Capacitance

We learned several videos ago that if I had an infinite uniformly charged plane-- let me draw one right here, and I won't draw it infinite and I'll tell you why in a second-- that if we had an infinite uniformly charged plane, and let's say this one's positive, that the electric field generated by it is constant. Those are the field lines. They should all be the same size. And the strength of the field, or the magnitude of the field, is equal to 2 times Coulomb's constant times pi times the charge density of the plate. So if this is infinite-- so what was charge density? We defined it when we proved that this truly is a uniform electric field, but what is charge density? Charge density is just the total amount of charge divided by the area, or charge per area. Well, if we have an infinite plane, the area's going to be infinite, and so if this is a constant number, this is also going to be infinite, so it's kind of hard to work with. But what we also know is that when we have a non-infinite plane that has some finite area, that near the center of it and fairly close to it, it approximates an infinite uniformly charged plane. So with that said, let's see if we can figure out some of the properties of the voltage and how the voltage relates to the charge. If we were to have two parallel-- let me draw it before I say it, because I think saying it'll just confuse it. So let's say I have two plates, that plate-- and then I'll do this one in a different color-- and I have that plate, and let's say they're the same size and they both have area A. Let's say that I place plus Q worth of charges here. So this is plus Q so this is positively charged, right? I could draw a bunch of charges here. Let's say this is minus Q, so this is negatively charged. So what's the electric field going to look like between these two? Well, it's essentially going to be the combination of the electric field generated by this plate on top of the electric field generated by this plate. And they're both going to be constant close to the center, assuming that they're reasonably-- and let's say that they're d apart. Assuming that d isn't too big, near the center we're going to have a constant electric field. For example, this green one is going to be generating-- its field lines are going to look something like this. Near the center, it's constant. These are meant to look constant. Near the center, it'll look like that, and it'll start to bulge out when you get to the edges. Once again, near the center, it's constant. That one should've been at an angle. They start to bulge out, and it'll look something like that. And similarly, this purple plate will generate a constant electric field, and since it's negatively charged, the field lines will be going towards it, not away from it, so its field lines are going to look something like this. Near the center, they'll be constant, and its field lines are going to look something like that. As you can see, they're going to be of the same magnitude and in the same direction, and they also will bulge out there. So the big picture is that you just kind of have twice the electric field as you would have if you just had one of these plates. So let's say we're operating near the center of these, where we have roughly a constant electric field and see if we can figure out the relationship between the voltage across these two plates and the area, and maybe the distance between the two plates. So we know that the electric field generated by any one of these charge plates-- I'll do it in the blue of this color. So for the bottom plate right here, what is the electric field generated? It's 2K pi times sigma. Sigma is just the total charge divided by the area. So Q/A, right? And we know that the total electric field generated by this one is going to be essentially the same thing. I mean, we could say it's a minus, a negative, because it's going towards it, but it's essentially the same thing. Because we see that they overlap just drawing the field line, so the electric field from that one, and we know that they go in the same direction. If this was somehow-- well, this is negative, so the field lines go towards it. So plus 2K pi and this is Q/A. We could have said minus and then had a minus Q/A, but we know that they go into same direction, so we know that they're going to be additive. And so we know that the total electric field is going to be 4K pi Q/A. So now we know the exact strength of the electric field. Let's see if we can figure out the voltage difference between this point and this point. What was voltage difference, just as a review? Well, voltage difference is the electrical potential energy per charge if the charge was here versus here. So how much more potential energy per coulomb is there for a charge to be here relative to here? So another way to view it is a charge here, a positive charge here, because by default we're always assuming a positive charge when we talk about positive numbers and the direction of the field lines or what the positive charge would do. So by default, a positive charge here really wants to go up to this negative plate, although we later learned that most of the movement in electronics and electricity, it's actually the negative charge that's moving. It's the electrons moving. But let's say we did have a positive ion or a positive charge. The voltage is a measure of if any charge is here, how badly does it want to move to this point if it has a way to move? If we have air here or if we have a vacuum here, it might be difficult or impossible for it to move up here. But maybe if we were to connect a wire where the charges could freely conduct, then it will move. And the voltage is just kind of how badly does it want to move? You could almost view it as electrical pressure. And maybe I'll do a whole video on trying to get an intuitive understanding of voltage, because that really is probably the most important thing to get an intuitive understanding of, if you ever want to study electrical engineering or whatever. But anyway, back to the problem. We know that the combined electric field is this, right? It goes upwards in that direction. So what is the electric potential at this point relative to this point or the potential difference from here to here? Well, that's the amount of energy per charge it would take to move a positive charge from here to there, right? Remember, electric potential energy is the amount of work necessary to move a charge from there to there, and then the voltage is how much to do it per charge. Let me write that down. So the work necessary to move a charge from there to there-- let's say a 1-coulomb charge, it will be 1 coulomb times the electric field, because we're always going to have to be going against the electric field. So we have to apply an equal and opposite force. So the force that is going to be the electric field-- so far, this just generates this force. coulomb times electric field, charge times electric field, tells us the force on the charge, right? That's force, and then we have to multiply that times distance. Force times distance. So we see the work necessary is going to be the electric field times d joules-- the J is joules-- and so what is the voltage difference or the electric potential difference between this point and this point? Let's call that point a. Let's call that point b. So Va minus Vb, which is the voltage difference, that's essentially the electric potential energy difference divided by the charge. Or, per charge. Well, here, the charge was just 1, so we can just divide by 1, and we see that it is equal to the electric field times the distance. And the units are going to be joules because we divided both sides by charge joules per coulomb, or volts, right? That's just the units. So what does that equal? So the voltage difference-- so we can say change in voltage. The voltage difference is equal to the electric field, which we know is constant 4K pi Q over A times distance. Or we could rewrite this. Let's see if we could write Q as a function of V. So if we just do a little bit of algebraic manipulation, we can get Q is equal to what? We would essentially divide both sides by 4 pi Kd and multiply both sides by A, so we would get A over 4K pi d voltage. And why is this interesting? Why did I go through all of this work to get this relationship? Well, what it shows you, if you look at this, if we assume that the area of the plates aren't changing-- that's a constant; this is definitely a constant-- and if we assume that the distance between the plates don't change, what we see is that there's a proportional difference between the voltage and the amount of the combined charge in the plates. And that's interesting because, before doing this, maybe voltage is somehow proportional to the square of the charges or to the square root, but now we know that it's directly proportionally. And actually, this term right here has a name, and it is called capacitance. And so another way of rewriting this, if we divide both sides by voltage, we get Q/V is equal to 1 over 4K pi area over distance. And so what it essentially says is that the amount of energy that-- well, actually, I don't want to go into that yet. But for a given configuration, and the configuration is defined by the area of the plates and the distance-- for a given configuration, if I know the amount of charge that I put onto the plates, if I did a minus Q here and a plus Q here, I know the voltage across the plates or vice versa. If I know the voltage across the plates and I know its configuration, I know how much charge there is, and this is called capacitance, and the unit for capacitance is called the Farad. And if you become an electrical engineer or even take a couple of electrical engineering courses, you'll become very familiar with this. And one other thing to point out; this term right here, just so you know a little bit of terminology. This term right here. This 1 over 4K pi, this is often called epsilon nought, or just epsilon, and that's called the permittivity of free space or permittivity of the vacuum. And maybe in a future lecture or a future video, I'll talk more about why it's called that. But anyway, I'm already well over the time limit. So I just wanted to give you a sense of, one, that you can calculate the voltage across what we call, in this case, a capacitor. It has capacitance. That voltage, you can kind of view it as the electric pressure. How bad does the charge here want to move here? And if you put a wire here, you'll learn in a second-- not in a second, in several videos-- that that charge will flow. Or actually the negative charges will flow this way and generate current. And we'll do that when we start learning a little bit more about electricity. For any given configuration, it has a corresponding capacitance, and then given that capacitance, if I put some amount of charge, I can figure out the voltage, or if I know there's some voltage, I can figure out the charge. Anyway, I will see you in the next video.