The diagram shows a circuit containing a capacitor and a resistor. The capacitor is discharging and there is an electric current through the circuit. What would happen if a capacitor were allowed to discharge through the same circuit, but without the resistor?
Okay, so first of all in this circuit, we can see that we’ve got a capacitor and a resistor placed in series with each other. Now, apart from the capacitor and the resistor, there are no other components in the circuit. And what we’ve been told is that the capacitor which was initially charged is now discharging. And current is flowing through the circuit. So before we even think about answering the question, let’s think about what it means for a capacitor to be charged.
Well, as we can see from the circuit symbol of a capacitor, the most basic kind of capacitor we can think about is the parallel plate capacitor. As the name suggests, a parallel plate capacitor consists of two parallel plates. So that’s plate number one and plate number two made of conducting materials. And as well as this, we can see wires coming off these two plates. Now, those wires will carry current to the capacitor. But charge cannot jump across the gap between the two parallel plates of the capacitor.
Therefore, what we can do is to connect up this capacitor with, say, a potential difference source. And we can then charge it up. The way to do this, as we said earlier, is to connect it to a potential different source such as a cell. And what that cell does is causes a movement of charge because negatively charged electrons will move in this direction around the circuit and be deposited on this plate here. And in fact, the electrons on this plate will be stripped from that plate and travel counterclockwise around the circuit as well.
What this means is that the plate on the right has more and more negative charge electrons dumped onto it. And so it becomes more and more negatively charged as time progresses. And at the same time, the plate on the left has negatively charged electrons taken from it. And, therefore, it’s left with a deficit of negative charge. In other words, as time progresses, this plate becomes more and more positively charged. So this is what it means for a capacitor to be charged. Now, once we’re happy with the level of charge on the plates of the capacitor, we can disconnect the capacitor from the circuit. And now, what we have is a charged capacitor. This is the kind of capacitor that was then connected into this circuit and allowed to discharge.
Now, the reason that it would discharge is because we can see, for example, on the right-hand side plate that there are lots of negative charges. And because we know that like charges repel each other, these negative charges will be trying to move away from each other as much as possible. Now in this state, when the capacitor is not connected to anything, there’s nowhere for the negative charges to go. But as soon as we connect the capacitor into the circuit, the electrons can stop flowing away from the negatively charged plate because of the wires. And of course, the same is true for the positive charges on the left-hand side plate.
Because it’s so positively charged, that plate will attract negatively charged electrons. Or we can, of course, think about it as conventional current which is the flow of positive charge. And so these positive charges will be flowing away from the positively charged plate. Regardless of how we think about it, this circuit now consists of a capacitor discharging. In other words, because we’ve connected the plates of this capacitor to wires and to a complete circuit, the charge on each one of these plates is decreasing. And because we’ve got moving charges in this circuit, we’ve, therefore, got a current for the circuit because, remember, a current is defined as the amount of charge flowing past a certain point per unit time.
And so if we’ve got flowing charges or moving charges within the circuit, we’ve got a current. Now, we can also realise that a resistor is called a resistor because it resist the flow of charge. In other words, it’s going to limit the amount of current through the circuit. And the way we’re gonna think about this is that, just as the capacitor starts to discharge, we can think of it as generating a potential difference which we’ll call 𝑉, across the resistor. In other words, we can basically think of the capacitor as a cell, except that a cell’s potential difference stays constant over time, whereas as the capacitor discharges, this value of 𝑉 will decrease over time.
But then let’s just think about the very beginning of the discharge phase of the capacitor at which point the potential difference that it generates is 𝑉. And as well as this, let’s say that the resistor has some resistance 𝑅. We don’t know what this is. But that’s not relevant. At this point, we can recall Ohm’s law, which tells us that the potential difference across a component in a circuit is equal to the current through that component, 𝐼, multiplied by the resistance of that component. And because we know the capacity will produce a potential difference 𝑉 at a certain point in time, in other words, this value 𝑉 is fixed for certain point in time, the current through the circuit 𝐼 is limited by the value of the resistance 𝑅 because the current multiplied by the resistance together must give the value of 𝑉.
And, therefore, if we have a resistor in the circuit, then that will allow a certain current through the circuit. However, if we were to decrease the resistance in the circuit, then the current through the circuit would increase because, remember, 𝐼 and 𝑅 multiplied together must give the same value 𝑉. But then if we were to decrease the resistance so much that the resistance of the circuit became zero which, in other words, means that we’ve got rid of the resistor entirely and we just had a capacitor discharging through a closed circuit, then the current would increase massively. And then if the current was very large, then this means that there’s a large amount of charge flowing per unit time. And this is, of course, the charge flowing through the circuit.
Therefore, coming back to our picture of the capacitor here, if a large amount of charge is allowed to flow per unit time, then that means that almost all of the charge that was on this plate initially would be allowed to flow very quickly away from the capacitor. And hence, through this circuit without any resistor, the capacitor would be allowed to discharge almost instantaneously. It would happen very very quickly. The only thing that would limit it will be the resistance of the wires through which this capacitor is discharging because, remember, in real life, even wires have resistance, admittedly a very small resistance but a resistance nonetheless.
So the question asks us, what would happen if a capacitor were allowed to discharge through the same circuit but without the resistor, in other words, if the capacitor were allowed to discharge through this circuit? And as we’ve seen, because the resistance of the circuit is very low, the current through the circuit would be very high which, in other words, means that there’s a huge amount of charge flowing per unit time in this circuit. And that charge comes about because the charge particles on the plates of this capacitor start flowing very very quickly. Hence, our final answer is that the capacitor would discharge almost instantaneously.