Video Transcript
The circuit shown in the diagram has a current of 0.5 amps through it. What is the value of 𝑅?
The diagram in this question shows a circuit in which a cell is connected to two resistors in series. The cell provides a potential difference of 10 volts. The right-hand resistor has a resistance of seven ohms, and the left-hand one is labeled with a resistance of 𝑅. We’re told in the question that this circuit has a current of 0.5 amps through it. We’ll label the potential difference provided by the cell as 𝑉 and the current through the circuit as 𝐼. We’ll label the total resistance of the circuit as 𝑅 subscript 𝑇.
At this point, we don’t know the value of this total resistance. However, we can recall that there is a law which links the three quantities potential difference, current, and resistance. This law is Ohm’s law, and it tells us that potential difference 𝑉 is equal to current 𝐼 multiplied by resistance 𝑅 subscript 𝑇. In our case, we know the values of 𝑉 and 𝐼. We want to find the value of resistance 𝑅 subscript 𝑇. So we need to take our Ohm’s law equation and rearrange it to make this resistance the subject.
To do this, we’ll divide both sides of the equation by the current 𝐼. On the right-hand side, the 𝐼 in the numerator cancels with the 𝐼 in the denominator. And so we have that 𝑉 divided by 𝐼 is equal to 𝑅 subscript 𝑇. Of course, we can also write this equation the other way around to say that 𝑅 subscript 𝑇 equals 𝑉 over 𝐼.
So now we can substitute in that 𝑉 is equal to 10 volts and 𝐼 is equal to 0.5 amps into this equation. Doing this gives us that the total resistance 𝑅 subscript 𝑇 is equal to 10 volts divided by 0.5 amps. Evaluating the right-hand side, we find that 𝑅 subscript 𝑇 is equal to 20 ohms. This value of resistance is the total resistance of the circuit. That is, it’s the resistance of both of these two resistors together.
The question asks us to find the value of 𝑅. So that’s the resistance of this left-hand resistor. We know that the right-hand resistor has a resistance of seven ohms and that both of them together have a resistance of 20 ohms. Let’s recall what happens when we have two resistors connected in series like these two 𝑅.
We’ll consider two general resistors connected in series. We’ll label them with resistances 𝑅 one and 𝑅 two. If we label the total resistance of these two resistors as 𝑅 subscript 𝑇, then we have that 𝑅 subscript 𝑇 is equal to 𝑅 one plus 𝑅 two. In other words, whenever we have two resistors connected in series, we add together their individual resistances in order to get the total resistance of both of them together.
We can use this general equation to calculate our unknown resistance 𝑅. In our case, we found that the value of 𝑅 subscript 𝑇 is equal to 20 ohms. And we know that this 20 ohms comes from two resistors connected in series, one with a resistance of seven ohms, the other with the resistance of 𝑅. So we can replace 𝑅 one and 𝑅 two with the values of our two resistors. Let’s let 𝑅 one be the seven-ohm resistor and 𝑅 two be the one with the resistance of 𝑅. This gives us an equation which says that 20 ohms is equal to seven ohms plus 𝑅. We need to rearrange this equation in order to make 𝑅 the subject.
To do this, we subtract seven ohms from both sides of the equation. Then, on the left-hand side, we have 20 ohms minus seven ohms, which gives us 13 ohms. And on the right-hand side, we have seven ohms plus 𝑅 minus seven ohms, which simply gives us 𝑅. And so we’ve reached an equation which tells us that 𝑅 is equal to 13 ohms. So our answer to the question is that the value of 𝑅 is equal to 13 ohms.
