Video: Ohm’s Law

In this lesson, we will learn how to use the formula 𝑉 = 𝐼𝑅 (Ohm’s Law) to work out the values of the potential difference, current, and resistance in simple circuits.

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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.

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