In this lesson, we’re learning about a particular type of electrical circuit known as a series circuit. Believe it or not, this circuit we see on screen is a series circuit. As we learn how they operate, we’ll see that the governing principles of series circuits are very useful in a broad sense. Let’s start out by defining just what a series circuit is.
A series circuit is one in which current never divides across different branches of the circuit. Here’s what we mean by that. Say we have this simple electrical circuit with a cell and a single resistor. We know that in this circuit conventional current will flow in a counterclockwise direction.
If we track with the current, following along with it as it goes around the circuit we see that it moves around the bends, goes through the resistor, and comes back to the negative terminal of the cell. At no point in its path around this loop did the current divide up. It never had the option too. There is only one path for it to follow at any given point.
That’s what we mean by a series circuit: the current only has one path it can possibly follow. So we can say that a series circuit is one in which current never divides or equivalently we can say that a series circuit is one in which current has only one path to follow. It never divides, and it never branches off. This definition brings us to the first important point to remember about series circuits.
In this type of circuit, current is the same everywhere. This has to be true; current has to be the same everywhere because in this type of circuit it only has one path to travel. That means the current in the circuit here is the same as that here and here and here and here and at every other individual point in the circuit. That’s a quite useful fact because it means once we solve for the current at any one point in the circuit, we now know it at every point in the circuit. So we can call this important fact number one about series circuits: current is the same everywhere.
Knowing this, let’s go back to our circuit and we’ll make a little change to it. Currently, we have one resistor in the circuit. But let’s add another one right next to it. And let’s say further that each of these resistors has the same resistance. We’ll simply call it 𝑅. We’ll leave it general for now. Now here’s a question: Does this change that we’ve made to our circuit impact the pathway that current has to travel when it flows around it? The answer is no. Current still only has one path to follow as it moves around the circuit and therefore it’s still a series circuit.
Before we added this second resistor, we just had this one. And we said its resistance value is 𝑅. That means that at the outset before we added our second resistor, the total resistance in this whole entire loop was simply capital 𝑅. That’s because at that point this was the only resistor in the loop. But of course, now we’ve added this second identical resistor to the circuit and the question becomes “what now is the total circuit resistance, now that we have these two resistors?”
This question brings us to what we can call the second important fact about series circuits. And that is that in a series circuit, the total circuit resistance is the sum of the individual resistors. We’ve written this in words and we can also write it as a mathematical equation. Let’s call the total resistance of a series circuit — not just this particular circuit we’re looking at, but any series circuit — let’s call this total resistance 𝑅 sub 𝑡.
In a general series circuit, we don’t know how many resistors we have. But let’s just say that we have some number — we’ll call it 𝑛 — so there are 𝑛 resistors in this general series circuit we’re imagining. In that case, we can say that the total resistance is equal to the value of the first resistor in the circuit plus the value of the second resistor in the circuit plus dot dot dot all the way up to the value of the 𝑛th resistor in this series circuit. This equation is very helpful and worth keeping in mind because like we said, it works for any series circuit with any number 𝑛 of resistors.
As we look back over to our particular circuit, we see that in this case 𝑛 is equal to two. We have two resistors. That means if we were to apply this general equation to our particular circuit, we would say that the total resistance of this circuit is equal to the value of the first resistor which is 𝑅 plus the value of the second resistor which is also 𝑅. Since that’s all the resistors this circuit has, we can see that the total resistance is two times 𝑅. This is what it means for the total circuit resistance to be the linear sum of the individual resistors.
So far, we’ve talked about current and resistance in this circuit. And the one missing piece that we haven’t mentioned yet is potential difference, voltage. In order to talk about that a bit, let’s assign a voltage to our cell in this circuit. Let’s just say it provides a voltage 𝑉. And let’s say — as we did before — that a current 𝐼 is running around this circuit in the anticlockwise direction. The fact that we have a voltage, a current, and a resistance in this circuit may remind us of a law that applies to electrical circuits.
That law is Ohm’s law. And it says that for a resistor of constant value, that value multiplied by the current running through the resistor is equal to the potential difference across it. Taking a look at our circuit, we see that we don’t just have one resistor though. We have two. So how does Ohm’s law apply when we have more than one resistor in a circuit?
This question brings us to the next important fact about series circuits. And that is that in series circuits, Ohm’s law applies to individual resistors and to the circuit as a whole. Here is what that means. Let’s say for a moment that we focus entirely on this single resistor in the circuit. Ohm’s law tells us that for this particular resistor, if we multiply the resistance value 𝑅 by the current running through it, then that will equal the potential difference dropped across this resistor in particular.
To see how this all works in a way that’s clear, let’s add subscripts to our two resistors. Let’s call the first one 𝑅 sub one and the second one 𝑅 sub two. Notice that we haven’t changed anything about their relative value: 𝑅 sub one is equal to 𝑅 sub two. We just wanna give them these subscripts so we know which one we’re talking about. And for the moment, 𝑅 sub one is the one we’re focusing on. Let’s apply Ohm’s law to this particular resistor.
This law says that if we take this resistor 𝑅 sub one and then we multiply it by the current running through it which we know must be equal to 𝐼 because the current in a series circuit is the same everywhere, then Ohm’s law tells us that that product is equal to the potential difference across that resistor. We’ll call it 𝑉 sub one. Now that we’ve got this far, it seems to make sense that if we instead focus on the second resistor 𝑅 sub two, we could generate a similar equation.
We could say that the resistor 𝑅 sub two multiplied by the current running through it which is still 𝐼 because that current is the same everywhere is equal to the potential difference across 𝑅 sub two. And we can call that 𝑉 sub two. So we now have these two equations for the potential difference across our two resistors.
This is what it means for Ohm’s law to apply to the individual resistors in the circuit. Just now, we’ve applied it to these two separate resistors. But we’ve also made the claim that this law applies to the circuit as a whole. Going back to our basic Ohm’s law equation, that means if we were to find the total equivalent resistance of these two resistors, which recall what we did earlier, and we were to multiply that equivalent resistance by the current in the circuit, then that will give us the total potential difference across the entire circuit.
Let’s do that then. Let’s apply Ohm’s law to the circuit as a whole. Starting off, we can say that the total equivalent resistance of these two separate resistors is found by using our addition rule: 𝑅 sub 𝑡 for our circuit is equal to 𝑅 sub one plus 𝑅 sub two. And remembering these two values are equal, we found earlier that this was simply equal to two times what we call 𝑅. But for now, let’s leave our total resistance in this form: 𝑅 sub one plus 𝑅 sub two.
Moving ahead, Ohm’s law tells us that if we multiply this total resistance by the current running through our circuit which is still 𝐼, the net product will equal the total potential difference across our entire circuit. And looking at our cell, we see that the total potential difference is capital 𝑉. Now we’ve got Ohm’s law equations for our individual resistors and our circuit considering it as a whole. And here is where things get interesting.
Let’s do this: let’s multiply out this term 𝐼 the current against both resistance values 𝑅 sub one and 𝑅 sub two. When we do that, we see that the overall potential difference for our circuit 𝑉 is equal to 𝐼 times 𝑅 sub one plus 𝐼 times 𝑅 sub two. But look at this: what is 𝐼 times 𝑅 sub one and what is 𝐼 times 𝑅 sub two? Well, the first term is equal to 𝑉 sub one and the second is 𝑉 sub two. When we substitute those two terms in, we find that the total potential difference in our circuit is equal to the sum of the potential differences across our individual resistors.
And furthermore, because 𝑅 sub one is equal to 𝑅 sub two, that must mean that 𝑉 sub one is equal to 𝑉 sub two. And if that’s the case, it means that this total potential difference 𝑉 from our cell evenly divides across our two identical resistors: 𝑅 sub one and 𝑅 sub two. As an equation, we can write that like this: we can say that 𝑉 over two is equal to 𝐼 times 𝑅 sub one. And as well 𝑉 over two is equal 𝐼 times 𝑅 sub two.
So then using Ohm’s law, we’ve solved for the potential drop across both of the resistors in our circuit individually. That’s just one of the advantages of being able to apply Ohm’s law to a series circuit both to individual components as well to the circuit as a whole. Okay, just one more bit before we get some practice with these ideas. There’s a term you may encounter when working with series circuits. And that term is equivalent circuit.
Here is what that term means. An equivalent circuit is a simplified version of a more complex circuit which keeps the characteristics of the original. That’s a long definition. But it actually means something pretty straightforward. And we can see that by using our example circuit. Let’s imagine that we wanted to create an equivalent circuit for this circuit that we have here in front of us. What will that look like?
According to our definition, this would be a simplified version of the more complex original which keeps the characteristics of the original circuit. So what are the characteristics of this original circuit? Well, it has a potential difference 𝑉, it has a current 𝐼, and it has two resistors: one of value 𝑅 one and one of value 𝑅 two. Looking at this circuit, the only way we can really simplify it is by combining our two individual resistors into one equivalent resistance.
We can refer to our series resistor addition rule to see how to do that. We see that we just add 𝑅 sub one and 𝑅 sub two together. This means that we could condense our two resistors into one which has a new resistance value of 𝑅 sub one plus 𝑅 sub two. And having done that, we’ve created an equivalent circuit for our original. This is a simplified version. Instead of having two resistors, we now just have one. But it keeps the characteristics of the original. We have the same overall potential difference, the same overall current, and the same overall resistance.
It’s often the case in working with series circuits that if we want to create an equivalent circuit that is a simplified version of the more complex original, then that simplification simply involves combining all the resistors in the circuit into one equivalent resistance. And as we said, that happens according to our resistance addition rule. So anyway, that’s what an equivalent circuit is. And we saw practically how to create one. Keeping all this in mind, let’s try out a series circuit example.
The circuit shown in the diagram consists of three resistors connected in series with a cell. The first resistor has a resistance of 𝑅, the second resistor has a resistance of two 𝑅, and the third resistor has a resistance of three 𝑅. The cell provides a potential difference of 24 volts. The current through the circuit is 0.1 amperes. What is the value of 𝑅?
Taking a look at our circuit, we see that indeed there are these three individual resistors of values 𝑅, two 𝑅, and three 𝑅 in series with our 24-volt cell. We’re told a bit more about the circuit. We’re also told that it carries a current of 0.1 amperes. We can call that current 𝐼. Knowing all this, we want to solve for the value of 𝑅, which we see all three of our resistors are expressed in terms of.
As we get to work, let’s clear some space on screen. Okay, to briefly recap, we were told that the current running through the circuit is 0.1 amps. And overall, we want to solve for the value of 𝑅. To do that, we would like to relate the two known quantities in our circuit, which are the voltage and the current, to the unknown quantity, which is the resistance. Whenever we name these three quantities, voltage, current, and resistance, that can remind us of a law for electrical circuits — Ohm’s law.
This law tells us that given a resistor of constant value, if we multiply the value of that resistor by the current running through it, the net product is equal to the potential difference across the resistor. Looking at our circuit, we see that we don’t have one resistor, but we have three. But really, we would like to have one. We would like to be able to express these three resistors as one equivalent resistance.
Thankfully, we’re able to do that because these resistors are arranged in series and we can recall the addition rule for resistors arranged this way. We can remember that for 𝑛 series resistors — that is 𝑛 resistors arranged in series with one another — the total resistance of all those resistors equals the value of the first resistor plus the value of the second resistor plus dot dot dot all the way up to the value of the 𝑛th resistor.
In our case, in our circuit, 𝑛 is equal to three. That’s how many series resistors we have. So we can say that the total resistance in our particular circuit is equal to the value of the first resistor 𝑅 plus the value of the second resistor two 𝑅 plus the value of the third resistor three 𝑅. Looking at the right-hand side of this expression, we see we can add all these terms together: one 𝑅 plus two 𝑅 plus three 𝑅 is six 𝑅. That’s the total resistance in this circuit.
It’s wonderful that we’ve solved for that because now we can return to Ohm’s law, where in this case, our resistor value is 𝑅 sub 𝑡, the value we’ve just solved for. When we apply Ohm’s law to our equivalent or total resistance in our circuit, we write that six 𝑅 which is our total equivalent resistance multiplied by the current in the circuit — we’ve called it 𝐼 — is equal to the potential difference across the entire circuit. Let’s call it 𝑉.
As we look though, we see that we know the value for 𝐼; it’s 0.1 amperes. And we also know the value for 𝑉; it’s given as 24 volts. This means we can substitute those values in. And now, we want to arrange this equation so that we solve for 𝑅. Dividing both sides by 0.1 amps times six, both those terms cancel out on the right-hand side. And we see that 𝑅 is equal to 24 volts divided by 0.1 amps times six. When we calculate this fraction, we find a result of 40 ohms. That’s the resistance value of 𝑅.
Let’s take a moment to summarize what we’ve learned about series circuits. In this lesson, the first thing we saw was that in a series circuit, current follows one single path. That’s basically the definition of a series circuit. This led us to our first important fact, which is that in a series circuit, current is the same everywhere. And we can see that this follows from the fact that current only has one path to travel.
Second, we saw that the total equivalent resistance in a series circuit — what we could call 𝑅 sub 𝑡 — is equal to the linear sum of the individual resistances. We also saw that Ohm’s law that 𝑉 is equal to 𝐼 times 𝑅 applies to individual resistors in a series circuit as well as the circuit in its entirety.
And finally, we learned about equivalent circuits that they’re simple representations of a more complex circuit that keeps it characteristics.