Video: Electromotive Force and Internal Resistance

In this video, we will learn how to relate the electromotive force (emf) of a battery to its terminal voltage and its internal resistance.

15:07

Video Transcript

In this lesson, we’re talking about electromotive force and internal resistance. This topic will give us inside information on how electrical circuits work. And specifically, we’ll be learning about batteries and cells in these circuits. As we get started, let’s consider this simple electrical circuit, consisting of a battery and a resistor with a resistance 𝑅. The term battery sometimes indicates a single cell in a circuit like this. Or it can indicate multiple cells strung end to end like this. For our purposes, in this video, we’ll consider a battery to be a single unit, a single cell.

Now, a battery, we can recall, is a device that converts stored chemical energy into electrical energy. Through a chemical reaction process, electrons are driven to one end of the battery called the anode. And this leaves the other end with a relatively positive charge. This end is called the cathode of the battery. So, when its internal chemistry is running well, lots of negative charges, electrons, build up on one end of the battery. And this means the other end has a large concentration of positive charge. If we take a battery like this and we insert it into an electrical circuit, then all the negative charge buildup at one end of the battery, called one of the battery’s terminals, tends to push away any other nearby negative charges. This means that a mobile negative charge, say, at this point in our circuit would experience a force from the negative terminal of the battery that pushes it to the right.

And ultimately, this charge will be pushed in a clockwise direction all the way around the circuit. It’s this flow of negative charge through the circuit that’s called electrical current. Just as a side note, the other terminal of our battery, the positive one, exerts the same kind of repulsive force on nearby positive charges in the circuit. But unlike the negative charges, the electrons, the positive charges tend not to be mobile. This isn’t true for every type of charge flow, but it is true for electric circuits like this one. This is why we say that electric current is actually made up of negative charges rather than positive charges. For circuits connected by metal wires, as this one is, it’s the negative charges that do the moving.

Now, let’s say that the total current in this circuit is given by the value 𝐼. Given this information, that the total current in the circuit is 𝐼 and that the total resistance in the circuit is 𝑅. If we were then asked to solve for the potential difference across our battery, we could call that potential difference 𝑉. Then, we might think of Ohm’s law and recall that the total potential difference across the circuit is equal to the total current in that circuit times the circuit resistance. So, we might then say that 𝐼 times 𝑅 is equal to 𝑉, the potential difference across our battery. This analysis is accurate, but there’s more going on inside the battery. Let’s consider that battery once more.

When this battery is connected in a closed circuit, then it drives the flow of negative charge around that circuit. These negative charges, electrons, move in a loop. But then we wonder, what happens when they reach the positive terminal of the battery? Electrically speaking, it seems like the electrons would want to stay here and not keep moving. That’s because the electrons, with their negative charge, would be drawn towards the positive terminal of the battery and repelled from the negative terminal. From this perspective then, we might imagine that once an electron has made its way from one terminal of the battery to the other, then it would stop here. And the electrons would just kind of pile up around this positive terminal.

If that happened, though, all these built-up negative charges would start to neutralize this positive battery terminal. The terminal itself would become less and less positively charged. And as that happened, it would mean that the other terminal, which was originally negative, would become more and more positively charged. This would be due to the fact that negative charge building up here would mean that there’s a relative positive charge buildup on the negative terminal of the battery. These conditions would neutralize the polarity, we could call it, of this battery. And that would quickly lead to the end of charge flow through our electrical circuit. It would cut off current.

Since we don’t observe this phenomenon in electrical circuits, something else must be going on. And indeed, something is. As the negatively charged electrons enter the positive terminal of our battery, a chemical reaction goes on inside the battery, which overcomes the electrons’ natural tendency to stick close to the positive terminal. Instead, thanks to this reaction, the electrons actually move left to right from positive to negative in our battery. When this goes on, charge does not build up as we’ve drawn it here. And therefore, ions carrying charge can continue to flow through the battery, and charge can continue to flow through the rest of the circuit.

What we’re saying then is that our battery itself is part of our overall circuit and that current exists in the battery too in the same direction and with the same magnitude as it has everywhere else. Not only does charge flow in a battery but as it does so, it also encounters some resistance. Often, we refer to that resistance with a lowercase 𝑟. The name we give for this is internal resistance. We call it that because it’s resistance that comes from the battery or the cell itself in our circuit. Internal resistance has an impact on several properties of an electrical circuit. First off, if our battery does indeed have some internal resistance, lowercase 𝑟, then as we apply Ohm’s law to a circuit, this is a resistance that we need to take into account.

In the case of this circuit we see here, let’s say that 𝑉 is the potential difference supplied by the battery. In other words, if we were to measure the electrical potential at the positive terminal of the battery and then measure the electrical potential at the negative terminal, those values would be different. And that difference is called the potential difference across the battery. Another name for this potential difference is the terminal voltage. And this is what we typically symbolize using a capital 𝑉. It’s the potential difference supplied to the circuit outside the battery.

So, getting back to our circuit here, this terminal voltage, the potential difference across our battery, already takes into account the internal resistance of the battery, lowercase 𝑟. We could say then that this terminal voltage is what the rest of the circuit, the circuit that starts here and ends here all of it outside the battery, experiences. Since capital 𝑉 is the terminal voltage of our battery that already reflects the diminishment of that voltage thanks to the battery’s internal resistance. Following along with Ohm’s law, we indeed can write that the terminal voltage of our battery is equal to the current flowing through our circuit multiplied by the external resistance, capital 𝑅.

But then, what about the voltage across our battery before the internal resistance has taken a chunk out of it, so to speak? There’s a name given to that particular amount of voltage; it’s the electromotive force. This force is also called emf for short, and it’s typically represented using the Greek letter 𝜀. And the first thing to notice about it is that even though we call it a force, we measure it in units of volts. So really, the electromotive force is a potential difference. Specifically, it’s the potential difference across a battery when no current is in it.

Here’s how we can think of this. Say that we have a battery that is not part of an electrical circuit. It’s just a battery all by itself. This battery, when it’s functioning properly, will nonetheless have a positive and a negative terminal to it. That is, negative charge will build up on one end and positive charge on the other. This charge buildup creates a potential difference across the battery. And this potential difference is the electromotive force 𝜀. But then, if we were to connect our battery so that it was now part of an electrical circuit, this potential difference 𝜀, the electromotive force, is not the potential difference that the rest of the circuit outside the battery would experience. That is, emf is not the same as the voltage 𝑉.

Recall that when a battery is connected in a circuit and charge is flowing, that means that negative charge moves across the battery. And this current that travels across the battery is the same as the current that travels everywhere else in the circuit. We’ve called that current capital 𝐼, so we can give it the same name over here. A current of capital 𝐼 is in the battery. Since the battery has some nonzero internal resistance though, 𝐼 times that internal resistance represents a voltage loss.

So, here’s what we do. We take our electromotive force 𝜀 and we subtract from it 𝐼 times lowercase 𝑟, the battery’s internal resistance. The current in the circuit multiplied by the internal resistance of the battery is called lost voltage. This is how much potential difference drops simply across the battery itself. So, we start out with 𝜀 and then we subtract away from it the lost voltage. And it’s that, then, that’s equal to 𝑉, the potential difference experienced by the rest of the circuit.

Now, if we consider this equation for electromotive force along with this one we found from applying Ohm’s law to the rest of our circuit, we can see something interesting going on. First, notice that we can write our equation for electromotive force like this. emf is equal to capital 𝑉 plus 𝐼 times the internal resistance of our battery. But then, we see that capital 𝑉 can be replaced by 𝐼 times capital 𝑅, where 𝐼 is the current in our circuit and capital 𝑅 is the resistance of the entire circuit outside the battery. We then see that we can group these two terms together since they have a common factor of the current 𝐼.

And now that we look at it, we see that this equation looks a lot like our Ohm’s law equation, 𝑉 is equal to 𝐼 times 𝑅. Here we have a potential difference, specifically, the potential difference across our battery when no charge moves through it. Here we have our total circuit current. And if we combine capital 𝑅 and lowercase 𝑟, we have the total resistance in the circuit. So, it’s Ohm’s law all over again. But this time, we’re accounting for the internal resistance of our battery. Knowing all this, let’s get a bit of practice with these ideas through an example.

Which of the following statements is a correct description of the electromotive force, emf, of a battery? (A) The emf of a battery is the voltage that it applies across a circuit to which it is connected. (B) The emf of a battery is the voltage required to overcome its internal resistance. (C) The emf of a battery is the current within the battery. And (D) the emf of a battery is the potential difference across its terminals when it is not producing any current.

Okay, as we get started figuring out which of these four options is the correct description of the electromotive force or emf of a battery, let’s clear some space at the top of our screen. Now, when we talk about a battery, sometimes that term refers to a single unit, or a single cell like this, or other times it can refer to multiple cells strung end to end. For simplicity’s sake, we’ll refer to this single unit as a battery. And we want to identify a correct description of the emf of this battery. A battery, we can recall, is a device that converts chemical energy into electrical energy. It does this by chemically separating out electric charges, sending negative charges towards one end of the battery called the negative terminal. And that leaves an abundance of positive charges at the other terminal.

We can see that this battery, as is, is not part of an electric circuit. That means that there’s no charge flowing through the battery. Under these conditions, if we were to measure the electric potential at the positive end of the battery, the positive terminal, and also make a measurement of electric potential at the negative terminal. We could call the potential at the positive terminal 𝑉 sub plus and the potential at the negative terminal 𝑉 sub minus. Then, the emf of our battery is equal to the magnitude of the difference between these two potentials. In other words, emf is a potential difference. Looking through our answer options, we see that this matches up with option (D). But let’s look through the other answer options to see why it is that they’re incorrect.

Option (A) says that the emf of a battery is the voltage that it applies across a circuit to which it is connected. So, getting back to our battery, say that we connect it up so that it’s now part of an electrical circuit like this. Option (A) is saying that the battery’s emf is the voltage that it applies across the circuit to which it’s connected. In other words, it’s the potential difference created by the battery across this external part of the circuit, we could call it. The problem with this definition is it ignores the fact that the battery itself may have internal resistance. We often represent this internal resistance with a lowercase 𝑟. And this internal resistance, combined with the current inside the battery, diminishes the emf so that the voltage the battery applies across the rest of the circuit is actually less than the emf.

If the current in this circuit is equal to capital 𝐼, then that current multiplied by the internal resistance 𝑟 must be added to a voltage that we typically call 𝑉 in order to add up to the emf created by the battery. Answer option (A) describes a voltage that is applied across the rest of the circuit to which a battery is connected. That voltage is represented by this capital 𝑉 here. And we can see that that’s different from the emf. The only way that 𝑉 would equal emf is if the internal resistance of our battery were zero. Practically speaking though, this isn’t the case. And this is why answer option (A) won’t be our choice.

Moving on to answer option (B), this says that the emf of a battery is the voltage required to overcome its internal resistance. Well, it’s true that emf is a voltage, which may be surprising considering its name is a force. But looking back at our equation for emf, we could say that the voltage required to overcome a battery’s internal resistance is equal to 𝐼 times lowercase 𝑟, that internal resistance value. We can see, though, that this isn’t the whole story when it comes to emf. emf also includes the voltage supplied for the rest of the circuit. When we only consider one of these two terms in our description of emf, that description is incomplete. We won’t choose option (B) either.

Option (C) tells us that the emf of a battery is the current within the battery. But we’ve already seen that emf is a voltage, so calling it a current can’t be a correct description either. For this reason, we won’t choose option (C). This confirms our choice of option (D), that the emf of a battery is the potential difference across its terminals when it is not producing any current. And this agrees with our equation for emf because if we set the current 𝐼 to be zero, then emf equals 𝑉.

Let’s take a moment now to summarize what we’ve learned about electromotive force and internal resistance. Starting off, we saw that when a battery is connected to an electric circuit, it creates current in that circuit, and also that batteries possess some amount of internal resistance, referred to using a lowercase 𝑟. Moreover, we saw that the potential difference across a battery when no charge flows through it is called its electromotive force. This is also known as its emf, and it’s represented symbolically using the letter 𝜀.

If we have a scenario where a battery is connected to a circuit and charge is flowing through the circuit, then the emf of the battery is equal to 𝑉, the potential difference across the rest of the circuit outside the battery, plus the current in the circuit multiplied by the battery’s internal resistance.

We saw further that if we can represent 𝑉 as the current in the circuit times the external resistance of the circuit, that is, its resistance outside the battery. Then, we can rewrite this equation for emf as follows: the current in the circuit 𝐼 times the quantity, the external resistance in the circuit, plus the internal resistance. And here, external and internal refer to outside and inside the battery. This is a summary of electromotive force and internal resistance.

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