Malala creates the potential divider shown in the diagram. 𝑅 one has a value of 30 ohms and 𝑅 two has a value of 90 ohms. What ratio of output voltage to input voltage does this produce?
Okay, so, in this diagram, we can see what’s known as a potential divider. What this consists of is usually two resistors, 𝑅 one and 𝑅 two in this case. And because the input voltage is across both the resistors that are placed in series, and the output voltage is only across one of the resistors, this device can essentially be used to produce an output voltage, which we’ll call 𝑉 sub out in this case, that’s a certain fraction of the input voltage, which we will call 𝑉 sub in.
Now what we’ve been told, first of all, is that the value of 𝑅 one is 30 ohms. And the value of 𝑅 two is 90 ohms. These are the resistances of these resistors here. What we’ve been asked to do in the question is to find out the ratio of output voltage to input voltage produced by this potential divider. In other words, we’ve been asked to find the ratio 𝑉 out divided by 𝑉 in. So, in order to work out this ratio, let’s start by finding an expression for 𝑉 out.
Well, 𝑉 out is the potential difference across this resistor 𝑅 two. Now because we know that the right-hand end of this circuit is at zero volts, this is going to work towards making life easier for us. Because the potential difference across the resistor 𝑅 two is simply the voltage at this point minus the voltage at this point. Or, in other words, that’s 𝑉 out minus zero volts, or just 𝑉 out. In other words then, the value of 𝑉 out that we’re trying to find out in our ratio is simply the potential difference across resistor 𝑅 two.
Now we can work this out using a law known as Ohm’s law. Ohm’s law tells us that the potential difference across a component in a circuit is equal to the current through that component multiplied by the resistance of that component. So, in this case, we can say that 𝑉 out is equal to, well, firstly, the current in the resistor, which we don’t know but we’ll just call 𝐼 for now, multiplied by the resistance of the resistor, which is 𝑅 two. And so, at this point, we have an expression for 𝑉 out, which is the top of our fraction here. Let’s then try and find an expression for 𝑉 in.
Now 𝑉 in is simply the potential difference across both the resistors 𝑅 two and 𝑅 one. So, if we want to apply Ohm’s law once again, then, first of all, we need to know the current through both of the resistors. And secondly, we need to know the resistance of both resistors combined. Now let’s start by discussing the current. This one’s the simpler of the two.
Because the resistors 𝑅 two and 𝑅 one are in series, we know that the current flowing through both of them must be the same. In other words, there must be a current 𝐼 flowing through both the resistors. Because that was the current we decided was flowing through resistor 𝑅 two. And once again, because they’re in series, that’s the same current that’s flowing through 𝑅 one. So, we found the current through both of the resistors, but what is the combined resistance of 𝑅 one and 𝑅 two?
Well, we can recall that the total resistance of resistors when connected in series is given by simply adding up all the resistances of all of the components until we’ve added the resistance of the final component in the circuit. Now in this case, we’ve only got two components, 𝑅 one and 𝑅 two. So, the total resistance of the two resistors together, which we’ll call 𝑅 sub tot, is simply equal to 𝑅 one plus 𝑅 two. And now armed with this knowledge, we can finally work out an expression for 𝑉 sub in.
Because 𝑉 sub in, that’s the total potential difference across both resistors, is equal to the current through both resistors, that’s 𝐼, multiplied by the resistance of both resistors combined, that’s 𝑅 sub tot. But we’ve just worked out an expression for 𝑅 sub tot. We know that 𝑅 sub tot is equal to 𝑅 one plus 𝑅 two. And now we have an expression for 𝑉 out and an expression for 𝑉 in. So, let’s divide 𝑉 out by 𝑉 in to give us this ratio.
We can say that 𝑉 out divided by 𝑉 in is equal to 𝐼𝑅 two, that’s 𝑉 out, divided by 𝐼 brackets 𝑅 one plus 𝑅 two, that’s 𝑉 in. And then, we can see that the factor of 𝐼 in the numerator and the denominator cancels. Hence, we’re simply left with the right-hand side 𝑅 two divided by 𝑅 one plus 𝑅 two. So, that’s essentially the ratio of output voltage to input voltage. But this is only an algebraic expression. We can go further than this because we’ve actually been told the values of 𝑅 one and 𝑅 two.
𝑅 one has a value of 30 ohms and 𝑅 two has a value of 90 ohms. We can plug in the values of 𝑅 one and 𝑅 two to give us an expression for 𝑉 out over 𝑉 in, as being equal to 90 ohms, that’s 𝑅 two, divided by 30 ohms, that’s 𝑅 one, plus 90 ohms, that’s 𝑅 two. The denominator simplifies to 120 ohms. At which point, the unit of ohms in the numerator and the denominator cancels. So, we’re just left with 90 divided by 120. And this simplifies down to 0.75. Hence, we can now say that the ratio of output voltage to input voltage produced by this potential divider is 0.75.