 Capacitors may at first seem daunting with their large numbers of electric charges separated by a spatial displacement. What happens if you have more than one capacitor-like object in a system? Let's begin to learn how to handle situations where more than one physical capacitor system is present. In this brief lecture, I'm going to explain how it is that you begin to handle capacitors when there is more than one capacitor present in a system. To begin with, I have to define what that system is. The system that I'm going to be discussing is a system that we're going to see over and over and over again, one that's very common and commonly represented with a set of well-defined standard symbols. The system is something known as a circuit. Now the heart of a circuit is some device that's capable of generating an electric potential difference. And we're used to thinking about these in common discussions about the world around us. We think of sources of electric potential difference like batteries or like the wall plug outlet which in the United States delivers about 120 volts. So I will start with an archetype and I will simply label it battery. We'll explore batteries real and otherwise in a bit in the course. But I can begin by defining a battery simply as something that delivers stable delta Vs, that is electric potential differences. That's the job of a battery. Whenever you see a battery represented in a circuit diagram, which I'll sketch in a moment, that is a device that delivers a constant, reliable, stable electric potential difference. And as we know from studying electric potential differences, that means that this is a device capable of providing work per unit charge. Now the symbol for a battery looks like this. It has one long line and one short line and then it usually has lines that come off of those at right angles. So this is a standard symbol for a battery and it's very common to label the side of the battery that when it's connected to another device delivers positive electric charges. The movement of positive electric charges will be our reference point for all discussions of circuits going forward. So the long line is labeled with a plus sign. Now let's look at a very oversimplified circuit. A circuit is simply a closed path through which electric charge can be moved, much like water through pipes in a home plumbing system. So let's look at a very simple circuit employing only a battery. So I will draw my battery, I will label the side that emits positive charge. Now these lines that come out the end, these are little bits of conductor. So I have here conductor that extends out of one side of the battery and conductor that extends out of the other side of the battery. Remember that a conductor is a material that allows charge to move freely through its volume or along its surface. Conductors are the heart of circuits. They are the highways that allow charge to move from one place to another under the influence of an electric potential difference. What we're going to do is we're going to simply make the simplest possible circuit by connecting one end of the battery back to the other end of the battery. And I've drawn this here in blue, but again all of this is just conductor. So you can think of the battery with its electric potential difference like a pump. It's capable of moving positive electric charges to a higher electric potential point and then the charges will move through the conductor to get back to the lower electric potential point. So this is very similar to a pump that takes water from a lake up to a hill and then there is an aqueduct or some other device that lets the water run downhill. And then the water at the bottom of the hill can do work by moving a paddle wheel for instance and then it could be allowed to flow back into the lake and then pumped back up again. And if you could make that a perfectly closed system with no losses of water then you would never need to replenish the lake. You could just keep pumping the water up, let it roll down, do some mechanical work on the way, let it coast back into the lake and then pump it back up again. This is what a battery does in a circuit. It's a pump that takes charges and it moves them to high electric potentials and then it allows the charge to move, for instance if you connect one end of the battery to the other like I've done here, it allows the charge to then essentially flow downhill through the conductor and then be pumped back up again. So the flow of positive charges in the circuit, which again will be our reference points, as I've drawn the circuit, will be clockwise. The battery emits positive charge at the top, the conductor then carries the positive charge without any resistance to motion to the other side of the battery and then inside the battery a chemical reaction will pump the charge back up to high potential energy and then let the whole thing repeat. This battery will have an electric potential difference delta V and that is the electric potential difference that allows for charges to be moved around. Okay, that's the simplest possible circuit. It's a closed system, positive charge flows out one end of the battery through the conductor and back into the other end of the battery and the whole process repeats itself. Now later on in the course we will explore why it is really stupid to plug one end of the battery straight into the other with just pure conductor. We'll see why that's bad and we'll explore the consequences of that later, but it's a good thing to know that you should never try to do this in real life. Don't go home and make a simple circuit like this right now by taking some recovered wires from an old electronic device that's just lying around gathering dust and then a 9 volt battery. If you hook a 9 volt battery's positive terminal to its negative terminal and you put what's called load in the circuit that is no thing on which the charges can do work to lose energy you run the risk of blowing the battery up. So we'll explore that in detail later. Let's put a device into our circuit. The next simplest thing that we can put into a circuit is merely a switch. This is a device that either interrupts or completes the circuit. It also has a simple symbol. It has a line like this, conductor coming in, a line that's raised like a drawbridge and then another line going out on the other side. So this looks just like a switch. If you were to push this thing down, if you were to shove this down and close the switch you would then create a perfect line of conductor that's uninterrupted. On the other hand, if you open the switch as it's represented here then you interrupt the conductor and you would prevent the flow of charge. So let's go ahead and draw a very simple circuit with a switch in it. So now I've just repeated the drawing above. Here I have my battery with its electric potential difference delta V and now I've put a switch into the circuit. The switch is open as it's drawn here. I could instead draw the switches closed like this. So very typically, if you want to indicate that the circuit begins in a state where charge cannot flow, you would draw the switch open and that would indicate that at time zero the switch interrupts the flow of any charge through the system. Later on you might redraw this and show the switch closed, in which case now you're indicating that the circuit allows for the movement of charge freely through the conductor without interruption. So those are two very simple devices that one could imagine putting into a circuit. And the next device we're going to put in is a capacitor. Capacitors also have a standard symbol. They have a line of conductor, a plate, a line that represents one of the plates in the capacitor, an equally sized line on the other side representing the other plate and you'll see there's a little gap in between them and then another line of conductor. So this is the standard symbol for a capacitor in a circuit and if we wanted to for instance we could go ahead and draw a very simple circuit right now with just a battery and just a capacitor. So what I'll do is I'll label the battery by its steady electric potential difference that it produces, delta V, and I will label the capacitor by its capacitance, C. Now you'll notice that the symbol looks like a parallel plate capacitor. That is one where you have two plates of equal area separated by a gap with a distance D. The plates have area A and so the capacitance is very easily given as being equal to for instance epsilon naught kappa for the dielectric that might be put into the gap A over D. That's for a parallel plate capacitor. That's a special case. You don't actually have to know whether or not the capacitors that are drawn in a circuit are parallel plate capacitors or not. All that matters is that they have a capacitance C and that every capacitor has a capacitor equation which allows you to relate the charge that's stored on the capacitor to the electric potential difference that is across the capacitor. In that case we just have Q equals Cv where V is the voltage across the capacitor, C is the capacitance, and Q is the charge stored on either plate of the capacitor in terms of magnitude of charge. That's the key feature of every capacitor is that it has one of these equations. So in this very simple circuit we can start doing some explorations of electric potential difference and figure out what the electric potential difference across the capacitor is. Now you might look at this and go, oh well professor it's just the electric potential difference provided by the battery. But let's explore that and make sure that we understand why the in this simple circuit with one capacitor hooked up to one battery the electric potential difference across the capacitor must be that provided by the battery. What I'm going to do is I'm going to imagine that I can drop little probes onto this circuit. I'm going to define one point in the circuit and it really doesn't matter where I choose it but I'm going to pick this corner down here and I'm going to define this to be ground. That is at this point in the circuit the potential is equal to zero by definition. Now why I do this is because I need to measure changes in electric potential through the system and so I have to pick a reference point. When we're dealing with point charges we put the reference point at infinity. When we're dealing with circuits we pick a point in the circuit that we declare to be ground or potential equals zero and we measure all potential changes relative to that point. You choose that first point anywhere you like and then after that what matters is that you measure the electric potential of any other point with reference to the first one. Every delta v needs a vi and a vf so you've got to pick your vi what's your initial point what's your initial potential and I choose this corner. I could have picked anywhere else in the circuit. You can play around with this if you like on your own and see that it doesn't make a difference but you have to make that choice first. Okay great so now what I'm going to do is I'm going to pick another point and I'm going to maybe use purple or something like that magenta in this case to label the other point that I'm going to measure. I'm going to choose just as an initial demonstration of how one does this. A point that happens to be right near the first one that I picked so I'm going to draw a little point to measure as my second potential location. This magenta dot just above the corner where I drew what I'm referring to as ground or zero potential. So in this case what I need to know is what are the changes in electric potential between the red point and the magenta point the corner and the second dot that I drew. So in this case if I label this as point one and this is point zero I would like to know v1 minus v0 that's the first delta v that I'm interested in. Well v0 is by definition zero and what I have to do is look at the circuit and figure out whether or not there are any places in between the two points along any path where there are devices that might potentially have a change in electric potential like a battery or a capacitor or as we'll see later other devices like points of resistance in the circuit which can generate electric potential differences also. We seem to have nothing but pure conductor in between v0 and v1 the points where I'm measuring these voltages. Because along the shortest path here that I'm indicating by moving the dot there are no devices that change the electric potential from ground to point one. I'm left to conclude that this must be zero. Well this is interesting because there's an alternate path in the circuit that I could take to get from ground to point one rather than taking the shortest path I could take a long path one that takes me through the capacitor then back through the battery and then down to point one. So this is the short path answer. What about the long path? Well in a closed system where there's no way that energy can get into or get out of a system like the circuit that I've drawn here any path I take through the electric fields that are present in either the capacitor or in the battery must yield changes in potential that are independent of the path that I travel just like in a point charge electric field just like in a uniform electric field if I go from point A to point B the changes in energy between those two points because the electric force is a conservative force must be the same independent of the path that I take. I could take a short path I could take a long path it doesn't matter. So I must find that going the long way this must still be true otherwise energy would not be conserved in a closed system and I would have free energy on my hands that would actually run out of control and possibly end the cosmos which would be a very bad thing. So conservation of energy gives us a principle by which we can analyze short paths or long paths between one potential measurement point and another potential measurement point in a closed system like a circuit. If I look at the short path there are no things that change the electric potential itself between zero and one and so the changes there must be zero. If I take the long path I have to go through two devices but it must be true that it's still zero which means that the net change in electric potential going from this corner through the capacitor and back through the battery must be zero. So I learned a very interesting thing by this exploration. I learned that the change in the electric potential across the battery must be equal to the change in electric potential across the capacitor such that if I do delta V battery minus delta V capacitor I get no net change in potential zero. So this is how we know for instance by doing a very simple assessment of electric potentials and changes in electric potentials that whatever the voltage provided by the battery in this simple circuit it must also be the voltage that's across the capacitor assuming that the capacitor has had enough time to fully get its maximum charge and set up its electric potential difference. Now what if I have two capacitors in a circuit? So let's imagine this circuit here's my battery here's some conductor I have one capacitor some conductor a second capacitor and then all conductor back to the battery. So here's my change in potential across the battery which I'll just call delta V little b and then I've got delta V c1 delta V c2 and corresponding capacitance is c1 and c2. What's known? Well we know the following we know that whatever c1 and c2 are each of those capacitors has a capacitance equation the charge on capacitor one must be equal to c1 delta V1 and the charge on capacitor two must be equal to c2 delta V2. So we've just learned a very important thing we've learned that every time we look at a problem that involves more than one capacitor the first thing we should write down of course are always the knowns in a problem and the known here is that each of those capacitors has its own relationship between the charge on its plates and the voltage across the plates the voltage change across the plates and those are related through the capacitance this constant that's determined by the geometry of each of the capacitors. I'm not assuming that c1 and c2 are the same capacitance I'm going to assume that these could be very different numbers from one another and proceed. Well what else is known? We know that in a closed system whatever the changes in electric potential they must all add up to zero. So we know that delta VB minus delta Vc1 minus delta Vc2 must be equal to zero this is a closed system or a closed circuit there's no external way for energy to get into or out of the system all the energy that can be provided is provided by the battery and any changes in energy must sum up to zero in a closed path in the system. So what I've just learned is that delta VB must be equal to delta Vc1 plus delta Vc2 and this means that if I pick a path let's say I define ground to be here and I pick another point some place else well I could even just come back to the same point right let's pick a simple path that takes me from ground through the circuit and back to ground if I go this way up through capacitor 2 to the left through capacitor 1 and down through the battery I've experienced all the changes in potential in the system and I come right back to where I started and in this particular case because I've come right back to where I started I know that the net change in energy must be zero and therefore the net change in electric potential must also be zero. So whatever the individual voltages across capacitor 1 and across capacitor 2 they must sum up to whatever is provided by the battery. Another way to look at this would be for instance to do the following. I could make my point of zero potential over here in the right corner and I could measure the electric potential difference over here at this point 1. Now there are two paths that I can take to measure these electric potential differences. I could go clockwise through the battery and then go to point 1 or I could go counterclockwise through the capacitors and go to point 1. What I see is that if I go clockwise I experience one potential change delta VB before I get to point 1. And by conservation of energy whatever changes I go through the capacitors they too must equal delta VB otherwise energy is not conserved in the system. So what I find of course is that while the sum of these voltages must equal delta VB there's no guarantee that the individual voltages themselves are equal to delta VB. They could be very different but their sum must be equal to delta VB and that's embodied in this equation. Okay great so let's go ahead and employ this. Let's make a hypothesis that C1 and C2 can be combined in some way mathematically and yield an equivalent capacitance of a single capacitor. That is the hypothesis is that if I have my battery, one capacitor and a second capacitor that through some mathematics I can re-represent this as my battery and just a single capacitor. So here's C1, here's C2, and here I will call this CEQ or the equivalent capacitance of somehow combining these two capacitors into one with its own single capacitance. That's a hypothesis. We don't know if it's possible yet but we can check. If this were true, if this hypothesis were correct then of course there would be an equivalent charge across this capacitor which is equal to CEQ delta VEQ. Well of course we can immediately see that this picture looks just like the original simple battery and capacitor circuit we drew and we already know what VEQ is going to be equal to. VEQ is just the delta V across the battery. So we have that delta VB is equal to delta VC1 plus delta VC2. We can now substitute in with the capacitance equations for each of the capacitors. So this is delta VEQ and if we now substitute by rewriting this equation we would find that this is equal to Q total over CEQ is equal to Q1 over C1 plus Q2 over C2. All I've done is take the capacitance equation for the hypothesized equivalent capacitor, the original capacitor 1 and the original capacitor 2 and use those to substitute for the delta V's that show up in this equation. So all I do is insert capacitor equations in this step. Now what about these charges? We have three of them to deal with. What are these going to be? We know that at the beginning before the capacitors are charged up. So imagine now I put a switch into the circuit. So here's my battery, here's my switch, here's capacitor 1, here's capacitor 2. At the beginning this is an electrically neutral system. So we start off with the total amount of positive charge. So Q positive plus Q negative is zero. That's a neutral system. Then we close the switch. Now the battery can exert it's electric potential difference on the two capacitors. The battery emits positive charge which builds up over here on this plate. A corresponding negative charge builds up over here on this plate. A corresponding positive charge builds up on this plate and a corresponding negative charge builds up on this plate. The way you can see this is if positive charge goes up the top and piles up on Then the positive charge here attracts the negative charges in this part of the conductor and capacitor system to the left, and the positive charges are attracted down here toward this plate on the bottom, separating the charge in between. If you wait long enough, eventually the net amount of positive charge on this plate, q1 positive plus q1 negative plus q2 positive plus q2 negative, by charge conservation must still equal zero. So if charge is conserved in the system, initially the net amount of positive and negative charge is zero. You add them together, you get zero. At the end, even though you've moved charge around in a closed system, no charges have been created or destroyed. And so you must still have that the positive and negative charge on capacitor one and the positive and negative charge on capacitor two are zero. And because no charge is moving around anymore, it must also be true that this is zero and this is zero. And the only way that all of this can be true is if the following relationship holds that the total positive charge in the system is equal to the positive charge that's built up on the plate, the positive plate on capacitor one is equal to the positive charge that's built up on the corresponding plate on capacitor two. And if the total amount of negative charge in the system is similarly equal to q1 negative and that's equal to q2 negative. Now keeping in mind that in the capacitor equation where you have q equals cv, this is the magnitude of charge on either plate of the capacitor, then for q1, this will just be equal to q1 plus which is equal to q1 minus. And for q2, this will be equal to q2 plus which is equal to q2 minus. And because q1 plus and q2 plus are equal, we find out that q1 equals q2 and that must be equal to the total charge stored on the equivalent capacitor in the circuit. So let's revisit our equation. We had that q total all over the equivalent capacitance is equal to q1 over c1 plus q2 over c2. We found out that this, this and this are all the same number. q total is equal to q1 is equal to q2. So these all cancel off of both sides of the equation. And we're just left with 1 over cEq equals 1 over c1 plus 1 over c2. So we found a relationship. We found a relationship between the equivalent capacitance in this particular circuit where I have battery, capacitor, then another capacitor in sequence back to battery, c1, c2. These are known as series capacitors. That is, they're in sequence and they divide the voltage that is in total across them. They also all have the same charge q on their plates. So if I add another capacitor in here and run through the whole exercise again, what you'll find is that 1 over the new c equivalent is 1 over c1 plus 1 over c2 plus 1 over c3. And if I add a fourth, then you just add plus 1 over c4. If I add a fifth, then you add plus 1 over c5 to this. So in general what I find out is that 1 over the equivalent capacitance for a bunch of series capacitors is the sum from i equals 1 to the total number of series capacitors that are directly in sequence with one another. It's the sum of 1 over their capacitances. And again, this is only for capacitors that are in sequence with one another. They all have the same charge. They all split up the total voltage across them. And this is how you relate their capacitances to an equivalent big capacitor. What about a different arrangement of capacitors? What if instead I have my battery and then I have a branch of conductor in my circuit and I have c1 to the left of c2. Well, I can play the same game. I can define zero potential to be down here in this corner. And then I can measure the electric potential difference with respect to a point one. There are three paths that I can take through this circuit. I could walk this way to point one. I could go this way to point one or I could go this way to point one. So I have path number, let's call this path A, path B, and path C. So what are the electric potential differences from zero to one in path A? Well, the only device that I go through is the battery. So it must be that this is equal to the voltage difference across the battery. What about path B? Well, in this case, I go through capacitor one and then get to point one. So it must be that this is equal to delta V c1. But since delta V10A and delta V10B have to be equal by conservation of energy, then we know that this must be equal to delta VB. What about path C? Well, again, here I take a path that goes through capacitor two and then gets me to point one. So this must just be the voltage change across capacitor two. And again, by conservation of energy, this must be delta VB. So what I find out in this arrangement, which is known as parallel capacitors, that is they're adjacent to each other in the circuit, they all have the same voltage. So what we find is delta V c1 equals delta V c2, unlike in series where they don't necessarily have the same voltages. In parallel, they do. And in this simple circuit, for the case that I've drawn here, this happens to be equal to delta VB. You might get a more complex circuit where these parallel capacitors are buried in a big nest of other capacitors. Be very careful about blindly relating changes in voltage to the voltage across the battery without first considering whether or not the battery is directly connected to the capacitors. In this case, they are. But they may not always be, and you need to be very cautious about this. It will simply be generally true that if you have one or two capacitors in parallel with one another, that they will definitely be at the same voltage, whatever that is. Well, the next thing we want to consider is the fact that the charge separation in the circuit has to be looked at. So we've used conservation of energy to relate voltage changes in the circuit. Let's look at conservation of charge now. So we have delta VB, we have c1, we have c2. So there's going to be some plus q1 and minus q1 charge separation here, and some plus q2 and minus q2 charge separation here. Now, if I hypothesize again that I can combine these in some way using math and get an equivalent capacitance, cEq, and this time for parallel capacitors, then here I will have some plus q total. We'll call this plus qEq and some minus qEq. By conservation of charge, it must be true that whatever q equivalent is on the plates of the capacitors, it's equal to q1 plus q2. That's the only way charge conservation can work out. So if this is possible, then this must be true. Since charge is conserved, it must be true that the sum of the charges on the two parallel capacitors, q1 and q2, will give me the total charge on the equivalent capacitor. So this is the equation I'm going to start from to try to solve for the relationship between capacitance 1, capacitance 2, and the equivalent capacitance. Each capacitor has its own capacitance equation, and since the equivalent capacitor is connected directly to the battery, we know that this is delta VB right here. There's also q1, which is equal to c1 delta V1, and we know from looking at the electric potential changes on the circuit that this is c1 delta VB. And then there's q2, which equals c2 delta V2, and again that's c2 delta VB. And so if I run this equation to its natural conclusion, I find out that substituting in with the capacitor equations, cq delta VB is equal to c1 delta VB plus c2 delta VB. The delta VBs cancel on both sides of the equation, and I find out that for parallel capacitors, all I have to do to get the equivalent capacitance is add the capacitances together directly. And in general, if I were to put a whole bunch of capacitors in parallel with one another like this, so here's 1, 2, 3, 4, for instance, then all I have to do to get the equivalent capacitance is sum from i equals 1 to n parallel ci. That would be the general relationship between the equivalent capacitance of this, for instance, 4 capacitor parallel network and the individual capacitances of each of the capacitors in the network. A final comment. What if you have a circuit that looks like this? Oh my, let's stop and think. Are there any capacitors that are only in series or only in parallel with one another? Well, if you've had a chance to think about that, you'll notice that these two right here are in parallel with each other. This is helpful because that means that we can write down a c equivalent for these two capacitors. We can change this circuit into this circuit using the rules of adding capacitances for two parallel capacitors. So if I label these now c1, c2, and c3. Here's c1 and ceq. What I find out is that because 2 and 3 are directly in parallel with each other, they're both at the same electric potential difference. Their charges, however, are different and I can use what I just did with parallel capacitors up here and I can add their capacitances together like so. This then gives me this circuit. I can go one more. I see that c1 is now in parallel with ceq. If I want to know the total capacitance of the network of three capacitors, now all I have to do is remember that 1 over c total will be equal to 1 over c1 plus 1 over ceq. I can use a little algebra and I can invert and solve for these total, c total here. If I do a little algebra, what I'll find out is for this two capacitor case, all I have to do is multiply c1 and ceq divided by their sum. We will encounter networks of capacitors that are very complicated. This is the simplest complicated example I can give you. You'll see more complicated ones than this, but the idea is simple. Once you know the rules of how to add capacitances in parallel and in series, study the network, find a pair or a triplet or whatever, a group of capacitors that are very obviously only in series with each other or only in parallel with each other and combine them into a single equivalent capacitance. Redraw your network. Look at it again. Now, what capacitors do you see that are most obviously and directly in series and in parallel with each other? Combine them using the rules of combining capacitances for the appropriate situation. Redraw your network and keep doing this until you get down to a simple one battery, one capacitor network. That's your goal in any of these. You might be asked, for instance, at the end of a problem to find out the voltage just across c2, knowing only the voltage across the battery. Once you get to this circuit, you now know the voltage across the total capacitor here is the voltage across the battery. And then you can use the rules of parallel and series capacitors to unpack the network and figure out what the charges and voltages are on each individual capacitor in the network. We'll work some problems in class that will help you to understand better how to do that kind of mechanics with problem solving for these style of problems.