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.
