 Configurations of electric charge require energy to be assembled, but once assembled, they can store that energy through the potential energy present in their configuration, due to the mutual forces they exert on one another through their electric fields. Let's take a look at a workhorse of common electric circuitry, the capacitor, which is nothing more than a configuration of charge that can store energy, or if disassembled, release energy. The topic that I'm going to cover today regards the first major electrical device that we're going to encounter, the capacitor, and it combines all of the elements that we've talked about so far in the class. There will be electric fields, there will be charges separated by a distance, there will be electric potential differences, and therefore potential energy available in this device, but before we get going I want to cover something which is also going to be present in this lecture as we move forward, and that is the battery. The battery is the major workhorse of circuit discussions, and by circuit what I mean is the following. If I can zoom in here for your convenience, it's convenient at this point to discuss the very basic idea of a circuit. Now we haven't quite talked about the movement of electric charge yet, we're going to get to that in detail shortly, but suffice to say that a circuit is basically a device constructed out of conductive material, so for instance here I've made a very simple loop, and it's made from a conductor, that is it offers essentially no resistance to the motion of electric charge, so no resistance to the movement or motion of charge, Q. That's the definition of a conductor, and that's in contrast with an insulator. An insulator is a material that doesn't let charge move around at all or very much. Circuits will be made from conductive material, and you can think of them as pipes, and if we imagine the copper or aluminum wires that usually make up a typical circuit as pipes that can carry electric charge, what we're going to be interested in knowing is what direction is positive charge plus Q flowing in the circuit. So as we build up our toolkit for understanding electric circuits, the first essential piece is merely the conductor. The conductor is the piping that allows us to hook up different components and allow charge to move freely between the components. Now having pipes by themselves is not particularly useful. If you think of electricity and the flow of electrons or the flow of positive charge in analogy to the movement of water through pipes, you need to have something that's capable of pumping the electricity through this conductor. On its own, if you just put charge in the conductor, it's not going to go anywhere. The conductor is electrically neutral because most matter that we encounter in everyday life is electrically neutral. There are equal numbers of electrons and protons in the material. And so you've got to entice the electrons to move in one direction and thus cause the positive charge to move in the other. So it's important to remember if we sketch a little picture of an atom, and out here in orbit around the central nucleus of the atom is an electron, we have to remember that when we are talking about electric charge moving in a circuit, the majority of the time what's actually moving is the electron. That is we're ripping it off of its parent atom and then moving it through the conductor. But because of the convention that when we talk about the motion of positive charge in a circuit, that's really what we mean by the motion of charge. Thank you Ben Franklin. We have to keep in mind that if the electrons go to the right, it means that positive charge is flowing in the opposite direction, that is to the left. So in a circuit, we're not so much interested in the direction that the electrons are moving, although they are the things that are typically actually moving in the circuit. We're actually interested in the direction opposite the way the electrons are moving. That's the direction that positive charge is flowing in the circuit. We need a pump. We need something that's going to entice these electrons to move in one direction and thus have the positive charge move in the opposite direction. And that is where we hit the concept of the battery. Now the battery can be drawn into a circuit diagram like the one I just showed you by putting a series of parallel lines, one big, one small, one big, one small, and then imagining a little line of conductor that comes out of each end of this little device. And then over here I have some components that go into the circuit. This is the battery and this is the pump. This is the thing that allows us to move charge. And honestly, if that's all you remember about a battery, you're already in pretty good shape, although we're going to drill down into this a little bit more deeply. Now it's convention to draw a positive sign on one end of the battery, and that happens to be the one where you have this longer terminal touching the conductor. So that's going to be the top side of the battery as I've drawn it here. And this implies by convention that the negative side is the other side of the battery. And the reason to do this is that when you draw the symbol this way, you immediately know that the flow of positive charge in the circuit is going to be out of the positive side through whatever the components are in the circuit and then back into the negative side over here. So that's why this symbol has this convention. The symbol for the battery tells you a lot once you write it down or once you see it in a circuit diagram. It tells you that this battery right here is trying to push positive charge up and around clockwise in this circuit. Now a battery is nothing more than an electric potential difference. So it contains some delta V. And if we imagine for a second this battery symbol, so here I'll draw it again, with its plus side and its minus side. Well if it's true that positive charge is moving this way through the battery, it must be true that the electric field inside the battery correspondingly points in this direction. I mean after all positive charge wants to follow the direction of the electric field lines in the way that they point. The electrons in the battery of course are going in the other direction. So if we wanted to imagine the actual things that are moving inside of these circuits, the electrons, we have to think of them as going down, not up. Now this is just the fact that we're stuck with this convention, that we think about the way the positive charge moves in circuits. And again you can thank Ben Franklin for the beginning of this convention. Alright well, so positive charge is moving up through the battery like this, being pushed through the battery, pumped like water through the circuit. This implies that the electric field in the battery must point in the same direction that the positive charge is moving. And so it must also be true that we have a situation where the initial and final electric potentials that are felt by the positive charge are in a sequence such that v-final is less than v-initial. And that is that the change in the electric potential must be less than zero because the change in the electric potential, which is v-final minus v-initial, would give you a number that's smaller than zero. So these are just some basic things that you want to keep in mind. Positive charge is going to move up through the battery from the negative terminal to the positive terminal. That's the word that's used to describe each end of the battery. And you're often used to seeing batteries that look something like this. They have a little copper top on them. And then there's a little nub up on the top. This is the positive side. This is the negative side. And so if we imagine that this is a double-lay battery, this has an electric potential difference between the two sides of the battery that's equal to 1.5 volts. That is the high potential side is at 1.5 volts, and the low potential side is at zero, as an example. Now there's one more essential thing that I want to note about this very simple little electric circuit that we've drawn here with our battery, with its positive terminal and its negative terminal. Let me redraw this one more time. Okay, so here we have a bunch of components in the circuit and we'll get into exactly what those are very shortly. We're going to visit our first component of an electric circuit in a moment. If we were to measure with some device the electric potential across this battery, we could do this, for instance, by hooking up a voltmeter, something that's actually capable of measuring an electric potential difference, to one side of the battery and the other. And that might have some kind of little gauge on it that indicates what the voltage is across the battery. Well, if we were to make a measurement, let's say that this is a AA battery, so it has a 1.5 volt potential difference across its terminals. If I were to put the voltmeter here and here, right on the top and bottom of the battery, and ask it, what is the potential difference? It would tell me 1.5 volts. Now imagine if I were to move the probes of the voltmeter down the conductor and then ask again, if we call this point A and this point B, this is the voltage at point A, and then we were to ask the question, what is the electric potential difference if I put the voltmeter 1 probet point B here and 1 probet point B here? Well, what I've done is I put a whole length of conductor in between me and the end of the battery on this side and another whole length of conductor in between me and the whole battery on this side. There are no other batteries in the way, and so the only potential difference that is being probed is still the potential difference between the battery terminals. There's no other change in electric potential as we move through this conductor and place our probe a little further down the line here and here. And so it must be true that the electric potential difference is still 1.5 volts. After all, we've encountered no other device that has an electric field than it besides the battery. And I can repeat this exercise and pick more points C and C and keep asking this question, what's the potential difference between this point and this point which really is just a bunch of conductor attached to the ends of the batteries? And I'll keep getting the same answer. Now it could be that there's some complicated mess of components inside here and inside of that we may experience a whole bunch of changes of electric potential. But because electric potential is just work per unit charge in a circuit, if work per unit charge is conserved because there are no external forces acting on the circuit, then it must also be true the potential difference is conserved. And so even if I were to stick the probes here at these points right on the edge of the components, I should still find that the potential difference across the components is 1.5 volts. So what's really great about this is because energy is conserved, I know that if this is my only battery that I'm considering and I've hidden all of my other components inside this blob and we'll explore what might be in here in a moment, I know that the electric potential difference across this blob is the same as the electric potential difference across the battery because all that's in my path is a whole bunch of conductor that doesn't impede the flow of electric charge. In and of itself it doesn't provide any more additional electric potentials that have to be crossed by charges as they move through the wire, for instance. And so if you're asked what's the electric potential difference across the components of the circuit in this picture, the answer is, well, it's just the electric potential difference across the battery, whatever that is. And so this will come in handy when we start thinking about capacitors in just a moment. So let's begin to think about capacitors and I'm going to pick up here where I left off at the end of my last actual in-person lecture. What we were doing at the end of my last in-person lecture was we were considering a very special situation. Imagine that I have a disc with area A equals pi times its radius squared. All right, so here's the radial line of that disc, some circular disc and I'm considering the electric field strength at some point P above the disc along this axis that goes through the center of the disc. Well, what we found out was as this distance above the center of the disc, which we can label as Z, as Z gets to be much, much less than R, that is, as Z gets very close to the center of the disc, we find something quite interesting happens to the electric field. The electric field is given merely by the surface charge density on the disc divided by 2 epsilon naught, where epsilon naught is a constant of nature. Epsilon naught is related to K by this formula here. Epsilon naught is 1 over 4 pi K, so if you know K, you can calculate epsilon naught. And the only thing left here is the direction in which the electric field points and if we imagine here that we have some positive charge Q spread out over this area A, then the electric field we expect to just point up toward the point P and we would expect in this case that it just points in some direction, for instance K hat straight up away from the disc. And what's really neat about this particular situation where you're very close to the surface of the disc, far closer than the radial width of the disc itself, you find that there's absolutely no dependence on Z in this formula. You can keep moving around very close to the disc and the electric field will always have the same strength, sigma over 2 epsilon naught, where again sigma is just the surface charge density, it's the charge divided by the area, which for a disc in this case is pi R squared. And this is a really neat result because we've just learned something really interesting about a disc of charge when we're very close to the surface of the disc. This gives us a uniform electric field, one that does not change with position. Unlike the point charge or the dipole field, which do change in some cases quite rapidly with your distance from the charges themselves, in this particular special situation where you're very close to the surface of a disc of charge, you actually find that the electric field is essentially uniform just above the disc. It isn't really changing at all with your distance from the disc. Now eventually you go far enough and that won't be true anymore. So we have to consider this limiting case. But what's great about this is, you know, it's essentially very straightforward in the lab environment or in an engineering environment to create a disc that's filled with a whole bunch of charge. You know, you could imagine making a disc out of some insulating material and through some means depositing a net charge on it. And the great thing about doing that is now right above the surface of the disc, you've engineered a situation where you have a uniform electric field whose magnitude is given by a very simple equation, the surface charge density divided by 2 epsilon naught. And this is why this is so powerful. Because what happens if we put two of these discs next to each other, one with a charge plus q and one with a charge minus q? So let's explore that possibility. So my picture will now be that I have a disc up here with a charge minus q. And then below it I put a disc with a charge plus q. And these discs are separated by some small distance d. And they both have radii r. So if I view this from the side, I can imagine that the sides of the discs look like this and I'm exaggerating the dimensions of this a little bit. The separation between them is d. We have plus q down here and exactly opposite amount, opposite sign but equal magnitude charge up on the top disc, negative q. They both have the same areas, a and a, which would just be given by pi r squared for the discs. And so we can think about what's going on inside of these plates, inside of these two discs. Well, we already have half the answer. We know that there's an electric field due to the positive charge from the lower disc. We can draw this by just a bunch of vertical arrows with equal spacing coming up from the bottom of the disc. We're using this approximation that if this distance here d is a whole lot smaller than the radius of the discs, then we essentially have a uniform electric field most everywhere inside of the space between the discs. Now if we consider the same situation for the top disc, well, we also there have an electric field, but rather than pointing out from the disc, it points in toward the disc. And again, for the same reasons that the lower disc's electric field was uniform, the upper disc's electric field should also be uniform very close to the surface of the disc. Well, now we can very easily calculate what the total electric field is inside of this space here between the two plates, the two discs. It's just going to be the electric field due to the positively charged disc plus the electric field due to the negatively charged disc. And we can very quickly write down what this is going to be. We know that this is sigma over 2 epsilon naught k hat. And this will also be sigma over 2 epsilon naught k hat. And so we find out is just the total electric field is sigma over epsilon naught k hat. And that's it. And the reason that this is remarkable is we can engineer in a very straightforward way a situation where we have a device with separated charge plus q and minus q, a small distance between the charges and in between them a uniform electric field. And what's great about uniform electric fields is it's really easy to do calculations with them. So let's do some calculations with them. And in particular, let's think about what it would take to move a little test charge, q zero, from the lower plate to the upper plate. So let me exaggerate this distance a little bit here. So here's the negative q and the positive q. And let's imagine this is my little q naught and it's going to take a journey. I'm going to move it straight across this gap and have it end up here. So this is its initial position. This is its final position up here. The electric field inside of the plates is uniform and points vertically upward from the positive plate to the negative plate. So there's our e vector and we already know what that looks like. E vector is sigma over epsilon naught k hat. And I'm going to grab this little test charge q naught and I'm going to move it across the gap. Now we know from the discussion of work that I don't just have to grab it and move it straight across the gap. I could grab it, whirl it down here, bring it back up and then eventually drop it on this point right across the gap that I've indicated here. Or I could take it on another path. I could move over here and then kind of zigzag back and forth and then eventually, boom, land on the top plate. But I know from energy conservation that it doesn't really matter which path I take. All that matters is the separation between the initial and final points. And that is what's going to define the work that's required for me to move this charge. So let's think about work. The work done by me, so the applied force in this situation, not the field, the work done by me, is going to be equal to the negative of the work done by the field in moving this charge across the gap. Okay, well, no problem. We can figure out what that's going to be. We know that, of course, the electric field would like to move the charge across this potential gap here. That's the direction that the field would love that charge to move. And so we know that the potential energy of this position, u initial, and this position, u final, must be related such that u initial is greater than u final and that delta u, which is u final minus u initial, must be less than zero. Well, where are we going with all of this? We can relate the applied force. It's the negative of the work done by the field and that's equal to the negative of the change in potential energy, which is just the change in potential energy. Well, we can find the work done by the field moving this charge across the gap. That's not so bad. The work done by the field is just going to be equal to the integral of the electric field dotted into little pieces of the path as we take our journey across the gap. That's the definition of the work done by the field in this case. It will be the integral of the force exerted by the field on the little test charge dq dotted into the little pieces of the path, ds vector, as we move across the gap. Well, let's think about our path. We're starting at some initial point i and ending at some final point f. We know that the electric field points straight across from i to f and if we imagine taking the simplest possible path, s vector, this could be made of a whole bunch of little ds vectors. Well, you can right away see that if e points in the k hat direction, ds vector can be simply written as k hat ds, where ds is just the magnitude of the step that you take in the k hat direction each time. And since the force exerted by the electric field is just going to be given by the charge, the test charge, times the electric field at each point. And because that electric field doesn't change at all for every step we take in the trip, we have a fairly simple integral to do here. We just wind up with q naught, the integral of e ds, and that's it. Since e is equal to sigma over epsilon naught k hat and ds vector is equal to k hat ds, the dot product of these two things is just going to be equal to sigma over epsilon naught ds k hat dot k hat, and the dot product of a unit vector with itself is always 1. So in the end we get a pretty straightforward integral that we have to do here. We're just going to get q naught, sigma over epsilon naught, those are just constants, integral of ds, and that integral is going to go from our initial position 0 to our final position, which is d, the distance across the gap. And so at the end of the day we're left with a pretty straightforward answer. The work done by the field moving a little test charge across the gap is just going to be equal to q naught, sigma over epsilon naught times d, and that's it. Well, this is nice and all, but the problem is that of course this work depends on the sign of the charge that we're moving, in this case it was a positive charge, and it's annoying to have to keep carrying this charge around. We don't want to have to keep doing that. And so that's why it's very important to calculate the work done by the field per unit charge, which is just equal to sigma over epsilon naught d. And hopefully we recognize this by now. The work per unit charge done in moving the electric charges across the gap is just going to be equal to the negative of the change in electric potential. And here we have a really important relationship. Now let's have a look at delta v. As a reminder, delta v is equal to the final potential minus the initial potential. And because we're talking about a positive test charge moving in the same direction as the electric field in this particular example, we're talking about going from a point of high electric potential to a point of low electric potential. So if this is the initial potential and this is the final, the final will be less than the initial. So we can decide where we're going to set the zero of electric potential, for instance in our system in this picture. And it's probably convenient to choose this one to be zero because it is the final position that we wind up at, and it's supposed to be the lowest potential point in the system. We have crossed the gap. We're on the other side of the electric field. That ends the electric field. And so that should also be the point where we reach our lowest electric potential. So let's call that zero. So then we have that delta v is equal to zero minus whatever the potential was on the other side, which we'll just call v. So we'll call the initial potential v. All right, well this is great because we see that the work done by the field per unit charge is equal to sigma over epsilon naught d, which is equal to the negative of the change in potential, which is equal to just v. Because I have a minus sign here and a minus sign here and so I get a positive sign. We can very quickly write a relationship between the electric potential on one side of the system, the charge density stored on the surfaces of either of the plates and the spacing between the plates. And if we go ahead and actually write out the charge density as q over a, and then we have one over epsilon naught and d, we have this relationship here. And so we arrive at the very first piece of electric component in a circuit that we can actually start to think about and analyze and see what it's going to do. And that is this device, which is called a capacitor. And a capacitor is nothing more than a set of plates where you can store charge and you have the same magnitude charge on either side of a gap. So here you have a gap. So far we have had a gap filled with absolutely nothing. That is not even air, just empty space or vacuum. On one side of the gap, we have a positive charge plus q. On the other side of the gap, we have a negative charge minus q. Each of these plates has the same area A and they're separated by a distance d. And this is the archetype for the capacitor. So capacitors are distinguished by the fact that the amount of charge that you are able to put on the either plate of the capacitor is proportional to the potential difference across the plates. And the relationship is given merely by a constant c, which is known as the capacitance of this device. Capacitance has units. And you can very quickly figure out what they are. The units of capacitance are coulombs per volt. And this is known as the ferrad, which is just going to be coulombs divided by volts. Remember volts are the amount of potential energy per unit charge. So in the end, you wind up with a very funny unit here, but it's given its own name and that's the ferrad. That's in honor of Michael Ferraday, one of the great physicists of the 1800s who is considered one of the people that really firmed up our understanding of the electric force and later the magnetic force. So armed with this definition, that for a capacitor there is this relationship between q and v, we can go back to our parallel plates with area a and separation d, and we can very quickly tease out what the relationship is between q and v. Specifically, we can figure out what the capacitance is for a parallel plate system like the one we've been looking at. Well, we have this equation, that v is equal to the charge divided by the area, one over epsilon naught and d. And all we have to do is move a bunch of these numbers over to the other side of the equation and solve for q. And what we find is that q is equal to a over d, epsilon naught, and that is multiplied by v. This is a very special case. This is for the parallel plate system, and this is known as a parallel plate capacitor. And so we can very quickly write down the capacitance for the parallel plate capacitor. It's just a over d epsilon naught. And we learn something relatively neat from this. If you can construct a system of parallel plates and you can put plus q on one of them and minus q on the other by some means, thus creating an electric field in between the plates and a potential difference between the plates as a result, you can control the exact details of the relationship between the amount of charge on the plates and the potential difference across the plates simply by adjusting the properties of the plates. You could increase their area. That would make the capacitance go up. You could increase the separation. That would make the capacitance go down. You could decrease the separation and make it go up. Epsilon naught is a very special constant of nature. And you can think about it as telling you about the permeability of empty space to electric fields. If we were to find a way to alter that permeability, we also could increase the capacitance. And it turns out that epsilon naught takes its smallest value, the smallest permeability of the vacuum that you can get is when you just have empty space. That is absolutely nothing there at all. We can actually change the properties of the gap in here. We can change the material inside this gap. And in doing so, we can also alter the properties of the capacitor. We can alter the capacitance of the parallel plate system. We'll explore that a little bit in a moment. So what I'd like to do now is actually use a simulator based on real observed properties of parallel plate capacitors and show you some of the behavior of a capacitor when we hook it up to a battery. And we can use the battery to create a potential difference to move charge from one side of the capacitor plate system to the other and create this charge imbalance. So let's take a look at that using a simulation. Now it's one thing to sketch on a little tablet computer equations. It's another thing to demonstrate using a simulation of the process. And so what I'm going to do now is walk you through a demonstration of a parallel plate capacitor in a circuit connected to a battery. We see from the sketches I was making earlier that we have the essential components here. We have a battery that is a source of electric potential difference that we can use to move charge in a circuit. We have conductors, these gray wires that stick out of each end of the battery from the top and now on the bottom down here. And then we have an arrangement of parallel plates. We have a rectangular plate up here on the top. We have a rectangular plate down here on the bottom. Well, okay, it's actually square but it's tilted at an angle so it looks a little rectangular. And we have the ability to do things. We have the ability to grab the plate and stretch it and make its area bigger. We have the ability to move the plates closer together or further apart from one another. And so you can already kind of see that this is a nice little laboratory for playing around with capacitance having built up some language from mathematics about what capacitance is and how it relates to the area and separation and the material in between the parallel plates. So one of the things that I can do just quickly to demonstrate what happens in a circuit is to actually take this battery and switch it on. So you see the battery comes with a little switch here and we can put on this battery. We can crank this thing up to a one and a half volt potential difference, 1.5 volts. And so, again, the convention is that the plus side is the copper top in the way that this is drawn here. The minus side of the battery is the other side below the copper top down here. And the plus side is supposed to be the part of the battery where positive charge is emitted from, so we would expect positive charge to come up here and then get on this plate and accumulate on this plate. And the bottom of the battery is the source of negative charge. Negative charges would move in the other direction and they would accumulate on this plate. So let's think physically about what's going on. In order to charge up the capacitor, I have to begin in a situation where the capacitor actually has no net charge stored on it. And in this case, I've essentially switched off the battery. Its voltage is set to zero. So there's absolutely no electric field anywhere in the system and there's no way that charges can be induced to move anywhere in the system. Well, we have plates. Let's imagine that they're made of something like copper. Copper is a conductor. What does that mean? Well, it means that its outermost electrons are very loosely held by the rest of the atom. And so if we could put a large enough electric field across a copper atom, we could induce those outermost electrons to leave the safe, comfortable orbit of their parent atom and move and go someplace else, end up someplace else inside of the material. And that's exactly what the battery lets us do. It lets us create an electric potential difference over here inside the battery, which sets up an electric field. That electric field can be set up to point up and by doing that, we can induce electrons to move off of the parent copper atoms. Alright, well, no, fine. So we are going to have an electric field that points up and this is going to cause a force on electrons. So let's imagine we're electrons up in this plate. There's an electric field here that points up into the top of the battery and it is established through the whole copper. There's no other source of electric field anywhere else to add or subtract from it. So whatever the electric field is at the top of the battery, it's just going to persist all the way over here into the top of the capacitor, the top plate. So the electrons in here are going to feel an electric field and they're going to want to move against it. So the electrons are going to start to move through this wire, down through the battery, out the bottom side, and then over here to the other plate. Now the key to a capacitor is that there's no way to jump the gap. This is a space that can't be crossed by the electrons. They simply don't have enough kinetic energy to get out of the material itself. So they can go through the wires. That's not a problem because wires are made of conductor. They're a nice free, essentially resistance-less path to the motion of electric charge. We have the battery. They can certainly move through that because that's where the electric field is. That's the source of the electric field. And then they can come out the bottom of the battery and they can get over to this plate and boom, you know, if they get here, they stop. And so we expect that as we crank up the voltage just a little bit. Do that again, not jitter so much. We'll begin to move a few electrons onto this plate and deplete them from this plate, leaving a net positive charge behind. So of course the convention is we're supposed to think about the way that positive charge is moving in the system. And here positive charge will be moving up out of the top of the battery and deposited here on the top plate. And that's exactly what we see. Now of course what's really going on is that electrons are being stripped from their parent atoms. They're going this way and they're piling up down here on this plate. And that's exactly what we see down here. And as we continue to crank up this potential difference in the battery, maybe taking it to about half full strength, we see that we can put more charge on the capacitor. Now what's going on inside the capacitor? Well, you can see here we also have an electric field. Here we have an electric field that points from the top plate to the bottom plate, from the positive charge plate to the negative charge plate. And charge will continue to move from one plate to the other. Electrons for instance will continue to go from the top plate to the bottom plate until the potential difference between the plates matches that of the battery. And charge motion will simply stop because now there's no more net potential difference in the system. So when thinking about a circuit, it's important to think about the electric potential difference induced by the battery setting electric potential difference induced on the capacitor. Those should be the same if you wait long enough because if you wait long enough enough charge electrons, for instance, will move from the top plate to the bottom plate until you get an electric field in here that exactly opposes the electric field in the battery and now there's no more net movement in the system. Okay, let's see what happens to capacitance. So I'm going to crank this battery all the way up to one and a half volts and I'm going to put a little meter on here that is a measurement of the capacitance of the parallel plate system. Now you could go ahead and calculate this yourself. You could use the known value of epsilon naught which is a constant of nature. You could multiply it by the area of the plates which for this setting is 158.1 millimeters squared. Be sure to convert your millimeters to meters correctly. And then the separation is 7.4 millimeters and again if you want to put this in meters at the end make sure you convert that correctly. From all of these numbers you can calculate the capacitance and the computer's done that for us. It's told us that for this configuration the capacitance is 0.19 times 10 to the minus 12 farad or about 0.2 picofarad. Now we can ask the question what will happen if I increase the area of the plates? Now remember the potential difference across the battery is 1.5 volts and at this point no more charge is moving. The potential difference across the parallel plate capacitor is also 1.5 volts and that isn't going to change. No matter what I do to this system I'm not going to change the electric field inside of this. I'm not going to change the potential difference across this. But what I can do, of course is I can change the capacitance and I can store more charge for the same potential. So if I increase the area I increase C, we expect that from the capacitance equation for a parallel plate capacitor and as I increase C for the same voltage charge on the same capacitor and that's in a sense where the term capacitance comes from. If I increase the capacitance of a capacitor I'm increasing its ability to carry that is its capacity for charge. I can shrink the area and I would expect the capacitance to drop and that's exactly what we see. Let me move this out to sort of an intermediate area again. Now what will happen if I decrease the distance between these plates? What happens to the capacitance if I keep the area fixed? I have the same potential difference across the capacitor. That's independent of the capacitance and I just move the separation to a smaller number. I shrink the distance between the plates. What should happen to the capacitance? If you said the capacitance should increase well let's go ahead and investigate and find out. So if I move this here the capacitance increases and the ability to hold more charge correspondingly increases. If I increase the separation I decrease the capacitance and I decrease the capacity with which the parallel plate capacitor can hold charge. So this demonstration is a very nice way to play around with capacitors and what I'd like to do now is show you what happens when you change the material inside the capacitor. Now on this tab of the demonstration lab I have the ability to take my parallel plate capacitor and shove a material into the space between the plates. Now this is not a conducting material it's a material that allows electric fields to move through it although it weakens them as they do. It does not allow charge however to move through it. These materials are typically made from a bunch of little dipoles so you could use water. Air actually works pretty well you could use styrofoam, polystyrene, plastics things that don't let charge move freely but which can still allow for the passage of an electric field although as the electric field passes through the material it's weakened by the presence of the little dipoles that line up along the electric field. Now there's this thing called dielectric constant and all you have to do to use the dielectric constant is take epsilon zero and multiply it by the dielectric constant. Now I said earlier that epsilon zero that's the smallest value you can ever set that number to because it's the one that tells you basically the way that the vacuum of space, empty space allows electric fields to propagate through it. Any material put in the place of the vacuum will always increase the value of epsilon not to something else called epsilon and the way you get from epsilon not to epsilon is you multiply by the dielectric constant. So if I wanted to increase this value with which the electric fields are permitted to move through space all I have to do is dial up or down my dielectric constant and I can do that over here on the right. So this is some custom material and you say I've put the capacitance meter up here and what I'm going to do is I'm going to zoom out a little bit here so we can see. Right now this configuration has a capacitance of about 0.4 picofarad and what I can do is I can alter the capacitance by grabbing this material and shoving it inside of the capacitor. I'm going to put it all the way in here and when I'm done you see that this material this capacitor is now capable of holding a whole lot more charge than it did before. Watch. Take it out, shove the material in and that's because I've taken epsilon not and I've multiplied it by a number greater than one called Kappa the dielectric constant. Here it's almost a factor of 3 bigger than 4 and that means this thing can hold 3 times more charge when I put this material inside of it. This is how you dial up capacitors to make stronger ones. You don't just have to keep increasing the area of the capacitor plates or the separation between them you could alter the material inside as well instead of having vacuum or air in there you could put in plastic or something like that and you'll see here that there's a number of different materials that we can play with so for instance glass is a very familiar material to most people and you can use it, it's an insulator you can stick it inside of a capacitor it has a whopping great dielectric constant of 4.7 and this really makes for beautiful capacitors. You can hold a whole lot more charge for the same electric potential if you shove glass inside of a parallel plate capacitor. So I hope that this has nicely demonstrated the ideas about capacitance and capacitors and there's one more thing we're going to do with these before we go into lectures on how to use capacitors in a circuit and then more about the detail of how electric charge moves inside of a circuit. We've gone through all this effort to engineer a really nice capacitor and that's great but one of the things we really care about as physicists is we care about the amount of energy that this device is now capable of storing for us in the form of charge separated by a distance and thus having an electric field between it that configuration allows us to store energy anytime we can separate charges and keep them apart we can store energy in the form of potential energy and then hopefully release it later by some other means. So the energy stored in our capacitor is something that we want to calculate and to do that we want to think about our little parallel plate system. So let us imagine that we have no charge originally stored on this device so it's capable of being hooked up to a battery but we've interrupted the circuit with something called a switch now you've heard about switches before you use them to turn on light bulbs all the time a switch is just a device that interrupts the flow of electric charge in a circuit and because this switch is open it's impossible for charge to be moved by the electric field in the battery because there is no electric field path that takes the electric field over to this side of the capacitor but of course if I close the switch I will be able to so I can indicate that here if I were to throw the switch and close it so that I create one nice path for this electric field to travel on from the battery I could begin to move electrons from the left plate over to the right plate and thus cause a positive charge to pile up over here and a negative charge to pile up over here and because I have some potential from the battery I expect, given enough time that the same potential difference will occur between my plates and the movement of charge will simply stop because there is no more net electric field inside of the circuit but we want to know now how much energy we have managed to store by moving all this charge and separating it and keeping it apart so that there's this electric field between the separated chunks of charge so we want to know the energy that was now put into the capacitor by moving this charge through the battery and to do this we can simply think about what it would mean to take a little charge maybe a little test charge Q naught and move it through this potential difference in the battery all the way over here and drop it on the positive plate so we're going to take a little positive test charge and we're going to move it through the circuit and deposit it on the other plate alright well in that case we know some of the relationships that are involved here we know that the relationship between capacitance and voltage across the capacitor is Q equals CV and we'll just leave C as a constant now we won't particularly stick in the parallel plate capacitor value here but we can later if we want to what we know is that every time we take a little bit of charge and we move it across the capacitor and we're going to correspondingly create a little V prime across the capacitor so the first charge we move maybe this is the first test charge moved by the battery across to the other side of the capacitor this is going to result in some new potential difference between the plates V prime it's not at the full strength of the battery yet but it will eventually get there given enough time so we know that our initial voltage across the capacitor given enough time, the final voltage across the capacitor will be whatever the voltage across the battery is that's our end point of this exercise and we've only started by moving an additional little bit of charge well imagine now that I move a little bit more so what I'm going to do is I'm going to increase the amount of charge that I moved by some DQ naught so I'm going to take another little bit of charge and I'm going to move it in the next step over here well in this case we actually have the capability of using calculus now to figure out the answer to the question what in the end will be the total energy stored in the electric field of the capacitor to get the total energy we want to figure out the total work done by the battery moving these charges across the plates so eventually we're going to want to get at the work applied by the battery to figure out about this problem incrementally in a calculus way we can already see that this is going to be equal to all the little bits of work that are needed to move the charges from one plate all the way to the other plate and so it's going to start off taking no work of course to move the first charge and in the end we're going to get to some final amount of work that will have taken to move all of the charges across this circuit alright well let's continue with this a little bit the little bits of work done by the battery are just going to be equal to the little voltage that's set up when we move each little bit of charge dq0 and now we have the ability to substitute in for this in terms of something that has voltage in it we can just use the capacitor equation dq0 is going to be equal to c dv' and that just comes from this capacitor equation up here so thinking about moving an additional little bit of charge dq0 across the circuit and to the other side of the capacitor so if we plug that now into the little bit of work that we are doing every time we move a little bit of charge we wind up with a very convenient equation here which is just cv' dv' and finally what we can do is we can integrate this so we can integrate the little bits of work applied from 0 up to the total and that's going to give us the total work done by the battery and that's going to be equal to the integral from 0 voltage up to the final voltage of the battery cv' dv' c is just a constant it's a constant of the geometry of the system area, separation of the plates material in between the plates that's it and so it won't participate in the integral at all and all we have to do is integrate dv' so this is just the integral of x dx which is not a bad integral to have to do and what we find if we work through this is that in the end the total energy that it takes for the battery to move all that charge from the one side of the capacitor system is given by one half times c times the voltage of the battery squared and remember the voltage of the battery squared is also going to be the final voltage across the capacitor plates so given that information we can do all kinds of things we can figure out what the energy stored in the field is we can figure out if we were to release that energy into kinetic energy how much kinetic energy you would get out of releasing the capacitor's stored energy from its electric field you can figure out the differences in potential energy you can do all kinds of stuff with this because energy is going to be conserved in the system and all you have to do is change it from one form into another you can change it from the potential energy stored in the field to kinetic energy of the motion of charges in the system so what I like to close with here right at the very end is just an example of where we find capacitors in the world around us and on the exam you saw a picture very similar to this you have the cell membrane there are different concentrations of ions inside and outside the cell membrane so for instance inside of here you have proteins which are anions and they can bond to for instance they carry a net negative charge they can bond to sodium and potassium ions there are pumps inside the cell membrane that can move ions from one side to the other and what we find is that the cell maintains an electric potential difference such that it's more negative in the intracellular part of the cell and so we have a separation of charge with a gap and that gap is the thickness of the cell membrane which is approximately 8 nanometers or so it varies of course but it's roughly 8-9 nanometers something like that this is a capacitor you have charge negative on one side an equal amount of charge but positively signed on the other side a gap and therefore an electric field between them is actually the situation that is described by even a parallel plate capacitor negative Q, positive Q a separation D and this thing has some surface area A that's a capacitor and in fact cell membranes are so difficult to measure in terms of their thickness using direct means the first way in which the cell membrane's thickness was ever actually measured was by measuring the capacitance to determine what separation of charge must exist that is what is the thickness of the cell membrane so the cell itself is a capacitor it may have an odd shape but it's effectively two parallel plates which have been bent into roughly spherical shapes and there's a separation of charge and that causes an electric field and that allows you to store energy that's a capacitor now of course human beings make capacitors custom built for different purposes in a wide array of such devices every one of these is a capacitor and what you can see if you look at them carefully in some cases they'll tell you for instance what the capacitance actually is so it's actually not incredibly obvious that it's written on any of these but usually you'll see a little number with an F, a capital F after it and that tells you in Farads ah, there's one of them so this is about a 5000 microfarad it looks pretty big but actually you can get capacitors that carry way more charge than this and are smaller than the thing that you see here microelectronic devices tend to have small capacitors in them like the ones down here and actually even these are big by modern standards these are the kinds of capacitors that when I was a kid yes, I would play with capacitors as a kid when I was a kid I would mess around with these are the ones you can actually handle by hand in the macroscopic world you have all kinds of extremely tiny capacitors that are all over the computer circuitry of iPhones, Android phones laptops, tablet computers car electronic components capacitors are an essential tool in storing and releasing energy for modern electronics an interesting place where you find capacitance and capacitors is actually every time you touch an iPhone touch screen most touch sensitive phones you use a technology known as a capacitive touch and as the name implies it essentially sets up a situation where the presence of your finger alters the way in which charge is distributed inside the bits and pieces of the screen the computer is programmed to detect and interpret those differences and by doing that it can determine where you've pressed even how firmly you've pressed on a surface and very accurately get pressure information and position information and translate that into responsiveness in the software in the operating system so for instance iOS 8 can respond to touch in very different ways swipe gestures touching, dragging and so forth and so all of that is made possible by capacitance and the ability to alter capacitance and detect those changes as they happen and interpret them as touch so that concludes this lecture on capacitors and capacitance please feel free to go back and review parts of this at your leisure the homework will cover material involving electric potential difference and then that concept applied to capacitors you'll have to do all kinds of things involving energy and capacitors calculating the capacitance and so forth so all of this should tie very neatly together for the next homework what we'll talk about in the next lecture in class is the discussion of capacitors by showing how you handle them when you see multiple capacitors in a circuit environment and then we'll talk a bit about how charge moves through a system and resistance to the motion of charge through a system this all leads up to something called Ohm's law which is an essential ingredient also in all electronics