Four identical resistors are placed in series in a circuit. The resistance of all four resistors together is 36 ohms. What is the resistance of each resistor individually?
Alright, so we’ve got a question about resistors here. Let’s see what we know about these resistors. First of all, we know that there’s four of them. And we also know that they’re all identical. Also, we know that they’re placed in series in a circuit. The resistance of all four resistors together is 36 ohms. We need to find the resistance of each resistor individually.
Now in order to help us do this, we can draw a partial circuit diagram. And that looks something like this. We’ve got the four resistors, which we’ll label 𝑅 one, 𝑅 two, 𝑅 three, and 𝑅 four. And we’ve got dots on either side because we don’t know what happens in the rest of the circuit. We don’t know how it’s wired up.
However, our main point of focus is the four resistors wired up in series. And that is quite important, the fact that they’re in series. Now, one of the most important things that we know from the question is that these four resistors are identical. Therefore, we can confidently say that the resistances of all of these resistors are equal. 𝑅 one is equal to 𝑅 two is equal to 𝑅 three is equal to 𝑅 four. So for convenience, let’s throw away the subscripts. Let’s just call each one of these resistances 𝑅.
Now we know the total resistance of the four resistors together. We know that is 36 ohms. And this is going to be the key to calculating the resistance of each one of these resistors. However, before we can do that, we need to know how the total resistance of resistors in series is calculated.
So the total resistance of a bunch of resistors in series is given the following way. If we’ve got 𝑛 resistors lined up in series, then the total resistance, 𝑅 sub tot, standing for total, is equal to the first resistance plus the second resistance plus so on and so forth until we get to 𝑅 sub 𝑛. And that is the resistance of the 𝑛th resistor, the final resistor in the series.
Basically, in other words, you just add up all the resistances together if the resistors are in series. And that ends up being the total resistance of the system. In our situation, we’ve got four resistors. Therefore, the value of 𝑛 is four. And so our total resistance is equal to 𝑅 one plus 𝑅 two plus 𝑅 three plus 𝑅 four.
Now there are two modifications to this equation that we can make. First of all, as we’ve already said, the resistances of each one of these resistors is the same. And we’ve called this 𝑅. So once again, we can get rid of the subscripts. And in fact, we can simplify this down further because 𝑅 plus 𝑅 plus 𝑅 plus 𝑅 is four 𝑅.
Secondly, we know the total resistance of all four resistors together. We’ve been given this in the question. That resistance is 36 ohms. So we’ll replace 𝑅 sub tot with 36 ohms. And at that point, we are very close to finding the resistance of one of these resistors. All we need to do is divide both sides of the equation by four. And so the fours on the right-hand side cancel, which means that all we need to do on the left-hand side is divide 36 by four, which leaves us with our final answer. The resistance of each resistor individually is nine ohms.