In this video, we’re talking about series circuits. These are electrical circuits where charge has only one path to follow as it moves through the circuit. It never splits up or divides across different branches, but rather it all moves within the same closed, continuous loop.
As an example of this, consider this circuit. This circuit consists of a cell, one resistor here, and a second resistor here. Conventionally, charge flows from the positive to the negative terminal of a cell in a circuit. So, we could draw that current in like this, counterclockwise, also known as anticlockwise, in the circuit. And since there’s only one pathway for the moving charge to follow, we know that this is a series circuit.
Now in this or any other series circuit, potential difference 𝑉, current 𝐼, and resistance 𝑅 all follow certain rules. If we start out by considering 𝑉, the potential difference, we know that, in general, a cell in our circuit will supply a potential difference that causes charge to move. As charge moves throughout the circuit though, whatever increase in potential difference we experience thanks to the cell must decrease as charge moves throughout the rest of the circuit. Whereas the cell is the place where potential difference is increased, the remaining circuit components, in this case, these two resistors, are the places where potential difference is decreased.
To see how potential difference changes throughout this circuit, say that we start at this point right here and we move in the direction of conventional current, counterclockwise. One way we could depict all this is this way. We open up our circuit, starting at that negative terminal of our cell, and then we stretch it out as though it’s in a line, being sure to include all the other components in our circuit. In this case, these two resistors. And let’s say that on a vertical axis, we have potential difference Δ𝑉 plotted. So, we’ll start here at the negative terminal of our cell, and then we’ll move in the direction of conventional current.
If we did this, then we would see that over the course of traveling across this cell, the potential difference would increase to this value 𝑉 supplied by the cell. Then, as we follow the direction of charge flow, our potential difference would be maintained at this value until we reach the first resistor. As charge crosses that first resistor, potential difference is lost. And in terms of the amount of how much is lost, if we assume that this resistor is the same as this one, then the potential difference lost across this first resistor will be equal to half the potential difference supplied by the cell. More on that later, but for now, let’s continue to follow charge as it moves anticlockwise through our circuit.
Once we cross over the first resistor, potential difference stays constant until we reach the second resistor. And moving across this resistor, our potential difference again drops. In this time, it drops all the way down to its original value at the negative terminal of our cell, zero. From that point on, until we reach what we can call the start point of our circuit, the negative terminal of our cell, the potential difference is zero. Notice that the potential difference starts and ends at the same value; that’s required in order for this circuit to be complete. And notice also that what we gain in potential difference from the cell is dropped across the various components of our circuit. This also is always true in an electrical circuit.
Now, so far, we haven’t given any labels to these two resistors. But let’s do that now. Let’s call this first resistor 𝑅 one and the second resistor 𝑅 two. Looking back down at our sketch, we could say that this amount of potential difference, that was dropped over 𝑅 one, is 𝑉 one. And we can call this amount of potential difference dropped over 𝑅 two 𝑉 two. And then with these values labeled, we can see there’s a relationship between 𝑉 one, 𝑉 two, and 𝑉 which is the potential difference supplied by the cell. Specifically, we can see that 𝑉 is equal to 𝑉 one plus 𝑉 two.
Now, earlier, we talked about the possibility that 𝑅 one is equal in value to 𝑅 two. That is, each resistor supplies the same number of ohms of resistance. But if 𝑅 one is equal to 𝑅 two, then that means the potential difference dropped over each resistor is the same. In other words, 𝑉 one is equal to 𝑉 two. And then, if 𝑉 one is equal to 𝑉 two and 𝑉 one plus 𝑉 two is equal to 𝑉. Then that means the potential difference dropped over the first resistor, 𝑉 one, is equal to the potential difference supplied by the cell divided by two. And that 𝑉 two, the potential difference dropped over the second resistor, is also 𝑉 over two. And this conclusion, we recall, requires that 𝑅 one be equal to 𝑅 two.
Now, in a series circuit with two resistors, it’s a bit of a special case when those resistors have the same exact value. When that happens, we were able to make this conclusion about the potential difference that’s dropped over each one. But even when that’s not the case, even when the resistors have different values, it’s still true that the potential difference supplied by the cell is equal to the sum of the potential differences dropped over the two resistors. And this equation here is very close to a general expression we can write for potential difference in a series circuit.
The expression we can write will look like this. We’ll say 𝑉 sub 𝑡, where 𝑉 sub 𝑡 is the potential difference supplied by the cell or battery in the circuit, is equal to the potential difference dropped over the first resistor. Plus the potential difference dropped over the second resistor, if there is one, plus the potential difference dropped over the third resistor, if there is one of those, plus dot, dot, dot. Meaning the potential difference dropped over any other resistors that might be in this series circuit.
If we happen to have a series circuit with only one resistor, then our equation would simply look like this: 𝑉 𝑡 is equal to 𝑉 one. But if we had two resistors, it would look like this, very much like the one we generated here. And then if we had three resistors, we would add together the potential difference dropped over those three. And that would equal the total supplied by the cell and so on, up to as many resistors as we might find in our series circuit. So, that’s potential difference and how it behaves in a series circuit.
Now, let’s move on to consider current. Remembering our definition of a series circuit, a closed loop where charge only has one path to follow. The implication of that is as charge travels around this circuit, the value of the current at any given point is always the same. That’s because any bit of charge that reaches this point will also reach this point and this point and this point and this point and this point and any other point in the series circuit. The charge has no choice, we could say, but to follow that same path. Let’s say that in this series circuit we’re considering, there’s some total value of the current we’ll call it 𝐼. And what we want to know is how does the current at any given point in the circuit, say, at the resistor 𝑅 one or at the resistor 𝑅 two, compare to this value of the total current.
From what we’ve seen so far, because any individual charge that travels through the circuit must cross all of it, that would mean that if 𝐼 one is the current through the first resistor, then that’s equal to the total current. And it’s also equal to 𝐼 two, what we could call the current through the second resistor. And it’s furthermore equal to the current at any given point, whether in a component or not, in this circuit. In a series circuit then, the rule for current is simple. It’s the same everywhere.
And we can write that as a general expression this way. We can say that the total current in the circuit is equal to the current in the first component, which is equal to the current in the second component, if there is one, which is equal to the current in the third component, if there is one of those, and so on. This fact that current is the same everywhere in a series circuit is often very helpful to us when we’re solving exercises. Okay, now, let’s move on to our last property that of resistance.
How do resistors behave in a series circuit? In particular, given a series circuit, like this one here, that has more than one resistor in it. What we may wonder is the total equivalent resistance of that circuit. For this circuit, the total equivalent resistance it has is equal to the sum of its individual resistors 𝑅 one and 𝑅 two. This may seem common sense. But there is another type of circuit known as a parallel circuit, where this sort of rule does not hold. But in this case, it does. In series circuits, to find the overall resistance of the circuit, and we can write this as a general equation where our overall resistance we’ll call 𝑅 sub 𝑡. To find that value, we take all the individual resistances that may appear in the circuit, and we add them together.
So if we have a series circuit with just one resistor, then the total resistance is equal to that resistor’s value. But if we have a circuit with two resistors like we do over here, then the total resistance is 𝑅 one plus 𝑅 two, and so on and so forth when we have progressively more resistors in our series circuit. These rules that we’ve discovered for series circuits are worth memorizing because they come up over and over again when we talk about these types of circuits.
Taking a step back now, let’s consider this simplified circuit diagram that we’ve drawn, which has been stripped of any labels of 𝑉 or 𝑅 values. When we’re first learning about series circuits, it may seem that there is some significance in the position of these different components, the resistors in the cell relative to one another. For example, we might think that this circuit is essentially different from, say, this circuit.
And the reason we might think that is because we can see that this resistor, which was originally here in the circuit, has now been moved to this side of it. Or to continue the example, we might think that this circuit is different in an important way from this one. And we could think that because the cell is reversed in polarity in this third circuit we’ve sketched in compared to the first one.
As it turns out though, the changes that we’ve sketched in, as well as many that we haven’t, don’t essentially change the properties of the circuit we’re considering. In all cases, we still have one cell and two resistors in series with one another. And whether charge moves clockwise or counterclockwise through these circuits doesn’t make a difference in terms of their performance. This means that there are many different ways of drawing the same circuit. So long as the various components in the circuit are all the same, in other words that, say, the cell supplies the same potential difference and the resistors all have the same value. Then these circuits are all what are called equivalent.
Equivalent circuits perform the same way and have the same characteristics. For this circuit and effectively it is a single circuit, even though we’ve sketched it three different ways, it doesn’t matter whether the resistors are on the bottom in the side versus the two sides, or whether the cell’s positive terminal faces left or right. In all these cases, the circuit’s performance is the same. So, these are all equivalent expressions of one and the same circuit. And that’s why we say they are equivalent.
Knowing all this about series circuits, let’s now get a bit of practice through an example exercise.
A battery supplying 12 volts is connected in series with two resistors. The potential difference across the first resistor is four volts. What is the potential difference across the second resistor?
Okay, so in this scenario, we have a battery connected in series with two resistors. So, let’s begin by sketching the circuit. Here is our battery, and we’re told that it supplies a potential difference of 12 volts. And then here are the two resistors it’s connected with in series. We’re not told about the values of these resistors. But just to give them names so we can refer to them, let’s call this one 𝑅 one, and this one we’ll call 𝑅 two. The problem goes on to tell us that the potential difference across the first resistor, across what we’ve called 𝑅 one, is four volts. So, let’s label that this way. Let’s say that 𝑉 one is the potential difference across this first resistor. So, 𝑉 one is four volts. And let’s call 𝑉 two the potential difference across 𝑅 two.
Now, it’s that value 𝑉 two that we want to solve for. And to do it, we can recall something about electrical circuits. In any electrical circuit, if we make one complete closed loop around the circuit, then the potential difference across that entire loop must be zero. It’s like saying that we need to start and end in the same place. And that’s true because we are talking about a circuit. In terms of potential difference, that means that whatever potential difference is supplied by our cell or battery, in this case, 12 volts, must be dropped or diminished across the rest of the circuit outside of the battery.
Now, in our case, the rest of the circuit consists of our two resistor components, 𝑅 one and 𝑅 two. These are the only places in the circuit where potential difference drops. This means we can now write an equation in terms of the various potential differences in the circuit. We can say the potential difference supplied by the battery 12 volts must be equal to the potential difference dropped across 𝑅 one. That’s 𝑉 one. Plus the potential difference dropped across 𝑅 two, what we’ve called 𝑉 two. And by the way, this expression we’ve written here is a specific case of a general equation. That equation says that the total potential difference supplied by a battery or a cell in a series circuit is equal to the sum of the potential difference drops across the various components of the circuit.
In our case, we only have two components, our two resistors 𝑅 one and 𝑅 two. So, our equation basically looks like this. And that’s what we see here with 12 volts in place of 𝑉 sub 𝑡. In this equation, it’s 𝑉 two that we want to solve for. And we already know that 𝑉 one is equal to four volts. So, when we substitute that in, we just need to subtract four volts from each side of this equation. And then, positive four volts minus four volts on the right cancels one another out. And we wind up with this expression; 12 volts minus four volts is equal to 𝑉 two, and 12 minus four is eight. And so, this is our value for 𝑉 two. The potential difference across the second resistor is eight volts.
Let’s summarize now what we’ve learned about series circuits. In this lesson, we’ve learned that a series circuit is an electrical loop in which there is only one path for charge to travel. We also learned a series of equations describing potential difference, current, and resistance in series circuits. These equations showed us that the total equivalent resistance in a series circuit is equal to the sum of the individual resistors. That current in a series circuit is the same everywhere. And that the potential difference supplied by the cell or battery is equal to the sum of the potential differences dropped across the other components of the circuit. And lastly, we learned about equivalent circuits, which are circuits that are arranged differently but that perform in the same way. This is a summary of series circuits.