### Video Transcript

In this video, we’re going to be
talking about Ohm’s law. This law was devised by the German
physicist George Ohm in the 1800s. As we’ll see, this law has to do
with electric circuits. And in particular, it connects the
current, voltage, and resistance in those circuits.

In Ohm’s day, these concepts of
voltage, current, and resistance in circuits were known. But they weren’t very well
understood. So Ohm devised an experiment to
better understand them. He set up a simple electrical
circuit involving a source of voltage, in those days called a voltaic pile. And then he gathered together a
collection of conductors of different lengths, thicknesses, and even material
types.

Using one of the conductors to
complete the circuit, Ohm would then apply a certain potential difference across the
circuit. And then installing a galvanometer
in this circuit to measure current, he would read out the current that flowed
through the circuit as a result of this particular conductor being in it under this
potential difference.

After that data was collected, Ohm
would vary the potential difference across this circuit by changing the height of
the voltaic pile and once more read out the current that flowed through this circuit
as a result. Then once he was finished making a
whole series of measurements for a given conductor, he would move on to another one
from the collection and do the same thing, run through a series of potential
differences across the circuit and record the current that would flow through it
each time.

After having done this with all the
conductors, Ohm had gathered quite a lot of corresponding voltage and current data
points. Ohm saw that these points could be
plotted on a graph. In his experiment, the independent
variable was the voltage applied across the circuit. The dependent variable was the
current that would flow through the circuit as a result.

Ohm found that when he plotted all
these data points considering each conductor separately, Ohm found that if he drew a
line of best fit through the data points from each individual conductor, something
interesting stood out. In each case, the line of best fit
was truly a line with a constant slope. And that line passed through the
origin. Ohm’s insight was to notice that
this implied a very particular relationship between the current in this circuit and
the voltage across it.

These straight line relationships
for each conductor that he tested implied that the current in this circuit was
directly proportional to the voltage across it. That means that if we were to
double the voltage across a circuit for a given conductor, then the current through
that conductor would also double. We can see that by taking a closer
look at one of these lines of best fit.

Let’s choose the pink line for a
closer look. Considering the line of best fit
for this particular conductor, if we move out from the origin two tick marks along
the horizontal axis, then that implies a particular value of potential difference
across the circuit. We don’t know what that value is
offhand. But we do know that if we trace it
up to the line of best fit for the pink conductor, then it corresponds to a current
through that conductor two tick marks up the vertical axis. So two tick marks out on the
voltage axis, however many volts that is, corresponds to two tick marks up on the
current axis, whatever current value that is.

But now let’s say we double the
voltage applied to this particular conductor. We go out four tick marks. If we then trace up from that point
until we reach the pink line of best fit and then trace over to the corresponding
current, we see that that’s now four tick marks up the axis from the origin. In other words, we’ve doubled the
voltage applied to this conductor. And we’ve also, as a result,
doubled the current through it. That’s what it means that current
is directly proportional to voltage.

And in fact, we can take this
relationship — 𝐼 is directly proportional to 𝑉 — and we can write it a different
way. A mathematically equivalent way to
write this is to say that 𝐼 is equal to some constant — we’ll call it 𝐶 —
multiplied by the voltage, 𝑉. And here this constant, 𝐶, is
called the constant of proportionality.

Now we’ve said that Ohm’s law
connects these concepts of voltage, current, and resistance in an electrical
circuit. Ohm saw that, for each of the
conductors he tested, so long as the line of best fit through the data points from
that conductor indeed formed a line, then that meant that this constant of
proportionality, 𝐶, was equal to one over the resistance of the conductor. That is, the slope of each of these
lines for the individual conductors is equal to one over the resistance of the
conductor.

It’s important to realize that that
slope, which we could refer to using the letter lowercase 𝑚, implies a different
resistance value for each particular conductor. They’re not all the same
resistance. But given that particular resistor
value for the conductor, that resistance stays the same regardless of how much
current we put across the conductor. That’s what Ohm saw. So that really is the secret to
Ohm’s law, that this resistance written in this equation is a constant value
regardless of how much voltage we apply across the circuit and therefore how much
current runs through it.

Now we may wonder, is this always
the case? That is, is it always true that,
regardless of the material our resistor is made up of, that when we plot the data
points from that material on an 𝐼-versus-𝑉 curve, we’ll get a straight line? The short answer to that is no. Not all materials behave like the
ones we see here. To see what this might look like,
let’s clear a bit of space on our graph.

Imagine that we find yet another
conductor of a different material and perform the experiment of recording the
voltage and current across it. And imagine further that when we
plot those data points, what we find is a relationship that looks like this. And when we fit this with a line of
best fit, we see that this line will have a curve to it. It won’t have a constant slope.

Recall that we’ve said that the
slope of this line that we saw earlier, the gold line, is equal to one over the
resistance of that conductor. And critically, since the slope of
this line is the same everywhere, that means the resistance of the conductor is the
same everywhere too. It’s a constant. Materials like this that have a
constant resistance value regardless of how much current is running through them
have a particular name. They’re called ohmic materials.

Now interestingly, in this other
case here, the slope of the line is still equal to one over the resistance. But clearly, for this line, the
slope isn’t constant throughout. It starts out fairly flat and then
increases until it’s almost a vertical line up at the top. Since the slope changes, that means
the resistance of this conductor changes as well. And that resistance depends,
therefore, on the current running through it.

You might guess that the name of a
material like this is nonohmic. That is, the resistance of the
material is not a constant. It does depend on the current
running through the material. When it comes to ohmic and nonohmic
materials, unless we’re told otherwise, it’s often safe to assume that the material
is ohmic. Therefore, it follows Ohm’s
law.

Speaking of Ohm’s law, we can
arrive at the most familiar form of that law by rearranging this equation just a
bit. If we multiply both sides of the
equation by their constant resistance, 𝑅, then that term cancels out on the
right-hand side. And we see that 𝑅 times 𝐼 is
equal to 𝑉, or equivalently 𝑉 is equal to 𝐼 times 𝑅.

And before we move on, let’s make
one quick note about the units in this expression. In recognition of all his
painstaking work, the unit of resistance is named after George Ohm. It’s called the ohm. And it’s represented by the Greek
symbol Ω.

So given a certain resistor, we
would say its resistance is five ohms or 10 ohms or 100 ohms or whatever the case
may be. We know that the unit of current is
the ampere and that the unit of voltage is the volt. So all this shows us that one ohm
is equal to a volt divided by an ampere. Or an ohm is equal to a volt per
ampere. Knowing all this, let’s get a bit
of practice using Ohm’s law through a couple of examples.

A student has a resistor of unknown
resistance. She places the resistor in series
with a source of variable potential difference. Using an ammeter, she measures the
current through the resistor at different potential differences and plots her
results on the graph as shown in the diagram. What is the resistance of the
resistor?

Looking at our graph, we see it’s a
plot of the current, in amperes, running through this resistor plotted against the
voltage, in volts, running across it. And based on the description in the
problem statement, we can make a little sketch of the circuit that generated the
data plotted here.

Let’s say that this is our resistor
of unknown value. We’re told that this resistor is
connected to a variable potential difference supply and that also in this circuit is
an ammeter for measuring current. The idea then is that we use this
variable supply of potential difference to apply two, four, six, and eight volts
across this resistor. And then using our ammeter, we read
out corresponding current values of 0.4, 0.8, 1.2, and 1.6 amperes.

With these values plotted on the
graph, we see they’ve been fit with a line of best fit that runs directly through
all four points and also passes through the origin. Now this line is indeed a line that
has a constant slope. And it’s that slope that will help
us answer this question of what is the resistance of our unknown resistor.

To see how, let’s recall Ohm’s
law. This law tells us that, for a
resistor of constant value, that resistance multiplied by the current running
through the resistor is equal to the voltage across it. In our case, we want to rearrange
this equation to solve for 𝑅. And we see that that’s equal to the
potential difference divided by the current. We aren’t given explicit values for
the potential difference or the current. But we can get those from the data
plotted in our graph.

Recall that those data points are
the basis for the line of best fit that passes through all of them. This means that, in order to supply
the voltage and current we need to solve for the resistance, 𝑅, we can choose from
among any of our four data points plotted in this graph. In fact, we could choose from any
point along this line of best fit line because it so happens to pass perfectly
through all these data points. But just to make things easier, we
may as well constrain our choice to these four. It doesn’t matter which of the four
we choose. Any of them will give the same
ratio and therefore the same overall result for the resistance of the resistor.

And just to pick one of the points
then, let’s choose the one at four volts. That voltage corresponds to a
current running through the resistor of 0.8 amps. So then to solve for the resistance
of the resistor, we’ll divide four volts by 0.8 amps. When we do this, we find a result
of five ohms, where ohm is the unit of resistance. Based on our graph and Ohm’s law,
we find the resistance of the resistor to be five ohms.

Now let’s look at one more example
of Ohm’s law.

A 10-ohm resistor in a circuit has
a potential difference of five volts across it. What is the current through the
resistor?

We see that, in this problem, we
want to connect these three things: resistance, potential difference, and
current. We can recall a mathematical
relationship that does connect all three, called Ohm’s law. This law tells us that if we have a
resistor whose value doesn’t change based on how much current is running through it,
then if we multiply that resistance by the current running through it, we’ll get the
potential difference across it. In this instance, it’s safe to
assume that our 10-ohm resistor indeed has a constant resistance value, that 10 ohms
won’t depend on the current running through the resistor.

Therefore, we can safely apply this
relationship that the potential difference across this particular resistor is equal
to the current through it times its resistance. As it’s written, this equation has
a solving for potential difference. But of course, we don’t want to
solve for potential difference.

We want to solve for current. To do that, we can rearrange this
equation so it reads 𝐼 is equal to 𝑉 divided by 𝑅. And from our problem statement, we
have values of 𝑉 and 𝑅 that we can substitute in. We’re working with a 10-ohm
resistor. And the voltage across it is five
volts. And when we calculate this
fraction, we find it’s equal to 0.5 amperes. Based on Ohm’s law, that’s the
current running through this resistor.

Let’s take a moment now to
summarize what we’ve learned about Ohm’s law. In this lesson, we’ve seen that
Ohm’s law relates current, voltage, and resistance in electrical circuits. When it’s written as an equation,
Ohm’s law expresses that, for a resistor of constant resistance, that resistor value
multiplied by the current running through it is equal to the potential difference
across it.

We also saw that while a great many
of the components in an electrical circuit are made of materials whose resistance,
𝑅, does not depend on the current running through them, this isn’t always the
case. If the resistance of a material
does not depend on how much or how little current is running through it, then that
material is called an ohmic material. On the other hand, if the
resistance value of a material does depend on how much current is running through
it, then that material is called nonohmic.

And we saw that, in general, unless
we’re told otherwise, it’s typically safe to assume that a given material and a
given resistor is ohmic. That is, it follows Ohm’s law. And lastly, we saw that the unit of
resistance is named after the discoverer of this law. It’s called an ohm. An ohm is symbolized using the
Greek letter Ω.

And we saw that, in terms of other
units, an ohm is equal to a volt per ampere. And with that, we’ve learned about
Ohm’s law, which is one of the most useful laws when we’re working with electric
circuits.