Video: Capacitors

In this video, we will learn how capacitors work in circuits, the property of capacitors known as capacitance, and the energy stored in a capacitor.

16:08

Video Transcript

In this video, we will be looking at an electrical circuit component, known as a capacitor. Now there’re many different kinds of capacitor that are used in electrical circuits. But today, we’ll be looking at a very simple construction of a capacitor, the parallel plate capacitor. As the name suggest, this consists of two plates, placed parallel to each other. Now, these plates are made up of a conducting material. And they’re separated by an insulating material. So in this case, the two plates are going to be made up of some conducting material, probably a metal. And we can see that, in between these two plates, we’ve left some space. The reason for this is that the insulating material in between the plates is actually air in this case, because air is a pretty decent insulator.

However, sometimes instead of air, a block of another insulating material can be used as well. But for simplicity, let’s just consider an air spaced parallel plate capacitor. Now, if we were to take this capacitor with the two parallel plates, which by the way have been connected to wires in this case, and we were to look at it side-on, then what we’d see is this. We’d see the wires either side of the two parallel plates. So here are the plates. And we can see that, in between them, there’s a little air gap. And because this is what we’d see when we look at a parallel plate capacitor side-on and it’s relatively simple to draw, this is therefore the circuit symbol for a capacitor.

Now, it’s important to note that not all capacitors are parallel plate capacitors. But we’ll still use this as the circuit symbol to represent any kind of capacitor generally. So we’ve just seen what a capacitor is. But the question is what does it do. Well, the function of a capacitor can be described in the following way. Well, let’s say we take a parallel plate capacitor and connect it to a DC cell. What the DC cell does is supply a potential difference and therefore causes charge to flow in the circuit. In other words, there is now a current in the circuit. And we can think of this current as positive charges flowing this way round the circuit away from the positive terminal of the DC cell. And that’s what we do when we’re talking about conventional current. Or equivalently, we can think about negatively charged electrons flowing the opposite way, away from the negative terminal.

By the way, what we can think about is positive charges flowing this way around the circuit or negative charges flowing this way. But then as soon as we get to the plates of the capacitor, there is a problem because, remember, in this particular capacitor, there is an air gap. Air is an insulator. And therefore, charge cannot flow across this gap. And so what actually happens is that, for example, if we’re thinking about the negatively charged electrons, they start getting deposited onto this plate here. It starts becoming negatively charged.

And similarly, for the other plate, we have positive charges being deposited onto this plate. In reality, what’s happening is that negatively charged electrons are actually flowing away from this plate, in keeping with our counterclockwise flow of negatively charged electrons in this circuit. And so this plate ends up being positively charged. And so the net result of all of this is that positive charges are building up on this plate. And negative charges are building up on this plate.

However, as we can imagine, as time passes and more and more negative charge builds up on the right-hand side plate and more and more positive charge builds up on the left-hand side plate, we can see that it would get more and more difficult to put even more negative charge on the right-hand side plate. This is because negative charges repel each other. And the more we try and isolate negative charges onto this plate, the more they will want to resist the further putting on of negative charges onto that plate. And eventually, at some point, the force provided by the DC cell that’s trying to push even more electrons onto the right-hand side plate will be exactly balanced by the repulsive force of all of these negative charges trying to push any electrons away from this negatively charged plate.

And of course, we can think about conventional current in exactly the same way. We can say that the cell is trying to push conventional current in the clockwise direction. But all of the positive charges on the plate are trying to push positive charges away from that plate. And of course, all of those charges can only go in this direction because they can’t travel across the air gap. Now at the point where the forces on the charges trying to push charges onto the plate are exactly balanced by the forces trying to push charges away from the plate, there will no longer be any more charge buildup on the plates.

At that point, we say that the capacitor is charged. There’s no longer a current in the circuit because there’s no flow of charge in the circuit. And it’s also worth noting by the way that the amount of positive charge on this plate is exactly equal to the amount of negative charge on this plate. This is because the only reason this plate is positively charged is because a certain number of electrons have left that plate, travelled around the circuit, and been deposited on this plate. And hence, the magnitude of the charges on the two plates must be the same. And this makes life easier for us because we can then conventionally say that the charge on this capacitor is 𝑄, where 𝑄 is the magnitude of charge on either one of these plates.

In other words, if there’s positive five coulombs of charge deposited on the positive plate, then 𝑄 is five coulombs. And by the way in this situation, there will also be negative five coulombs of charge on the negative plate. But anyway, so 𝑄 is the magnitude of charge deposited on one of the plates. Now let’s stop and think for a minute. What we’ve got in this situation now is a conducting plate with positive charge and a conducting plate with negative charge of the exact same magnitude, placed parallel to that first plate. So if we zoom in slightly, here’s our positively charged plate. And here’s our negatively charged plate.

Now, we can recall that, in between charged plates, we have an electric field. This is because electric field lines flow from positive charges to negative charges. And so in between parallel charged plates, the electric field looks something like this, flowing from the positively charged plate to the negatively charged plate. And of course, before the capacitor was charged, there was no electric field between the plates because there were no charges on those plates. This electric field has only developed as the charge on these plates has increased. And when the capacitor is set to finally be fully charged, the electric field between this plate has reached its maximum value.

Now, it’s taking some amount of energy, supplied by the DC cell, to actually push the negative charges onto this plate or the positive charges onto this plate. In other words, some amount of work has had to be done in order to push those charges onto those plates. And so a capacitor is said to store energy in the electric field. This is because work was done to push the charges onto these plates. And as those charges were pushed onto those plates, this electric field grew larger and larger. And so we can say that a capacitor stores energy in an electric field.

Now, coming back to our charged capacitor, with a charge 𝑄 on each one of these plates, we can imagine that, due to the massive electrostatic repulsion on each one of these plates, negative charges want to flow away from the plate. And positive charges want to flow away from this plate. But as we saw earlier, this tendency was balanced by the force supplied by the cell. So let’s imagine now if we disconnect the capacitor from the circuit. Let’s say we take away all of the wires and the cell. And then, what we’re left with are simply two charged parallel plates. Now in this case, there’s no way for charge to flow. And so these plates remain charged.

But then, we can imagine hooking up this parallel plate capacitor to a resistor. Now in this case, there is somewhere for charge to flow. The charge can flow along the wires and through the resistor. And there’s no cell to stop this flow from occurring. So the electrostatic repulsion of all the negatively charged electrons on the negative plate will take over. And electrons will be repelled from this plate now. They will start to flow around the circuit in this direction. And similarly, positive charges will be repelled from this plate flowing in this direction. And as that happens, this plate becomes less and less negatively charged. And this plate becomes less and less positively charged. In other words, at this point, the capacitor is said to be discharging. And it does this by setting up a current in this circuit, which in other words is a flow of charged particles away from the plates of the capacitor.

Right to the beginning of the discharge process when there were lots of electrons on this plate, the electrons would’ve begun to flow really quickly away from this plate. But then, as the charge depletes on this plate, the electrostatic force of repulsion between negatively charged electrons decreases as they’re getting more and more spread out over the circuit. And so the electrons are not so strongly repelled by this plate. In other words, the initial current when the capacitor beings to discharge is very high. But as the capacitor discharges, the current decreases, until, eventually, we get to a point where there’re basically no more charges on either one of these plates. And this means that there’s no current in the circuit. And therefore, there’s no electrostatic repulsion to try and push electrons away from the right-hand side plate or, equivalently, positive charges away from the left-hand side plate. At this point, the capacitor is said to be discharged.

So now that we’ve got a discharged capacitor doing nothing in this circuit, let’s take out this resistor and once again replace it with a cell. Now, we’ve put the cell in with the same orientation as before. So once again, this cell is trying to push positive charges onto this plate and, equivalently, negative charges onto this plate. Now, we’ve seen that as we do this, charge slowly starts to build up on these plates. And therefore, as we’ve seen in this diagram, we slowly build up the electric field between these plates. Now, if we were to take a charged particle and place it in this electric field, then a positively charged particle would be attracted to the negative plate. And a negatively charged particle would be attracted to the positive plate.

Now, remember, these are not the charges that are on the plates. These are just charges that we’ve placed into the electric field from outside. But the point is that if we were to take a charged particle and place it in this electric field, it would move to either the positively charged plate or negatively charged plate. In other words, this electric field would cause charges placed in the field to move. And moving charges can be thought of as a current because, remember, current is the rate of flow of charge, charge divided by time or the amount of charge flowing past a point divided by the time taken for that charge to flow.

But then, because an electric field causes charges to flow, this means that a current would be set up where a positively charged or negatively charged particle to be placed in this air gap. And this therefore means that there must be a potential difference across this air gap. Now, the stronger the electric field in between this air gap, the larger the potential difference because a stronger electric field means that charges placed in this electric field will experience a stronger force. So the stronger the field, the larger the potential difference. But then, we’ve seen that the electric field between these plates gets stronger as more and more charges are placed on the plates of the capacitor. In other words, the charge on these plates of the capacitor is directly proportional to what we can say is the potential difference across the capacitor. And that potential difference is directly proportional to the strength of the electric field.

And so we can take this equation, 𝑄 is directly proportional to 𝑉, and introduce a constant of proportionality. We can say that this constant of proportionality is 𝐶, which we’ll call the capacitance of the capacitor. And it can be thought about as the constant of proportionality between the charge stored on these capacitor’s plates and the potential difference across them. Or it can be thought of as the charge per unit potential difference across this capacitor. Now naturally, each capacitor has its own value for 𝐶. And 𝐶 is dependent on a couple of things, firstly the geometry of the plates being used in the capacitor. So whether they’re squares or rectangles or long or thin or placed next to each other in a particular way, all of these factors will affect the capacitance of the capacitor.

And the other factor affecting 𝐶 is something known as the permittivity, represented by the Greek letter 𝜀, of the insulator in the gap between the plates of the capacitor. Now, that’s not something we need to worry about. But something we should know is that the unit of capacitance is known as the farad, represented by a capital F and named after Michael Faraday. We can also see that one farad is equivalent to one coulomb divided by one volt or one coulomb per volt. And this is because the unit of capacitance is equal to the unit of charge divided by the unit of potential difference.

And finally, one more thing we should be looking at here is the fact that we said earlier that a capacitor stores energy in an electric field. This energy is stored in the field because work was done to push charges onto these plates. Now, it can be shown that the energy stored in the electric field of a capacitor is equal to half multiplied by the charge on the plate of a capacitor multiplied by the potential difference across it. But then, as we’ve already seen, the charge on a capacitor is equal to the capacitance multiplied by the potential difference across it. And so we can say that the charge on the plate of a capacitor is equal to 𝐶𝑉. And so this gives us an alternative expression for the energy stored in the electric field of a capacitor.

But then, another thing we could do is to substitute for 𝑉 and say that 𝑉 is equal to 𝑄 divided by 𝐶. And this would give us a third expression for the energy stored in a capacitor’s electric field. We see that this ends up being half multiplied by the charge squared divided by the capacitance. And these three expressions are equivalent ways of writing the same thing. They’re all expressions for the energy stored in the electric field of a capacitor. So if we ever need to calculate this energy and we’re given the charge on the plates of a capacitor and the potential difference across it, then we can use this expression.

Otherwise, if we know the capacitance and the potential difference, then we can use this expression. And if we know the charge and the capacitance, then we can use this expression. But the easiest thing to do is to simply memorize one of these expressions as well as this equation, which defines capacitance, and then use that equation to derive these two expressions. But anyway, so at this point, we’ve looked at lots of different features of capacitors. So let’s have a go at an example question.

As a capacitor is charged, the amount of charge on it blank, and the potential difference across it blank.

Okay, so in this question, we’re talking about charging a capacitor. And one way to do this is to connect a DC cell to the capacitor in series. Now, the DC cell applies a potential difference across the circuit that sets up a current in the circuit. And so if we can set a conventional current, then what that means is that positive charges are flowing away from the positive terminal of the cell and being deposited onto this particular plate of the capacitor.

Similarly, negatively charged electrons are flowing this way around the circuit and being deposited onto this plate. This is what it means for the capacitor to be charged, because as the current is present in the circuit and charges cannot flow across the gap in between the plates of the capacitor, we see that there’s a buildup of positive charge on the left-hand side plate, as we’ve drawn it, and a buildup of the same amount of negative charge on the right-hand side plate, as we’ve drawn it.

And so what we’ve got here is a plate where there’s an increasing amount of positive charge being deposited onto it and another plate parallel to this, where an increasing amount of negative charge is being deposited. Now, if we were to zoom in slightly to the setup, we can see that there’s the positively charged plate and the negatively charged plate. Now, we can recall that, in between two oppositely charged parallel plates, an electric field will be set up. And that field will be going from the positively charged plate to the negatively charged plate. So we can draw in the electric field lines between these two parallel plates.

Now, what this electric field means in practice is that if we were to take an external electric charge so, for example, a positively charged particle from somewhere else and place it into this electric field, then that charged particle would experience a force. And that force would be in the direction of the plate with the opposite charge to that particle. In other words, a positive charge would flow in this direction. And a negative charge would flow in this direction.

Now, it’s important to note that the charges that were placing in the field are not the same as the charges on the plates of the capacitor. Those cannot flow across the gap between the plates. But anyway, so what we’ve got is an electric field between these plates. And external charges placed between these plates will be moving towards one plate or another. In other words, this electric field is causing a flow of external charge. Or another way to think about this is that the external charges are forming a current, even if there’s just one charged particle. The fact that there’s a charged particle moving means that there is momentarily a current because, remember, current is defined as the rate of flow of charge.

And we can see that if we build up the charge on these plates, so we increase the amount of charge on each plate, then the electric field gets stronger, which in other words means that the force on any of these charged particles will be larger. And so a positively charged particle with this increased electric field strength will experience an even larger force towards the negatively charged plate. And similarly, the negatively charged particle will experience a larger force towards the positively charged plate. In other words then, as the charge on these plates increases, the strength of the electric field increases. And therefore, the potential difference across the plates increases as well.

Therefore, coming back to our original statement, we can say that as a capacitor is charged, the amount of charge on it increases, and the potential difference across it increases as well.

Okay, so now that we’ve had a look at an example question, let’s summarise what we’ve talked about in this lesson. We firstly saw that the simplest construction of a capacitor consists of conducting parallel plates, separated by an insulator. And commonly, this insulator is just air. We also saw that the circuit symbol for a capacitor consists of these two parallel lines. And we can connect it to wires if necessary. Secondly, we saw that capacitance is defined as 𝐶 is equal to 𝑄 divided by 𝑉, where 𝑄 is the magnitude of charge on one of the plates of the capacitor. And 𝑉 is the potential difference across the plates of the capacitor.

We also saw that capacitance has the unit farad, where one farad is equivalent to one coulomb per volt. And finally, we saw that the energy stored in the electric field of the capacitor is given by multiplying half by the charge on the plate of a capacitor by the potential difference across that capacitor. But this is equivalent to half multiplied by the capacitance multiplied by the potential difference squared or half multiplied by the charge squared divided by the capacitance. So capacitors are circuit components with the ability to charge and discharge. And in doing so, they store energy in the electric field between the plates of the capacitor.

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