Video: Finding the Resistance of Components in Series

Video Transcript

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.