Capacitors

Alright now let's talk about a very, very important circuit element called a capacitor alright now we know batteries what are they meant to do? Well they got to impose a constant potential difference and drive current through a circuit. What's a resistors job? A resistors job is to use energy that's what a resistor does anything that uses energy can be thought of as a resistor. What's a wire's job? A wire's job is to get current from one side of the battery to another to go, to allow current to flow between circuit elements. So what's a capacitors job? A capacitor is meant to store charge and energy so just like a resistor uses power a capacitor stores energy. Alright, so the way that capacitors look they are 2 plates, they're not always actually like this but you can think of them like this. 2 plates separated by a distance and what we're going to do is we're going to put some charge on this plate.

Alright now the big question is how good of a capacitor is it? In other words what's its capacity? Well here's the idea the more charge that I store here, the more of a potential difference there'll be across the plates. Because if I were to take another charge let's say it was a positive charge it would rather be on the negative plate than on the positive plate because positive charges don't like other positive charges. So the more charge I store the bigger the potential difference will be. And what we're going to do is we're going to define the capacitance as the amount of charge that I'm able to store divided by the potential difference. So that means if I'm able to store a lot of charge without having a huge potential difference then I'll get a very, very nice large capacitance. If on the other hand even storing a little bit of charge requires a huge potential difference then I'll have a really small capacitance. So that's the definition of capacitance, we write it in terms of symbols c equals charge which is q divided by the potential difference and sometimes people will write delta v there but let's just leave it with a v for right now.

Alright, so what should the unit be, well just like everything in SI everything is in SI now we're going to call it a Farad but 1 Farad is equal to 1 Coulomb stored per volt of potential difference across. So that's the idea capacitance charge divided by potential difference. The more charge you got the bigger the potential difference imposed across the capacitor. Alright now let's think about this I want to think about this parallel plate capacitor and what its capacitance should be. Well here's the idea the reason that we have limits to the capacitance is because this charge is force to congregate next to itself and it doesn't like it. Positive charges like negative charges, they don't like other positive charges, but I can kind of make that okay if the area of this plate is really big. The bigger the area the more room that, that charge has to spread out and the less it cares that it's sitting next each other.

You can think of the potential difference kind of as how much the charge cares alright, so our capacitance ought to be proportional to the area of the plates of this parallel plate capacitor. Alright another Geometric property of this parallel plate capacitor is how far apart are the plates? So we've got area and then we've got distance apart. Now let's think about this the further the plates are apart the larger distance we're separating this charge over and the charge doesn't want to be separated. So if we put the 2 plates real close together, then they say you know what yeah we don't like being separated but it's almost as if we're not separated. So we can store more with a smaller potential difference, so that means that when the distance between the plates get smaller the capacitance gets bigger I can store more because they don't care as much they're not really separated by that much. So that means that the capacitance should be proportional to 1 over the distance between the plates. So we can write the capacitance of a parallel plate capacitor as a constant times the area divided by the distance between the plates.

This constant has a numerical value 8.854 times 10 to the minus 12 Farads per meter. Now how did I work that out? Well jeez if I take that constant multiply it by meters squared the area and then I divide by meters the distance then I got to get Farads so it's Farads per meter. Alright this is called the permittivity of free space and it plays a major role in more advanced discussions about capacitors and electric fields. Alright now from our definition of capacitance the potential difference across any capacitor is equal to the amount of charge that it's holding divided by the capacitance. Now the neat thing about the capacitance is that it's just a Geometrical property. It's like resistance, it doesn't change depending on what situation the capacitor is put in, charge will, potential difference will but the capacitance will remain the same.

Alright, so the potential difference is charge divided capacitance. Now we said that the purpose of the capacitor was to store energy as well as charge so how much energy is stored? Well I'm not going to derive this for you it takes a little bit of work to derive it but what we end up with is q squared over 2c so it's the square of the charge divide by twice the capacitance. Now we can use this delta v equals q over c to rewrite this into 2 different or 3 different ways overall one half c delta v squared and one half charge times potential difference. So these 3 formulas are all saying the same thing but depending on what information you have, you might use this one as opposed to that one. Alright if you know the capacitance and you know the charge alright let's do q squared over 2c. Alright so let's do an example, I've got a 3 Farad capacitor and it's got a potential difference of 4 volts maintained across it. And I want to know how much energy does it store, alright so what have they told me? Well I've got the capacitance and I've got the potential difference.

Alright capacitance, potential difference so I want to use this formula one half capacitance potential difference right, squared. Okay so 16 divided by 2 is 8 and 8 times 3 is 24 joules. Alright so that's just the way it goes energy is stored in a capacitor, potential difference across the capacitor and the meaning of capacitance. Just think about it as a Geometrical quantity that will be big when the charge can spread out and will be small when the chargers are all confined to one space and it's not allowed to be next to its friends the opposite charge. Alright that's capacitors.