Lesson Video: Using Potential Dividers Physics

Video Transcript

In this video, we will be discussing a particular type of circuit known as a potential divider. We will see how this circuit is built, why it’s called a potential divider circuit, and how it is used. So let’s start by looking at how we would actually build this kind of circuit.

We start by connecting two resistors in series. Let’s say that the resistance of the first resistor is 𝑅 one and the resistance of the second resistor is 𝑅 two. These two resistors can then be connected to, say, a power source, in this case we’ve drawn a battery, or to some other more complicated circuit. In fact, we’re not particularly interested in what we find in this part of the circuit, which is why we’re going to draw two terminals, one here and one here, to show that we can connect multiple different things between these terminals. All we care about is that it needs to be something that supplies a potential difference.

But what really turns the circuit from just being two resistors in series to a potential divider is when we connect some components in parallel with one of the resistors. In this case, we’re using 𝑅 two. For example, we could choose to connect a light bulb here or for that matter any other components. But the important thing is that the components here are connected in parallel with only one of the resistors, in this case 𝑅 two.

But once again, we’ll draw some terminals here and here because we don’t really care what’s connected across this gap. What we’re interested in is the effect that all of this has. So let’s imagine that we did connect a battery in this position here, with the positive terminal over here and the negative terminal over here. This means that the battery sets up a potential difference across the rest of the circuit. Let’s call this potential difference supplied by the battery 𝑉 subscript in for input voltage. And the result of all of this is that a current is set up in this circuit.

Specifically, if we think about conventional current, which is a flow of positively charged particles, then we can see that this current would be moving away from the positive terminal of our battery and, therefore, be moving clockwise through our circuit until it arrived at the negative terminal again. Let’s say that the magnitude or size of the current set up in this circuit is 𝐼. And to keep things simple, we will consider only the circuit consisting of our battery, resistor 𝑅 one, and resistor 𝑅 two.

In other words, for now, let’s ignore everything that was connected in parallel with resistor 𝑅 two because this way all we’ve got is a series circuit. And we can recall that for a series circuit, firstly, the current through all components is the same. In other words, if there’s a current of magnitude 𝐼 moving this way through our wire in the circuit, then this means that the current through our resistor 𝑅 one is 𝐼 as well and the current through the resistor 𝑅 two is also 𝐼.

Secondly, we can consider the fact that there’s a potential difference of 𝑉 in across our circuit. What this means is that the battery that we’ve drawn here as the placeholder for whatever is actually connected in this part of the circuit is providing a potential difference of 𝑉 in across both of the components in our circuit, in this case the resistors, at which point we can erase our placeholder battery. Because like we said earlier, we’re not that interested in what’s actually connected here, just the fact that something should be connected here. And that will provide a potential difference of 𝑉 in across both resistors.

But let’s also recall that for a series circuit, the potential difference across the entire circuit is shared across the components. And what we mean by this is that the total potential difference across, in this case, both our components 𝑅 one and 𝑅 two is shared in such a way that if we say that the potential difference across resistor one is 𝑉 one and the potential difference across resistor two is 𝑉 two, then we know that 𝑉 one plus 𝑉 two must be equal to 𝑉 in. Intuitively, this tells us that the potential difference across the first resistor plus the potential difference across the second resistor is equal to the potential difference across both resistors combined, which sounds like a very common-sense thing to say. But remember that this is not true for parallel circuits. And we’ll have to worry about those in a moment.

But for now, let’s also recall 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 its resistance. And we can apply this law to components in our circuit, firstly the resistor 𝑅 one and secondly the resistor 𝑅 two.

For the resistor 𝑅 one, we can say that the potential difference across that resistor, 𝑉 one, is equal to the current through that resistor, which we know is 𝐼, multiplied by its resistance, which is 𝑅 one. And similarly, for the resistor 𝑅 two, we know that the potential difference across that resistor is equal to the current through that resistor multiplied by its resistance.

And lastly, we can apply Ohm’s law to the entire circuit itself. In this case, we know that the potential difference across the circuit is 𝑉 in. So that’s the total potential difference across both resistors combined. And we know that the current in the circuit is 𝐼, because it’s a series circuit, which means the current is constant throughout. And we can link this to the total resistance of the circuit. In other words, we can say that the total potential difference across the circuit, 𝑉 in, is equal to the current through the circuit, which is 𝐼, multiplied by the total resistance of the circuit, which we will have to call 𝑅 subscript tot.

Now, we’re gonna have to go about finding what 𝑅 subscript tot actually is. And to do this, we’ll have to recall that for resistors in series, the total resistance of a set of resistors, 𝑅 subscript tot, is equal to the resistance of the first resistor connected in series plus the resistance of the second resistor connected in series plus so on and so forth, however many resistances there are connected in series.

In our scenario, we’ve got two resistors, 𝑅 one and 𝑅 two. And hence, we can say that the total resistance of our circuit here, 𝑅 subscript tot, is equal to 𝑅 one plus 𝑅 two, which means that we have an expression for 𝑅 subscript tot in terms of 𝑅 one and 𝑅 two. And we can substitute that expression into this equation here. Specifically, instead of 𝑅 subscript tot, we can substitute in 𝑅 one plus 𝑅 two, which now tells us that the total potential difference across the circuit is equal to the current through that circuit multiplied by the sum of the resistances 𝑅 one and 𝑅 two.

Now, at this point, we might be wondering, what’s the point of all of this? Why are we writing down all of these mathematical expressions? Well, the reason is because a potential divider circuit does something very special. To see this, we’re going to have to do one more step of mathematics. Specifically, we’re going to want to find an expression for the ratio between the potential difference across the second resistor and the potential difference across the whole circuit. We want to find 𝑉 two divided by 𝑉 in.

But we know from this equation here that 𝑉 two is equal to 𝐼 multiplied by 𝑅 two. That’s the current through the resistor multiplied by its resistance. And we know that 𝑉 in is equal to 𝐼 multiplied by 𝑅 one plus 𝑅 two. So that’s the current through the whole circuit multiplied by the sum of the two resistances, at which point we can see that both in the numerator and the denominator we’ve got the current 𝐼. And 𝐼 divided by 𝐼 is equal to one. And so the expression on the right-hand side simplifies to 𝑅 two divided by 𝑅 one plus 𝑅 two.

And what this means is the following. Despite the fact that we have a fixed potential difference across our circuit, which is 𝑉 in, that is provided by whatever battery was connected here, which we said we don’t really care about, we can still modify the potential difference across resistor 𝑅 two simply by changing the values of resistances 𝑅 one or 𝑅 two. In other words, the potential difference 𝑉 two across resistor 𝑅 two is only dependent on the resistances 𝑅 one and 𝑅 two, as we see in this expression.

And this is the point at which we bring back in the parallel connection to resistor 𝑅 two. We connect some components over here. Again, it doesn’t matter what those components are. The point is that by simply modifying the values of the resistances 𝑅 one and 𝑅 two, we can change the potential difference that is across 𝑅 two and, therefore, change the potential difference that’s dropped across the connected components here. Because now we can recall that in a parallel circuit, the potential difference is the same across parallel branches. In other words, if the potential difference across resistor 𝑅 two is 𝑉 two, then the potential difference across whatever is connected here is also 𝑉 two. And as we’ve seen from this equation here, the ratio between 𝑉 two and 𝑉 in is only dependent on the values of the resistances 𝑅 one and 𝑅 two. Those are the only things in our expression.

Now, if 𝑉 in is fixed, as we’ve been assuming up until this point, 𝑉 in being the potential difference provided by, say, the battery, then this equation is most useful to us if we rearrange it slightly. If we multiply both sides of the equation by 𝑉 in, then we find that the potential difference 𝑉 two across the resistor 𝑅 two is equal to this ratio, 𝑅 two divided by 𝑅 one plus 𝑅 two, multiplied by what we’re saying is a constant potential difference provided by the battery, 𝑉 in. And so this is telling us that we can tune the value of 𝑉 two, the potential difference not just across resistor 𝑅 two but across whatever components are connected here, simply by changing this ratio of resistances. And we can see that 𝑉 two can take any value between zero volts and 𝑉 in.

So despite the fact that we only have a battery that provides 𝑉 in of potential difference, we can actually tune our circuit so that we can choose any potential difference between zero and 𝑉 in to be across any components that we’re connecting over here. And we can see that the range of 𝑉 two is between zero and 𝑉 in if we consider the two following scenarios.

Firstly, if we consider a scenario where the value of 𝑅 two is zero, in other words, there’s not actually a resistor connected here, it’s just a piece of wire, or equally we’re thinking about it as connecting a resistor with zero resistance in this position. In that case, our expression for 𝑉 two becomes 𝑉 two is equal to zero, because that’s what 𝑅 two is, divided by whatever 𝑅 one is plus zero. And we multiply this by the potential difference provided by the battery, which is 𝑉 in.

But then because we’ve got zero in the numerator, this whole fraction becomes zero. And therefore, this whole right-hand side expression becomes zero. In other words, we find in this particular scenario, where 𝑅 two is equal to zero, the potential difference across this resistor is also zero. And therefore, once again, the potential difference across all components connected here is zero.

If we now instead consider a scenario where 𝑅 one is equal to zero and 𝑅 two has some nonzero value, so in this case there’s no resistor connected in the 𝑅 one position or a resistor with zero resistance is connected there, then our expression for 𝑉 two becomes 𝑅 two divided by zero plus 𝑅 two multiplied by 𝑉 in. And this fraction simplifies to just 𝑅 two divided by 𝑅 two, which is equal to one. And so we find that 𝑉 two is equal to one multiplied by 𝑉 in or simply 𝑉 in.

And so considering these two extreme cases, we’ve found the range of 𝑉 two. 𝑉 two can be anything between zero and 𝑉 in depending on the values that we choose for the resistances 𝑅 one and 𝑅 two. But the point is that this equation is an important one to be able to use.

Now, we’ve looked at the basics of a potential divider circuit and specifically what it does. But why is it a useful circuit to know? How can it be used practically? Well, one practical use of a potential divider circuit can be found if instead of having a box standard constant resistor here, we replace it with a different kind of resistor known as a thermistor.

Thermistors are temperature-dependent resistors. In other words, the value of the resistance 𝑅 two will change as the temperature of the environment surrounding this resistor changes. Now, there are different kinds of thermistors. But a common type of thermistor will behave according to this plot shown.

Specifically, we’ve plotted the temperature of the environment surrounding our thermistor on the horizontal axis. And we’ve plotted the value of the resistance of the thermistor 𝑅 two on the vertical axis. And what we see here is that if the value of the temperature is relatively low, then the resistance of the thermistor 𝑅 two is quite high, whereas if the value of the temperature is high, then the resistance of the thermistor is quite low.

And we can summarize this in the following way. If the environment around the thermistor is cold, the value of 𝑅 two is large. If the temperature surrounding the thermistor is hot, the value of 𝑅 two is small. But then based on this equation over here, we see that for large values of 𝑅 two, the value of 𝑉 two is large as well. And for small values of 𝑅 two, the value of 𝑉 two is small as well.

And based on this circuit, what we can do is to connect some sort of heating element in this part of the circuit here. Because this way when the temperature of the environment around our thermistor is low, when it’s cold around our thermistor, this results in a large value of 𝑉 two, a large potential difference across a heating element, which will be connected here. And therefore, the heating element will have a current through it and start heating up the environment, whereas if the environment is hot around our thermistor, then the value of 𝑉 two will be small. And so our heating element will not actually heat up the environment.

So a potential divider circuit is essentially acting as a very primitive thermostat. It detects the temperature in its environment by using a thermistor and switches on or off a heating element connected here accordingly. Now, real thermostats are much more complicated than this. But still, we can use a thermistor in a temperature-dependent circuit. And in this case, it’s specifically a potential divider circuit.

Another practical use for a potential divider circuit is instead of using a thermistor here, we could use what’s known as a light-dependent resistor or LDR. Now, an LDR displays similar behavior to the thermistor we discussed earlier, except that it depends on the intensity of light falling on it rather than the temperature of the surrounding environment. We can even plot a similar graph showing how the resistance of the LDR changes. But on the horizontal axis, we’d have to plot light intensity rather than temperature.

And so what we find here is that when the light intensity falling on our LDR is low, in other words, when the surroundings of the LDR are dark, the resistance of this particular LDR, 𝑅 two, is high, which means that the potential difference 𝑉 two across that part of the circuit is also high, whereas when the light intensity falling on our LDR is high, in other words, when it’s light surrounding our LDR, this results in a small value of the resistance 𝑅 two and, therefore, a small value of 𝑉 two. And so instead of a heating element, this time we can connect some sort of lighting element, some sort of light bulb, for example, which would then light up if the surroundings of the LDR are dark. And it would switch off if the surroundings are light.

And so at this point, we’ve seen a couple of practical uses of a potential divider circuit, which means we can take a look at an example question.

A potential divider has an input voltage of 48 volts. The resistance of the second resistor, 𝑅 two, is 100 kiloohms. The output voltage is drawn across the second resistor, 𝑅 two. What resistance must the first resistor 𝑅 one have in order to produce an output voltage of 32 volts?

Okay, so in this case, we’re dealing with a potential divider circuit. So let’s first start by drawing our potential divider circuit. Let’s first start by recalling that a potential divider circuit consists of two resistors connected in series. We’ll call the resistance of our first resistor 𝑅 one and the resistance of our second resistor 𝑅 two, as we’ve been told to in the question. We also recall that across this part of the circuit, some sort of power source is connected, often a battery, but not necessarily so. And this power source is what provides what’s known as the input voltage. Let’s call this input voltage 𝑉 subscript in. And in this case, in the question, we’ve been told that this input voltage is equal to 48 volts.

Now, additionally, we’ve been told that the resistance of the second resistor 𝑅 two is 100 kiloohms. That’s 100000 ohms because the prefix kilo- means 1000. We’ve also been told that the output voltage in our potential divider circuit is drawn across the second resistor. What this essentially means is that we connect a pair of wires here and here so that we can connect some components in parallel with our resistor 𝑅 two. In this case, we’re not really worried about which components are connected here, but just that the output voltage, which we will call 𝑉 subscript out, is drawn across our second resistor.

Now, the reason that we know that this is the output voltage is because the voltage across our second resistor must be the same as 𝑉 out. The reason for this is that our resistor 𝑅 two and whatever components are connected here are connected in parallel. And components in parallel have the same potential difference across them. Therefore, to reiterate, the potential difference across resistor 𝑅 two is 𝑉 subscript out. That’s the output voltage. And that’s also the potential difference across whatever components are connected here and is therefore the output voltage.

We’ve been told in the question that the output voltage must be equal to 32 volts. And we’ve been asked to try and work out what the resistance of the first resistor 𝑅 one must be in order for this to be true. So in order to answer this question, we will need to recall the potential divider equation. This equation tells us that the potential difference across resistor 𝑅 two, which we’re calling 𝑉 two, is equal to the resistance 𝑅 two divided by the sum of the resistances 𝑅 one plus 𝑅 two all multiplied by the input potential difference 𝑉 subscript in.

Now remember, the potential difference across our second resistor 𝑉 two is the same as our output potential difference because they’re connected in parallel. So we can replace 𝑉 two with 𝑉 out here, at which point we see we’ve already got a value for 𝑉 out, a value for 𝑅 two, and a value for 𝑉 in. We just need to rearrange to solve for 𝑅 one. We can do this by multiplying both sides of the equation by 𝑅 one plus 𝑅 two and dividing both sides of the equation by 𝑉 out. Because this way on the left-hand side we’ve got 𝑉 out over 𝑉 out, which is equal to one. And on the right-hand side, we’ve got 𝑅 one plus 𝑅 two both in the numerator and denominator.

Then we simply subtract 𝑅 two from both sides of the equation so that we’re left with 𝑅 one on the left, at which point we simply substitute in the values on the right-hand side, taking care to notice that in this fraction the unit of volts is both in the numerator and denominator and that if we’re going to stick with kiloohms, then our final answer for 𝑅 one is going to be in kiloohms as well. When we simplify all of the right-hand side, we find that our final answer is 𝑅 one is equal to 50 kiloohms.

So let’s summarize what we’ve talked about in this lesson. We’ve seen that potential dividers consist of resistors connected in series, with an output voltage drawn across one of the resistors. We saw the relationship between the output and input voltage and how it links to the resistances used in the circuit. And finally, we saw that thermistors and light-dependent resistors, or LDRs, can be used in these circuits for practical applications.