacements, there are also more free electrons at higher temperatures because the electrons also have more energy and so could more easily escape the confines of the nucleii. This presents us with two competing processes. More displacements means a higher collision frequency, which would tend to increase the resistivity of a material. But more free electrons means a larger free-electron density, which would tend to decrease the resistivity of the material. So which is it? Does resistivity increase or decrease with increasing temperature? It turns out that it depends on whether our material is a conductor or a semiconductor or insulator.

Because conductors have such a large free-electron density even at the absolute zero of temperature, there is not much increase in this density even as temperature increases. As a result, the atomic displacements play a much bigger role, and the resistivity of conductors tends to increase as temperature increases. On the other hand, although semiconductors and insulators have different free-electron densities at room temperature, at the absolute zero of temperature, both semiconductors and insulators have no free electrons at all. This means that the effect of the increasing free-electron density far outstrips the effects of the increasing number of atomic displacements.

This is because we go from no free electrons at all, in other words, a perfect insulator, to something with enough free electrons to have a measurable current. So the resistivity of semiconductors and insulators tends to decrease with increasing temperature. In fact, these opposite relationships are one of the ways to tell conductors apart from semiconductors and insulators.

Okay, now that we’ve seen how resistivity arises at an atomic scale, let’s extend some of these ideas to the length and cross-sectional area of a block of material. Here again, we have a regular atomic lattice hooked up to a voltage source, which causes the electrons to move across the lattice. As we saw before, the resistivity of the material is determined by the collision frequency and the electron density of the lattice. The electron density determines the charge flowing through the lattice, and the collision frequency determines how long it takes that charge to flow. Both of these in turn determine the current at the given voltage and thus determine the resistance.

There is, however, another way to increase the total charge flowing through the lattice and also another way to increase the total time it takes the charge to move from one end of the lattice to the other. Here are two more lattices of the same material with the same potential difference across them. In this lattice, we’ve doubled the dimension that is parallel to the direction of current. In other words, we’ve doubled the length. On the other hand, in this lattice, we’ve doubled the size of the dimension that is perpendicular to the direction of the current. In other words, we’ve doubled the cross-sectional area.

Let’s now see what happens as the electrons move across these lattices. In the lattice that is twice as long, we would expect the electrons to undergo twice as many collisions. So because the electrons will have to travel twice as far with the same collision frequency, the time to cross the lattice will be approximately doubled. But as we saw with resistivity, doubling the time means doubling the resistance, which confirms physically what we already knew experimentally that resistance is directly proportional to the length of the block of material. It’s worth mentioning that the number of free electrons available to make the full trip across the lattice hasn’t changed, which means the total charge moving from end to end is the same.

For the lattice with twice the cross-sectional area, the time it takes each electron to cross the lattice will be approximately the same as it was for the original lattice. However, there are now twice as many free electrons available to make the full trip across the lattice. With twice as many electrons crossing the lattice in the same time, the charge per unit time is doubled which, as we know, halves the resistance. This again confirms physically what we already knew experimentally that the resistance of a block of material is inversely proportional to its cross-sectional area. Now that we understand the physical basis for a formula, let’s actually use it to calculate the resistance of a piece of copper.

A copper wire is 2.5 meters long and has a cross-sectional area of 1.25 times 10 to the negative fifth meter squared. Find the resistance of the wire. Use 1.7 times 10 to the negative eighth ohm meters for the resistivity of copper.

We’re asked to find a resistance, and we’re given a length, a cross-sectional area, and a resistivity. Recall that we have a formula that relates these four quantities. The resistance of an object, that is, the ratio of an applied voltage to the current through the object, is equal to the resistivity of the material making up the object, that is, its intrinsic opposition to charge flow, times the length of the object divided by the object’s cross-sectional area. Since we are given the length, resistivity, and cross-sectional area, all we need to do to find the resistance is plug in values.

So we have that the resistance is 1.7 times 10 to the negative eighth ohm meters times 2.5 meters divided by 1.25 times 10 to the negative fifth meter squared. 1.7 times 10 to the negative eighth times 2.5 divided by 1.25 times 10 to the negative fifth is 3.4 times 10 to the negative third. For the units, meters times meters in the numerator divided by meter squared in the denominator is just one. And so we’re left with ohms, units of resistance, which is what we’re looking for. To simplify our result a little bit, recall that 10 to the negative third ohms is just one milliohm, so the resistance of the copper wire is 3.4 milliohms. It’s important to recognize the difference between mΩ, which is milliohms, a unit of resistance equal to one one thousandth of an ohm, and Ω times m, which is ohm meters, a unit for resistivity.

Alright, let’s now review what we’ve learned about resistance and resistivity. In this video, we learned how to calculate the resistance of a block of material from the resistivity, a property of a particular material in a particular temperature, as well as the length of the block of material and the block’s cross-sectional area. We also understood the dependencies in this formula on the basis of free electrons moving around inside an atomic lattice. Resistance increases with increasing length because it takes the electrons longer to cross the lattice. Resistance decreases with increasing cross-sectional area because, with increasing cross-sectional area, there are more electrons available to cross the lattice in the same amount of time.

The resistivity is a numerical factor for each material, and it depends primarily on the free-electron density and the structure of the atomic lattice. Finally, we also learned that the resistivity of all materials changes with temperature. For conductors, an increase in temperature results in an increase in resistivity. For semiconductors and insulators, an increase in temperature results in a decrease in resistivity. This is because as temperature increases, the free-electron density increases, which tends to decrease the resistivity. At the same time, though, the increased temperature leads to more lattice deformations, which leads to a higher collision frequency and tends to increase the resistivity.

For conductors, the free-electron density is so large, to begin with, that the increased lattice deformations having much more pronounced effect, resulting in an increased resistivity with increasing temperature. For semiconductors and insulators, the free-electron density is zero at the absolute zero of temperature, so the increase in free-electron density with increasing temperature has a much more significant effect than the increased lattice deformations, and the result is a decreasing resistivity with increasing temperature. In fact, measuring how the resistance of a block of material changes with temperature is one of the ways to distinguish between materials that are conducting and materials that are semiconducting or insulating.