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 and if we had an infinite uniformly charged plane and let's say you know this one is positive that the field the electric field generated by it is 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 conscious constant times pi times the charge density of the plate so if this is infinite so what was charge density we define 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 the total amount of charge divided by the area or charge per area well if we have an infinite if we have an infinite plane the area is going to be infinite and so if this is a constant number this is also going to be infinite and so it's kind of hard to work with but what we also know is that when we have a when we have a a non infinite plane that has some finite area that near the center of it and fairly close to it it approximates a infinite uniformly charged plane so with that said let's see if we can figure out some of the properties of the voltage and and how the voltage relates to the charge if we were to have two parallel let me let me draw it before I say because I think saying it'll just confuse it so let's say I have two plates let's say have two plates that plate and then I'll do this in a different color and I have that plate and let's say they're the same size and they both have area a and let's say that I place plus Q versus worth of charges here so this is plus Q so this is positively charged right I could draw a bunch of pluses here and let's say this is minus Q so this is negatively charged right so what's the electric field going to look like between these two what'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 clean 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 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 in the center it'll look like that and I'll start to bulge out near you when you get to the edges once again near the center it's constant that one shouldn't be an angle start to bulge out and look something like that and similarly this this purple plate will will generate a constant electric field and since it's negatively charged the field lines will be going towards it right not away from it so it's field lines are going to look something like this and near the center they'll be constant and it feels like it's field lines are going to look something like that as you can see they just 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 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 where we're operating near the center of these where we have a 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 let's say for this 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 2 pi 2k pi times Sigma Sigma is just the total charge the total charge divided by the area so Q over a right and we know that the total of the 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 going because it's going to the other direct towards it but it's essentially the same thing because we see that they overlap destroying 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 right the field lines go towards it so plus 2 K pi and this is Q over a alright we could have said minus then had a minus Q over a but we know that they go in the 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 4 K PI Q over a so now we know the exact strength of the electric field let's see if we can figure out the voltage difference between between this point let me draw it in a color that you can see between this point and this point and 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 we by default we're always assuming a positive charge when we talk about positive numbers and and and what you know the direction of the field lines are what's the positive charge would do so by default a positive charge here really wants to go up to this to this negative plate although we later learned that most of the movement in in in electronics and and electricity it's actually the negative charge is moving is the electrons moving but let's say we did have a positive ion our positive charge the voltage is a measure of if any charge is here how badly does it want to move to this point so if it has a way to move if we have air here maybe it's very or 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 that where the charges could freely conduct then it will then it will move and the voltage is kind of how badly does it want to move you can almost view it as electrical pressure and maybe able to 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 wanted 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 what is well 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 for a charge it would take to move a charge from a positive charge from let me keep switch colors from here to there right remember electric potential energy is the amount of work necessary to move something from a charge from there to there and then the voltage is how much to do it per charge let me let me write that down so the work necessary to move a charge from there to there let's say a 1 Coulomb charge right it will be 1 Coulomb 1 Coulomb times the electric field right because we're always going to have to be going against the electric fields we have to apply an equal and opposite force so the force that we're is going to be the electric field so 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 necessarily not necessarily necessary is going to be the electric field times D joules the J's joules and so what is the voltage but the voltage different or the voltage difference or the electric potential difference between this point and this point let's call that VA 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 or the charge or per charge well here the charge was just one so we can just divide by one 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 by both sides by charge joules per Coulomb or volts right that's just the units so what is that equal so the voltage difference so we could say you know change in voltage the voltage difference is equal to the electric field which we know is constant for K PI Q over a times distance or we could rewrite this let's see if we could write I don't know 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 K D and multiply both sides by a so we'd get a over 4 K PI D voltage and why is this interesting why did I go through all of this work to get to 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 place don't change that what we see is that there is that there is a proportional difference between the voltage and the amount of charge in the in the combined charge in the plates and that's interesting because you know before doing this we might maybe it was you know maybe voltage is somehow proportional to the square of the charges or or to the square root but now we know that's directly proportional and actually this 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 over V is equal to 1 over 4 K pi area over distance and so what it essentially says is is that the amount of energy that well I actually don't want to go into a bad yet but for a given configuration and the configuration is defined by the area of the plates and the distance if 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 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 units for capacitance is called the farad and if you become an electrical engineer 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 one over for K PI 1 over 4 K pi this is often called this is often called epsilon not or just epsilon and that's called the permittivity of free space or permittivity of the vacuum and maybe in the future in a future lecture or future video I'll talk more about why it's called that but anyway I'm already well over time limit so I just wanted to give you a sense of one that you can conduct that you can calculate the voltage across what we call in this case this is a capacitor it has capacitance that voltage is 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 that not in a second in several videos that that charge will flow or actually that 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 and then 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 if I know there's some voltage I can figure out the charge anyway I will see you in the next video