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The diagram shows a potential divider containing a resistor and a capacitor. Initially, no charge stored in the capacitor. What is the potential difference across the capacitor immediately after the switch is closed?

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Video Transcript

The diagram shows a potential divider containing a resistor and a capacitor. Initially, no charge stored in the capacitor. What is the potential difference across the capacitor immediately after the switch is closed?

Okay, so first of all, let’s take a quick look at the diagram. We can see that we’ve got a resistor, which has a resistance of 15 kiloohms, and we’ve got a capacitor, which is placed in series with a resistor. As well as this, we’ve got a switch that is on a branch that is at 10 volts. And this end of the circuit is at zero volts. Now the reason this circuit is called a potential divider is because of this branch over here, because we can measure the potential difference across this part of the circuit and this potential difference is going to be some fraction of the potential difference across the entire circuit. And so in essence, it’s dividing up this potential difference into two chunks: the potential difference across here and the potential difference across here, hence a potential divider.

Now we’ve been told that initially no charge is stored in the capacitor, and we’ve been asked to find the potential difference across the capacitor immediately after the switch is closed. So initially because the switch is open, there’s no current running for the circuit. However, as soon as we close the switch, there is now going to be a current flowing through the circuit. So as soon as we close the switch, the capacitor starts to charge up. However, immediately after we close the switch, the capacitor still has zero charge on it because it was uncharged before we closed the switch. And so as soon as we close the switch, the charge starts building up, but it’s still zero immediately after we closed the switch. As a result of this, there are no charges on either plates of the capacitor, and so there is no electric field between the plates of the capacitor.

What do we mean by this? Well, let’s zoom into the capacitor little bit. So here’s our capacitor in the circuit. Now normally in a circuit, the charge carriers are electrons. In other words, when there is a current flowing through the circuit, what that actually means is that it’s a flow of electrons moving around the circuit. And so as soon as we close the switch, there is a current flowing through the circuit. And hence electrons start to move away from one of the plates of the capacitor. And lots of electrons that were flowing around the circuit end up being deposited on the other plate. As a result of this, the first plate which is losing electrons ends up with a net positive charge and the other plate which is gaining electrons ends up with a net negative charge.

So this is what happens when a capacitor charges up. But then as a result of this, we’ve now got lots of positive charges on one plate and lots of negative charges on the other. This means there’s going to be an electric field between the two plates of the capacitor. And this electric field, which we will call 𝐸, happens to be directly proportional to the potential difference across the capacitor. In fact, it’s also related to the distance between the plates of the capacitor, which we’ll call 𝑑, but that’s not really relevant here.

All we need to know is that the electric field is directly proportional to the potential difference across the capacitor. And we’ve been asked to find the potential difference across a capacitor immediately after the switch is closed. But then, as we said earlier, immediately after the switch is closed, the capacitor is still uncharged. Therefore, there is zero electric field between the plates and so the potential difference across the capacitor is zero as well, but only immediately after the switch is closed. Because as charge starts to build up, the potential difference increases as well. So anyway, our answer to this part of the question is that the potential difference across the capacitor immediately after the switch is closed is zero volts.

Moving on then, let’s now consider the resistor in the circuit. What is the potential difference across the resistor immediately after the switch is closed? Okay so to answer this question, we need to realise that the voltage at this part of the circuit is 10 volts, and here it’s zero volts. This means that the potential difference between the two parts of the circuit is 10 volts minus zero volts, and that’s just 10 volts. Now, we also know that the resistor and capacitor are in series. Therefore, this 10 volts of potential difference get split across the resistor and the capacitor. However, we’ve just seen in the previous part of the question that as soon as the switch is closed, the potential difference across the capacitor is zero volts. Therefore, the remaining 10 volts must be across the resistor. And so that is our answer to this question.

Once we leave the circuit for some time however, the charges on the capacitors start to build as we said earlier. And so the potential difference across that capacitor starts to increase. So for example some time later, the potential difference could be one volt across the capacitor, and so the remaining nine volts must be across the resistor, and so on and so forth. But anyway, we don’t need to worry about that bit. Let’s now move on to considering this output voltage here. What is the output voltage immediately after the switch is closed? Okay, so as we mentioned right at the beginning, because we’ve got a potential divider, the output voltage is related to this part of the circuit and the potential difference across it.

Specifically, the output voltage minus whatever the voltage is here is the potential difference across the component in that part of the circuit, in this case the capacitor. But then we saw earlier that as soon as the switch is closed, this potential difference is zero volts. And so what we have is the output voltage minus zero volts, that’s this zero volts here, is equal to the potential difference across the capacitor, which is zero volts. And this means that the output voltage must also be zero volts. So that’s our answer to this part of the question. Now up until this point, we’ve only been considering what happens as soon as we close the switch. Let’s think about what happens after we’ve left the circuit alone for some time.

The capacitor takes 30 seconds to fully charge. What is the output voltage after 30 seconds? Alright, so to answer this question, we need to think about what happens when a capacitor gets fully charged. Coming back to a picture of the capacitor, we said earlier that one of the plates becomes positively charged as the current flows through the circuit, and the other plate becomes negatively charged. Now as the current continues to flow through the circuit, the top plate in this case will become more and more positively charged, and the bottom plate will become more and more negatively charged. And this makes it harder and harder for electrons to flow through the circuit, because now we’re trying to force even more electrons onto the negatively charged plate and even more electrons away from the positively charged plate. And this is a problem because negative charges will repel negative charges. And so all of these negative charges already on the plate are going to be trying to repel the electrons that have been forced onto this plate.

Similarly, negative charges are attracted to positive charges. So all the negatively charged electrons leaving this plate have a stronger and stronger force of attraction to this more and more positively charged plate. The only thing that’s forcing the electrons to keep moving is this potential difference here. But then as it becomes harder and harder for electrons to move, eventually current stops flowing. And at that point, we say that the capacitor is fully charged. In other words, there is no more flow of charge onto this plate or away from this plate. And that actually is key to answering this question, the fact that there is no more current flowing. Because if there is no more current flowing in the part of the circuit that has the capacitor in it, then because the resistor is in series with it, there is no current in the resistor either.

Therefore we can say that 𝐼 res, the current through the resistor, is equal to zero amps. But then a resistor we can recall follows something 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 to that component multiplied by the resistance of that component. Now we’ve been told that the resistance of the resistor is 15 kiloohms, but this is not important because the current is zero at this point. So we can say that the potential difference across the resistor, 𝑉 sub res we’ll call it, is equal to zero amps, that’s the current through the resistor, multiplied by 15 kiloohms that’s the resistance of the resistor. But then zero times another number is zero, so the potential difference across the resistor is zero volts. Hence, we can say zero volts in our diagram as the potential difference across the resistor.

Now, we still need to remember that this branch of the circuit is at 10 volts and this branch of the circuit is at zero volts. This means that there is still 10 volts of potential difference across this part of the circuit. But then if there is zero volts of potential difference across the resistor, then all of those 10 volts must be across the capacitor. And so we know that the potential difference between output voltage and zero volts, in other words the potential difference across this part of the circuit, is 10 volts. So this time we can say that output voltage, that’s this voltage here, minus zero volts, that’s this voltage here, is equal to the potential difference across the capacitor; that’s 10 volts. And so we find out the output voltage is equal to 10 volts. And that is our final answer to the final part of the question.

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