Lesson 1 - The Capacitor (Physics Tutor)

hi and welcome to the physics 3 tutor volume 2 now in this course we're going to continue right where we left off in the end of the physics 3 tutor volume 1 you know there's a tremendous amount of material that you need to cover in order to really understand electricity and magnetism and in the first DVD in the sequence the volume 1 of physics 3 we covered a tremendous amount of ground we covered you know what is an electric field what is an electric force how does it arise we talked about using Gauss's law to calculate the electric field we talked about the electric potential and how it kind of relates to our concept of voltage that we've kind of you know hear in everyday language right we did a lot of calculations and talking about when things are spherical II symmetric when things are cylindrical II symmetric when things are planar Li you know have a planar arrangement and we use Gauss's law extensively to talk about calculating the electric field in those cases and along the way we got comfortable with the concept of the electric field and so on in the electric potential now in this course in this section and this volume we're going to continue that journey we're going to obviously need everything that we've learned before so if you haven't viewed that you need to stop now and go back and review that because you know it's it's a building block process you have to know one thing before you move to the next in this section I mean in this volume we're going to continue and we're going to talk about the more practical side of how do you use these things capacitors we're going to talk about capacitors we're going to talk about what the electric field is inside of a capacitor in an electric circuit we're going to talk about electric circuits and resistors and figuring out what the electric current really means what the voltage really means how do we calculate those quantities in an electric circuit using our knowledge that we've already gained from understanding what an electric field really is and then we're going to move on toward the end of this volume to some more advanced topics here before moving on to the next topic well into the next volume where we're going to really hit the magnetic field and begin to tie electricity and magnetism together into one nice pretty package but you can't do that too early because you can't really tie electricity with magnetism till you really fully explore what each of those topics are and so we're building our way to that process so enjoy the course we're going to take it one step at a time we're going to work a lot of example problems to make sure that you're comfortable and so by the end of this I promise you if you stay with me on this journey you'll understand that you'll do well on your test and I think you'll have a really great appreciation for how cool this stuff really is so here we're gonna talk about the concept of capacitance and what is a capacitor capacitor you know you've seen them all you know before in your life I mean they're in every little circuit that you see if you open your computer case up and look inside your computer and look at the circuit board you'll see capacitors all over the board you open up a radio you'll see capacitors all inside of a clock radio so capacitor is one of those one of the three big three or four big you know components that you have you know in a circuit that really can make electronics come alive but before you can really understand how to use them it's really instructive to know how they're constructed what they do and a little bit of the theory behind it and so that's what we're going to talk about in this section on capacitance so briefly let me read you a little definition and I'm gonna write it up on the board for you basically a capacitor stores charge it's a charge reservoir it lets you it's sort of a vault that you can put inside of an electric circuit you know you have a battery in a circuit and you know battery supplies electric current and we're gonna draw pictures of circuits and all these things coming up but you know from your basic experience that you have batteries and they store electric charge and they deliver the charge to whatever's in the circuit well a capacitor is like sort of like a temporary battery it's like a temporary charge reservoir so you can use the battery to charge up the capacitor and then you can take the charge out of that capacitor later and you can kind of take money out of the bank so to speak in terms of electric charge and we'll see that there's a little bit more to it than that but that's the basic idea of what a capacitor is it stores charge and anytime you have charge present net charge and a positive or negative charge you have electric field presence so there's gonna be an electric field inside of this capacitor that's going to be always present any time the capacitor has charged and so that's why it's makes sense to introduce capacitance now because we just talked about the electric field in great detail now we're going to show you a practical device that you use in every circuit you know this microphone here the video camera anything has a capacitor in it and we're gonna show you a practical demonstration or sort of a discussion of how the capacitors work why they're useful and then in the next few sections we're going to talk about circuits with capacitors and then we're gonna move on into some other topics related to circuits as well okay so let's talk about the concept of a capacitor capacitor right it has a sort of a lengthy scary-sounding definition but it's exactly what it told you it's sort of a charged reservoir so it's a device and a circuit I mean a capacitor really doesn't make any sense if you just put one on the board here I mean it has to be hooked into a circuit to do something so it's a device of a circuit that stores electric charge right so you can almost think about it like a battery they're different though I mean a battery has a chemical reaction in there generating these electrons that come out of the battery into your circuit a capacitor is a passive thing there's no chemicals in it it's just that it's sort of like a bank vault you can put money in to mate in the form of electric charge and then the circuit can pull the charge out and so it's really good for oscillating circuits where you're like maybe trying to build a radio where you have an oscillating wave coming in it's putting money in the bank taking money out of the bank I'm getting ahead of myself a little bit but capacitors and later on inductors we'll talk about are really used quite frequently in oscillating circuits and we'll talk about all the reasons why so it's a device that stores electric charge let me write one more thing I didn't tell you so I'm gonna take this period off stores electric charge that allows the cap capacitor and I'm going to just call it the cap the capacitor you're going to see me write that out a lot when I say cap it means capacitive is an abbreviation to store energy as potential energy you know I'm gonna draw a lot of pictures here in a second but you know it's good to see things in words so let's read this real quick it's a device in a circuit that stores electric charge and it allows the capacitor to store energy in the form of potential energy so it's basically a device that you can basically use to store electric charge now when you think about it now when you think about it we're gonna draw a lot of pictures here in just a second so don't feel like if you don't quite understand what I'm saying we're not gonna talk about it more cuz we will but when you think about it anytime you have charge present you know like we talked about in free space and the last volume you have an electric field and we talked a lot about the electric potential and we talked about the fact that this field just by the virtue of it being there sort of has energy associated with it in other words it has the potential to do work because if I have an electric field here and I take a charged particle and I put it in the field what's going to happen it's gonna begin to move and that doesn't could come from nowhere it comes from the fact that the field is doing work on that charge just like if I push something I'm doing work on that charge or that massive object or whatever it is so the electric field can actually do work and so we say that it has energy associated with it I mean otherwise how would it do work so it has to have energy associated with it it has potential energy it's the same thing as a rubber band you stretch your rubber band you can think of that like an electric field and if it's always stretched it always has potential energy now at the moment that you decide to let it go that potential energy has begun you know begins to be used well an electric field always has the potential to do work but it can only do work on charged particles on particles that have an electric charge on it so if this capacitor stores charge we'll talk about drawing some pictures in a minute and if those charges that are in there sort of set up an electric field inside this capacitor then it stands to reason that that capacitor must be a sort of like a storage device for potential energy and that's why it's useful really in the details of the circuit theory of why you would use them and we're not going to design radios in this course but I'm telling you it's sort of like an advance preview that when you get into oscillating circuits you might be oscillate and dumping electric charge in and out in and out in and out you might be trying to generate a wave to be sent at a radio wave for instance or any number of other other reasons that you would use capacitors but that's basically sort of a preview so it's a storage device it stores the charge and because of that it's storing energy in terms of that potential energy of the electric field as present there so if we were going to draw a picture the simplest capacitor you could have it's called the parallel plate capacitor and so I'm gonna write this like this it's the parallel plate parallel these are two lines parallel to each other so I'm not gonna write the word parallel plate you're gonna see that I'm gonna write these things over and over again parallel parallel so it's just silly to write that I'm gonna write parallel plate capacitor now this kind of capacitor is just sort of like for discussion you're never gonna see a parallel plate capacitor inside of a radio because there just be two physically large but to get the idea across you can really understand it really easily by thinking of two big giant plates so you can sort of think of here's one plate literally it's like a sheet of aluminum foil you can think of right and then right underneath it there's another one right and it would go something like this like that so they're stacked on top of each other and and they're not touching that's very very important the capacitors are always going to have two sheets of a metal of some kind right in this case we're drawing it as two planes but they cannot touch there always has to be a gap always always in a capacitor and so we draw it with a gap here and so this gap here I'm gonna draw it like this that's a distance D right between them and then the capacitor itself has some kind of surface area associated with the top plate and the bottom plate they have the same surface area right and so that is a capacitor ladies and gentlemen there's air between them or you could even bring it into outer space you don't need any air you could just put the two plates separated by a very small distance there in space there doesn't have to be anything between them and this capacitor can store charge it seems kind of amazing it's a very simple device but what happens is if you take this capacitor let's say you built it like this and you hooked it up to a battery then you would you know you would solder some wire here and you would maybe connect another wire down at the bottom and you brought these two wires out and connected it actually to a battery and the battery can supply a voltage and you know this you know you know batteries can supply voltage so I'm not gonna draw the symbol of a battery here we'll get to that in just a few minutes but let's just say it supplies a voltage right it supplies a voltage right so let's say this is a five volt battery or something then what's going to happen is this battery is going to begin to supply electrons over to this plate of the capacitor or maybe if I hook the battery up another way the electrons are going to come out this way and charge at this side of the capacitor so what's going to end up happening in the net result no matter how you hook it up is that this top plate is going to be charged up because we say it's stores charge so we'd say let's say it charges up to positive Q so many coulombs I don't know because I don't have any numbers here but it charges up to so many coulombs and if it charges up to so many positive coulombs on the top and then the bottom plate must be charged up to the same number of coulombs but negative coulombs because if you think about it let's just pretend that this battery is hooked up in such a way that the electrons are coming down and then they're flooding into this plate right so there it's beginning to charge this plate up negatively the beginning to charge that plate up negatively now if this entire plate is negative then what's going to happen up here is that these plates are really close together so if that bottom plate becomes negative and these things are made of metal and remember metal has a lot of free electrons in there that can be easily induced into moving around that's why we use metal in our wiring because we took a battery and we can easily ask those electrons to move along that's because metal has a lot of free electrons in the outer shells of the atoms plastic doesn't have a lot of free electrons out there all bound really tightly so plastic isn't conducting electricity but if this guy comes up to a negative charge because of the battery that pushed the electrons there then what's going to happen is this negative charge is going to begin to attract or I should say repel the electrons in the top plate farther away so you're gonna have a net positive charge on the bottom there so you kind of have to use your imagination a little bit but if these plates have a certain thickness to it which they always will if the bottom plate becomes negative then it's going to sort of repel the electrons in the top plate away farther up and that's gonna leave right near the right near the gap there it's going to leave a net positive charge so when you're looking at capacitors the easiest way to think about it is that one of these plates is always going to be charged positive and one of these plates is always going to be charged negative always always always right it's just the way it works so on a battery if you look on a battery you'll see like if you look on a double-a sized battery or something you'll see that one of the terminals has a positive plus sign label and one of the terminals has a negative sign so if we put that battery let's just say we put the battery in like this whoops like this and let's say this side was labeled positive and this side was labeled negative then this would exactly be the situation this negative terminal is going to charge this up negatively and this positive terminal is going to correspond to the plate being charged up positively right and that's basically what a capacitor is so you have charge sort of stored there that batteries deliver charge and both plates have an equal and opposite charge that's sort of stored there and then we're going to draw some more pictures here in just a second but the net effect of that is that in between these plates is going to be an electric field because you've got these two giant charge distributions one on the top one on the bottom so you're gonna have an electric field there and so when you haven't anytime you have an electric field there you're obviously have potential energy and so on and so on we talked about the fact that capacitors store or potential energy storage devices so let's hang onto that because we don't want to throw that away you know let's look at a side view of what this might look like so this could be the top plate of capacitor so I'm drawing a thickness to it because in real life they're always going to have a thickness and this can be the bottom plate of the capacitor right it's gonna be the top and the bottom plate of the capacitor now we're gonna pretend for just a second that we've hooked this up to a battery I'm not going to draw the battery here because I don't want to clutter the diagram up but we've hooked it up to a battery and so you're gonna have to trust me a little bit here the top plate has charged up to positive Q so many coulombs and then the bottom plate has charged up to you know so many cool but negative coulombs so you're always gonna have a positive charge on one plate and a negative charge on the other just like a battery always has positive charge on one side of the negative charge on the other it's just the waste circuits are you're always gonna have that charge in balance and that's how the circuit ends up moving moving electricity around in the circuit so your o it when you hook a battery up like that you're always gonna have one side charge positive one side charge negative now to help you remember that I'm gonna just go ahead and put positive signs up here so this whole plate has sort of net positive charge and this this plate here has net negative charge and that's done just because of the battery so in between what do you think is gonna be happening in between this gap now don't forget this gap is really really tiny I mean you would you know very very tiny fractions of a millimeter so it's huge here that in reality the gap between the plates on a capacitor is very small in between you're gonna have an electric field and we always said electric field goes from positive and the arrows are gonna be pointing outward from the positive sources and into the negative so here's your electric field that exists right there and I'm gonna go and put a little label here this is the electric field now of course this is not infinite plate I mean it capacitors aren't infinite objects right so it's a little small thing so in between the plates you can pretty much take this electric field to be constant when you start to get toward the edge of the plates you know the electric field if you actually had a magic camera that could look at it you would start to see that it would start to kind of curve around I can't really draw this very well on the board but it would it would do something like that and then on this side it would kind of go in like like this and you know something like that right so the electric field is going to be doing things like this but in between the plates is what we're really most concerned about so we've taken this battery we've hooked it up to a capacitor we've charged the plates up an electric field now exists between these plates right now the interesting thing is if we you know we're gonna get into this here in just a few minutes with some pictures but once you charge that capacitor up if you disconnect the battery what happens well you have a positive plate up here in a negative plate here and if we disconnect it from the circuit there's nowhere for these extra charges to go because I mean very very slowly I guess they'll leak you know through the air but and maybe they'll leak across through the plates here if they're close enough together but in a perfect capacitor you know in the vacuum of space or whatever then the leakage is gonna be really really really small so if you disconnect the capacitor that's charged up it's gonna hold it to charge so that's why you got to be really careful if you work on a television not so much the big televisions that we have now the flat panels but you know the big tube televisions with the giant glass screens because inside there are very large capacitors and those capacitors are used as part of the circuitry to generate and to steer the electron beam that's inside of that television tube and if you turn off your television and then open it up and then begin to start messing around with the capacitors in there those capacitors probably still have very large electric charge stored in there and you could easily shock yourself you've got to be really careful working on electronics just because you unplug it doesn't mean anything and I'll give you one more example if you look even on your laptop computer or that's probably the best example most laptop power supplies like you plug it in the wall and you have like the black piece that there and then the plug goes into your computer well there's usually an a light and some kind of LED light that tells you when that power brick is active when you take and unplug that power brick from the wall if you look closely that LED you will see stays illuminated for probably one second after you've unplugged from the wall but you would expect if you want from the wall powers gone light should be out but see inside that brick it are capacitors and transformers and other things that store electricity temporarily and so it stores that electricity and you can see that light gradually go dimmer and dimmer and that's because the circuitry in there probably has some other circuit path that bleeds out that capacitor but in a perfect capacitor if you charged it up pulled it out of the circuit held it out in free space it would retain the charge that's in there and then you could hook it back into a circuit and believe the charge out later on okay positive charge on one plate negative on the other electric field in the middle that is how it's storing storing that and by the way that is the potential energy because if I take this capacitor and hook it up in another circuit and this charge begins to then flow out of this capacitor I have stored that electricity and now I'm pulling it out of the bank vault so that's how capacitors store energy so in a circuit we've danced around the concept of a circuit but let's begin to draw things just to give you an idea of how it would look in a circuit so in a circuit you all know that you have a battery and you all know we're talking about capacitor so the symbol for our capacitor is very nice actually because it looks just like this it's too little parallel plates this is a capacitor and it reminds you of a couple things it reminds you that a capacitor is basically like two parallel plates which is what we're talking about here and the other thing it reminds you of is that these capacitors have a gap in there so it sort of mentally reminds you when you look at the symbol okay here a parallel plate so there's a gap in other words there's no touching here and this can kind of get a little bit sort of confusing to you a little bit because most of you guys in advanced physics here you kind of have an idea of what an electric circuit is you kind of know that you have to have a full circuit to a battery to have anything happen so this capacitor has a big fat gap between it they're not connected there's no connection in here all right so we're going to talk about the mystery of how that works here in just a few minutes but let me let me continue on so here's the symbol for capacitor and then over here let's say we have a switch and here I'm just going to draw sort of general symbol that you would see in a circuit for a switch it's literally what you would think it's just the circuit can be open or closed that's the switch now a battery can get a little bit confusing a battery is drawn like this notice it looks almost exactly like a capacitor almost exactly like a capacitor the only difference really is that one line is longer and one line is shorter and you put a plus symbol on the longer line and a minus symbol on the shorter line and there's really not a great way to remember that honestly but you're just gonna have to remember it sometimes you might see if you're looking at a circuit you might see a bunch of these things like like drawn like this but always always always you're gonna have a long line on the top and a short line on the bottom and always always always the positive terminal is the the long line is the negative negative terminal it's always the short line so that's just sort of a universal truth and that's just something you're gonna have to remember so what this means is that when the switch is closed this battery is going to begin to supply electric current in the circuit but you might ask yourself well how can it do that if there's a giant big open circuit right there inside the capacitor how can any electric current flow well the reason it can flow when you think about it is because as soon as you shut the you know shut the switch the battery has no idea that there's no like there's an open circuit right there I mean what's going to happen is the electrons are going to attempt to come out and they're gonna see and they're gonna go and follow the path of least resistance and what is going to happen is that these electrons are gonna flow out of this negative terminal and they're gonna come over here and there we're going to begin to charge up this capacitor and depending on the value of the capacitor if I have a very large capacitor it can store a lot of charge if I have a very small value capacitor and we're gonna talk about the values and then it can store not so much charge just a little bit of charge but eventually I'm gonna put enough electrons in there war we can't store anymore and after that point the electrons are gonna stop flowing here because there's nowhere else for them to go because they can't go into the bank anymore so if there's no charge here at all then when I shut the switch off the electrons are gonna begin to flow this is going to charge up negatively on the bottom and this is gonna charge it positively it's gonna match the battery terminals right here there and eventually I'm gonna fill up that bank vault full of charge depending on how I've constructed my capacitor and when it's full its full you can't hold any more and then it's gonna stop and this current in the circuit is gonna cease but up until that point in which everything stops there is an electric current here so so even though there's an open circuit there there's there's no connection between them the current can flow for a short period of time until the capacitor charges until the capacitor charges if this capacitor were replaced by a wire just connecting the two then yeah you'd have current going around and around forever until the battery died right you would have everything going round and round in an electric circuit like this but because you have this capacitor here the current can only flow for a short time until the capacitor is charged and that's basically sort of the function and you could disconnect the capacitor and boom you have sort of like a little storage device you've got electric you know charge stored in there so I've already said all this stuff but I'm gonna write it down so this is the capacitor this is a switch and now mainly teaching you this so that you can get an idea of the symbols and this is a battery and you have to understand that the battery is looks just like a capacitor except one line is longer one line is shorter positive terminal is always in the longer line negative terminal always on the shorter line all right and this battery supplies a voltage voltage V now the other thing is I'm I need to bridge the gap a little bit between talking about pure electro magnetic theory which is what we talked about before Gauss's law electric fields all these things now we're starting to talk about circuits in you know before we always talked about a potential difference voltage difference you know an electric field between two points Delta V the difference in potential right nothing is different here it's just that you're writing circuits you don't want to write Delta V every where Delta V here is that the change the change of potential across the battery terminals there you don't want to write Delta's everywhere you don't want to write Delta V everywhere so basically you just know that when you're talking about circuits and you see a voltage V it always means that change of potential in fact anytime you see voltage how many volts you know between you know in a circuit somewhere it's always the change of potential between some point in a circuit and another point in the circuit always in fact that's the definition of of the electric potential and sort of the voltage it has to be talking about between two points here it's the terminals of the battery but we'll talk about later when we're talking about more complicated circuits you could take a voltmeter and put it between any two parts in a circuit over here over here across this piece of the circuit across that piece of the circuit and you're measuring the potential difference in volts between those two points but I'm just telling you that when you see it in circuit you're not gonna see Delta V you're just gonna see the letter V but that that's the same exact thing it's the potential difference between the battery terminals okay so let me redraw this one more time and say a few more things I don't want to clutter up my drawing so this guy is connected here the battery terminal is always positive on the long line and negative on the short line right so once the switch is closed let's go ahead and draw it closed like this so we close the switch what's going to happen in the net end of the net effective of everything after you see a long time pass is that this capacitor which has a capacitance C we'll talk about it how to calculate that in a second this terminal charges up to positive cool number of coulombs and this terminal charges up to negative Q coulombs and it's easy to know what is what because it has to it has to be the same of the battery if you have a positive terminal up here then this terminal must be positive if you have a negative terminal here and in this terminal must be negative and the value of how many coulombs is stored here is going to be dependent upon the the capacitance that we're going to calculate in a minute and it's also going to depend on the voltage that this battery supplies in other words the battery is going to supply the current so obviously how much charge stored depends on the battery it's also going to depend on the capacitance C which is how your capacitor is constructed that's going to determine actually how much charge is in there now there's a couple things I want to explain your book may not explain this this early but you gotta understand in the deepest part of my body I'm an electrical engineer I mean I love physics I have a degree in physics but I'm an engineer so I'm trying to teach you these practical things that your book may not tell you in a physics book quite so readily up front I'm gonna teach you a little bit about circuits earlier than in your physics book as it gives what I'm really trying to say so let me tell you a couple of additional things and the reason I'm going to tell you these now is because it's going to help you as you start reading the other sections in your book because of these things I'm telling you well uh point number one when you have a circuit like this connected like this capacitor is connected to the battery terminals this voltage V is let's say it's 10 volts whatever could be have a 10 volt battery there in nivel battery there because it's connected directly across the capacitor if I measured the capacitor that'd be a voltage between these two points like if I've got a meter out and stuck it across the capacitor it would also be 10 volts let's say let's say it's a 10 volt battery then it would be 10 volts across here in other words it sounds so simple but you can get really confused when you get into more complicated circuits but if you have a battery source that's giving so much voltage potential across the terminals and it's connected directly to something like this then the voltage across this guy has got to be exactly the same as the voltage across the battery because they're physically connected together along on wire and in that sort of the case so later on let's say you know I'm getting ahead of myself a little bit here but let's say I add another capacitor here and let's say I added another capacitor here and let's say I added another capacitor here I added like three capacitors and the question on your test was well this is a 10 volt battery what is the voltage across these terminals you see you might be tempted to do some complicated calculation with all these capacitors and we're going to talk about lots of different kinds of problems later that you how you could simplify this significantly but you need to realize that by looking at the circuit you know that this capacitor is directly connected to this battery so the voltage the potential difference has to be exactly the same because they're physically connected together it doesn't matter what's over here it's just that when you're talking about voltage if you're connected to the source like that the voltage that you see at those terminals way at the end of the circuit is exactly the same as what you see across the battery that's something that's important for you to know now the next thing I want to talk about is the charge up process a little bit this is the charge of process a little bit so what's going to basically happen when you turn the switch on is the current is going to start to flow and obviously it can't go across this guy but it can pile up and begin to be stored inside of this guy and the electric field begins to be you know present there between the the capacitor plates so what's gonna happen at the moment you switch on that circuit is a relatively high current is going to start to flow but as the capacitor begins to charge it's going to begin to fill up and so it's going to begin to be able to accept less and less charge because it's going to be getting closer to full so as it gets closer and closer to full the current is going to to slow down so to speak it's gonna begin giving begin to bleed off until eventually no current is flowing at all so I don't feel the need to draw a bunch of graphs here and confuse the situation but I just want to tell you that when you turn the switch on that current is going to begin to flow and then as the capacitor charges up that current is going to go down down down down because the capacitor is filling up and then once it's totally full and it can accept a no more charge and then the current goes right down to zero or basically it's closed so close to zero that it might as well be zero so no more charge no more current flow so if you let this guy charge up fully you'll see the 10 volts across the capacitor and if you had a current meter in here like if you put a meter in here to measure the current you would see no more left current flowing because this capacitor is fully charged all right that is basically all I wanted to say about that now let's get a little bit into the bread and butter of how to calculate capacitance so we know that we if we put a potential difference across a capacitor we know that it charges up to have you know charge charge piled up on the plates we know that but what we want to know is how to calculate that and how it's related so basically without further ado we're gonna answer that question now the main equation that deals with capacitors is the following so the capacitor charge right is equal to the following Q which is the Q that's piled up on the plates there because we've stored it is equal to the capacitance which is the value of that capacitor times the potential across the terminals that's what you need to know it's a very simple algebraic equation there's no integral here there's no derivative it's very simple if you want to know how many coulombs are stored on this plate as a positive charge and of course they would have the negative number of coulombs down here you just need to know the value of the capacitor right that we're going to show you what that means in a minute multiply it by how many volts you have you're always going to be dealing in volts you multiply those things together and you're gonna get coulombs which is the unit of charge that we're always talking about so for the units I mean switch colors here so for the units the units of capacitance is called a farad it's called a ferret so one farad right and that's equal to let me show you something here if I solve this for capacitance the capacitance is equal to how many charmix charge I have on the plates divided by the voltage that's applied right so one farad is going to be the number of coulombs we write it a farad as F and that's going to equal to one Coulomb per volt cool and provoked no surprises so one Coulomb per volt but you never write Coulomb per volt you just write ferrets now of course you know just like a lot of other units in electricity magnetism a farad is a very large unit of capacitance I mean a farad capacitor might be the size of this room so you never I've never seen a farad capacitor most capacitors that you look at all capacitors that you look at inside of electronic devices you know when your computers or in your radios they're nowhere close to a farad affair it's huge so you're gonna be having micro farad's is usually unit that you're talking about 10 to the minus 6 ferrets or even nano ferrets or pico farads maybe maybe maybe you might see a milli farad capacitor for a big you know project and you know some kind of thing that's not in a consumer device but something you might have in a lab somewhere but definitely never a farad capacitor so if you ever calculated capacitance that's that you know whole farad then you probably did something wrong or the problem was just set up to give you a huge answer for no great reason but most capacitors that you see are always gonna be micro farad's so you're gonna see this a lot micro farad you're gonna see that a lot ok so just like we said before it's the same thing with Coulomb's you never hardly see a Coulomb either you you know you might see a milli Coulomb or a micro Coulomb but you hardly ever see a whole Coulomb of charge written down there okay now what we're gonna do now is erase the board and we're going to go back down memory lane talk about Gauss's law to show you how you would actually use the stuff and calculate the capacitance for like a parallel plate capacitor you if you know the charge on the plate you know the voltage great you can you can calculate that but there's a couple little things I want to talk about before we get to that point though I want to kind of take you on a little bit of a tangent and it's a tangent that's gonna be very useful for you to understand I didn't learn this until way later in my studies you know a couple years maybe after what you're studying here in this course it's very very important though here's the deal in real life when we have a circuits we know that the electricity in the circuit is the electrons flowing in the circuit we know this because you know it's called electricity so electrons are the things that are moving and that's just because the atoms like we said have a lot of free electrons in their shells so they can move around and all of that stuff now here's the deal you can look at the circuit you can label positive and negative and if you wanted to you could totally calculate the electric current in terms of the electrons that are actually flowing here that would be fine to do but because electrons are negative you would have a bunch of negative signs running around in every equation you've ever wrote down for electric current you would always have a negative sign there and it just gets cumbersome so some smart people a long time ago decided that when we're talking about electric circuits we're not going to talk about negative current the electron current we're going to instead talk about the positive kind of equivalent current and the easiest way to think about that is is with a picture if you have a bunch of atoms here this is the nucleus one proton let's say and here's the only free electron there and here's another proton in this wire and another electron here's another proton and here's another electron right now we know we know that when we hook the battery up to this wire you get to pretend and user imagination and say this is a wire when we hook the battery up we know that this electron jumps to this atom and then this electron jumps to this atom and then this electron jumps and jumps and they all end up jumping in a chain reaction all the way back around to the battery and that's how an electric circuit actually works right that's great but that is the negative electrons that are flowing but if you think about it for a second for every let's say I have an atom here with my electron if my electron leaves this atom I have left behind a positive nuclear or a positively charged nucleus so for every electron that moves that way it's equally mathematically the same as sort of a positive charge flowing in the opposite direction it might be more you know kind of easy to understand if you just think about one single atom if you have one atom and here's your electron if he jumps away that current is sort of flowing that way it's exactly the same as a equivalent positive current flowing in the opposite direction and that positive current is what we actually use in our equations and in our electric circuits and all of that stuff so I've been careful so far to talk about electrons coming out here and charging up capacitors but in a few sections we're gonna talk about the definition of the electric current which is what what we're sort of hinting about right now and what I'm trying to tell you is that electric current is always gonna be positive current coming out of this positive terminal of the battery so it would always be coming out of the positive terminal of the battery even though protons are you know positive charges don't actually flow in a circuit we know this physics wise but mathematically it's the same exact thing and I think if you think hard enough about that and visualize it I think you can understand it if you have a whole string of these atoms and all the electrons are jumping from one guy to the next over this way then mathematically it's exactly the same thing as pretending that positive charges are going the other way even though they're not right so when we calculate the current we're gonna always be talking about the positive current coming out of the positive potato of the positive terminal the positive current coming out of the positive terminal even though in real life we know that it's not actually the positive charge is moving it's the negative charge is going in the opposite direction okay I think that's enough of that so let's go ahead and erase the board and talk a little bit more about how to calculate the capacitance so what we're gonna do is just move right along here so how would we how would we calculate the capacitance what we said here's how you calculate it if you know the charge on your capacitor and you know the voltage that you apply well then great you can calculate the capacitance but that doesn't really help me too much if I have a parallel plate capacitor I mean yeah I guess I could measure the charge on it and then I can I know what voltage I'm applying to it so you can always calculate the capacitance that way but you know if you had another arrangement let's say you built a cylindrical capacitor which we're gonna talk about in a minute where instead of two flat plates it's wrapped around the two things that are parallel to each other are kind of wrapped around like that how would you compare how would you calculate the capacitance of a different configuration so the general way that you do it it is exactly this it's just that you have to take a few steps here to calculate the capacitance you have to do the following what you don't have to do Phong but you you're gonna end up following the same step so the we know that the charge is equal to the capacitance times the voltage applied so the capacitance is definitely gonna be equal to the charge divided by the voltage so this is what we need to use it's just that usually you don't really know exactly what potential you know is gonna be over there and so unless you're building a circuit to actually measure I'm talking about when I say how to calculate the capacitance I'm talking about if I draw a picture of a capacitor how would I calculate it I mean of course you can build it in the lab and measure the capacitance but I'm talking about how would you mathematically calculate based on your design of a capacitor so here's what you're basically going to do and I'm only gonna do one simple example like this you're probably you may be asked to do this on a test but you're not gonna use this too much in real life so I'm not gonna I'm not gonna dwell on it the first thing you do is you assume a charge Q whoops Q exists on the capacitor on the capacitor plates right so you have to make an assumption there so you start at that point and then you calculate the electric field between the plates because you know there's gonna be electric field between the plates by Gauss's law and just for reference because you may not remember that off the top of your head Gauss's law is the permittivity times the surface integral of the electric field dotted through da and that's gonna equal to the included charge and we'll do an example to show you but basically if you know the charge then you know this and if you know the geometry you can sort of figure out where the electric field is gonna be and you can basically try to calculate the electric field now once you know the electric field then you can calculate this potential because the electric field remember is related to the potential so we use using e we calculate the potential across the plates and how do we do that if you remember the potential is going to be equal to whoops the change in potential is going to be equal to the final value minus the initial value of the two points that you're actually dealing with which is gonna be from one plate to the other because I'm trying to find the voltage across the two plates and that's gonna be equal to negative integral from initial to the final position which is across the plate of the electric field dotted with D s which is just the path that's gonna give you the voltage and then at the final step you know what Q is cuz you assumed what Q is you've calculated what V is so then you just say the capacitance is equal to the charge you assume divided by the voltage that you calculate so like I said I mean you can build the capacitor and you can hook it up to a 10 volt source and you can measure the charge on the plates and you can hack you can calculate the capacitance that's not what this is really intended to show you this is trying to say let's design a capacitor let's design it like a battleship you know total weird shape to it or whatever how would we calculate the capacitance without actually building it so you make an assumption you say okay I have five coulombs to charge on each plate I calculate the electric field that's between those guys and I use Gauss's law or whatever other tool I have to figure out what the electric field is once I know the electric field I can calculate the potential across the plates because I can just integrate the electric field between those two points and we covered all of this in volume one so all of this is stuff you already know how to do once you know the voltage across the plates and the thing that you assume there for the charge then you can just simply divide the two things together and calculate the capacitance so I think it's gonna be nice to show you this one an example and again we're gonna learn about a few different configurations of capacitors here I'm only going to derive the capacitance for the first configuration the easiest one the parallel plate capacitor because you could derive it for all the other ones but really it's diminishing returns once you know how to do it for one of them it's good to know you're not gonna be doing this very often unless you're maybe on a test but in real life you certainly wouldn't be doing this because when you buy a capacitor off the shelf it tells you what the value of the capacitance is this is sort of more of a theoretical thing to build your foundation and sort of teach you where this stuff's coming from so let's look at the parallel plate capacitor parallel plate capacitor and just like we drew a picture before it's basically just like you would think it's two plates you draw it kind of big so we can see here it's two plates like this and the top plate has positive charges let's just say cuz it's charged up we're gonna make an assumption that was step one remember step one is we assume we have a charge on the plate so we make that assumption and so we say okay this is positive Q and we say okay this is negative Q okay that was step one and step two is basically or step one I should say after we make an assumption that the plates have a certain charge on it is to try to calculate the electric field so let's go ahead and do that we're going to use Gauss's law which is the electric field dotted with da which is our Gaussian surface and the this is the charge enclosed so there's a couple things we need to add to our drawing in order to get this we know just from the previous drawings that there's an electric field here I mean it has to be there always is inside of a capacitor because you have these charges as are piled up on the plates so there is there is a there's an electric field inside there and in order to calculate that with Gauss's law we're going to have to have a Gaussian surface now you have to think about this and remember that this is a parallel plate capacitor so it extends out of the board it's a three dimensional object so here's one plate there's another plate underneath it and it kind of goes into the board there and so it has a surface area to it so this top plate has a surface area a which is the planar area and the bottom plate has exactly the same area so this is the this is the area and so if I'm going to calculate a Gaussian surface the best one or you use a Gaussian surface a really good one is this one right here I'm gonna cut this plate kind of in half I'm gonna come over and I'm gonna go right into the middle of my electric field I'm gonna come back up and the reason that's a good plate is because in a minute we're gonna be integrating this electric field through this Gaussian surface area that I have but there's no electric field pointing out through the sides here now in a real life capacitor there would be those fringing fields around the edge but when you're used to doing these theoretical calculations like that you assume that those fringing fields are very very small so they don't count so there's no field going through the sides there's no field inside of this metal we talked about that from before there's no electric field inside of the metal the only field is really out here and it's cutting across perpendicular to our Gaussian surface so this is a great one to use alright so let's go ahead and and do it so what we're gonna have is this is the productivity the electric field is a constant inside of here you know if you go back and look at your derivations of the parallel plate parallel plates when you have two parallel plates like that the electric field assuming that assuming that these things are really close together compared to how big they are then the electric field is going to be relatively constant there so you can pull the electric field out and that basically simplifies the integral totally all you're left with is some integral over the and equal to the charge and clothes but inside of this Gaussian surface this plate that's extending out like this has a total charge of positive Q so I'm gonna put positive Q negative Q on the other plate has nothing to do with it because it's not inside the Gaussian surface only what's inside the Gaussian surface is the charge enclosed or included so this integral here over da is nothing more than the surface area of the plate there that that you know because you're building it and so you have this result right here so you have the permittivity times whatever the electric field is times the surface area of the plate is equal to the charge now this isn't the answer this is just really allowing you to calculate you know sort of the electric field there so let's stop at this step we know it that charge on the plate is there and so remember the end result of what we're trying to go is the capacitance is equal to the charge on the plate divided by the potential we know what the charge on the plate is in terms of all of our other variables so let's stop there the next step is let's use this electric field where the electric field that exists I should say between there to calculate the potential so let's go and have a negative integral from over the two plates you've done it with the S so basically I'm going to take this electric field and I'm going to integrate from one plate to the other I'm going to integrate this electric field over a path from one plate to another and that integral is going to yield the negative of that number I get is going to be the voltage that's basically across the potential difference between those two plates so if I'm going from one plate to another I can simply say that this plate is at zero and I could say that this plate is at D because these plates are spaced apart by D units we talked about that in the very first drawing so I can go from 0 up to D and so this is a very simple integral because this is a constant so you get negative e integral D s which is just a linear integral over distance 0 to D okay and so what you're going to end up having here in the end is negative II D so the potential across is equal to negative II D and I think you can see that this integral does reduce to D because this integral is basically just having a 1 here so you have a dummy variable S evaluated at D minus 0 so you're basically just going to get D so you've Jacek aliy all you've done here is you've you've added up the distance between 0 and d and of course you get the the distance that you had to begin with this is a simple integral here so you have this guy so we know what the charge on these plates is in terms of everything and we know what the potential is across the place that we've calculated and so when we calculate the capacitance Q over V we have epsilon naught times e times a from here over this guy which is e D and I could carry a negative sign there but really we're only really care for this particular calculation we only care about the absolute value of the potential because if I put a negative sign here you remember how we talked about this in the last section when you have negative and positive in terms of potentials it just depends on which what's your whole what you're looking at is your initial position and what you're looking at is your final position it just means that one side is higher than another in terms of voltage if you have a negative here you know I could put a negative here but then I would would give me a negative capacitance and you never have negative capacitance so we basically just strip the sign out and when we're doing these calculations over the potential we're only really interested in the absolute value of the potential difference between the two another way to think of it is I did my integral like this if I had flipped my limits of integration around and integrated from top to bottom I would have got a positive number anyway and that's exactly what I want so the order of integration isn't really going to matter for this so you can see that the electric field cancels and so the capacitance of a parallel plate capacitor is the permittivity of free space multiplied by the area of the plates of these plates divided by the distance between them divided by the distance between them and that's it you'll see this in your book this is the parallel plate capacitor it's the capacitance so if on a test you have a problem and it's like you have a parallel plate capacitor the surface area of each plate is equal to so many square meters and the distance between them is so many millimeters of course you have to convert everything the meters divide them multiply by the permittivity boom you got a number in farad's that's the capacitance of that capacitor right now one more thing I want to point out well first I want to review a little bit then I want to point a couple things out all we did is we said okay we have a charge on our capacitor equal and opposite like always we use Gauss's law over a Gaussian surface that only includes one of the plates and the electric field can come out because it's constant in there and so what we're left with is this guy is equal to the charge so we hold that in our back pocket and then we go and look and see all right if we're gonna calculate the potential between the plates we integrate over the distance the electric field dotted would be s but again because the electric field is constant we can pull it out and this integral reduces down to something really simple so now we have the potential across and we have the charge we divide the two we calculate the capacitance for any other configuration in real life the electric field may not be perfect well even in this case a real parallel plate capacitor the electric fields not totally constant inside so pulling it out of the integral isn't really legal in real life but it's a very good approximation whenever you have plates that are really close together and very large so that's what we're gonna go with on this is sort of a theoretical calculation just think back to physics one when you had your massless pulleys or your your frictionless ice or something like that so that's what we're doing here the other thing I want to talk about is when you look at the capacitance here if the distance goes down between the two plates the capacitance goes up right if the distance goes down the capacitors goes up and also if the distance goes down if you look over here at this equation if the distance goes down then the electric field also goes up because if you solve this for electric field you're going to have a D on the bottom so what I'm trying to get at is if you have your the distance between the plates goes down electric field goes up so more higher intensity electric field in there and capacitance goes up so you're allowed to store more charge in that capacitor if you can get the plates closer and closer together and if you can also make them physically larger they can also hold more charge because there's more atoms in there to store the charge and if they're closer together you can get that electric field to higher intensity and you can also store more charge in there because of that because the electric field is inducing this electric field is inducing these negative charges on the other side here so the higher you can get that electric field the more charge are allowed to really store there another way to think about it is if you can get a higher electric field inside of this capacitor it'll have more potential energy because the field will be stronger okay so what we're gonna do now we've given you the equation for a parallel plate capacitor and I'm going to draw a few other kinds of configurations of capacitors that that are going to be in your book and we're going to give the values of the capacitance for those okay so let's write down the equation that we just derived for a parallel plate capacitor and then we'll write down a few other configurations so for parallel plate right the capacitance is equal to the permittivity times the surface area divided by the distance between the plates so this is sort of one to remember and notice that we derived it using Gauss's law so if you were to write down any configuration of two plates maybe spherical cylindrical however you want to do it you could use Gauss's law in exactly the same way to calculate the capacitance of any of those configurations you would assume a charge on there set up a Gaussian surface calculate the electric field calculate the potential divide the the charge by the potential and then you get your capacitance out and your book does that for some of these configurations but I'm not going to drive every one of them because I don't think it's a great use of time I think to know how to do one of them tells you where this is from the rest of it should just sort of be given I think cylindrical capacitor now cylindrical capacitor you actually do see quite a bit in you know in in real circuits so what you would have here is let's say you had a cylinder something that looked kind of like this has a length to it right and so here is another central core kind of so basically you've got sort of the same thing as the parallel plate it's just that it's wrapped around on itself you've got an inner surface and an outer surface and they're separated by a distance so this distance is D right that distance is actually the way I want to label it is a little bit different it is distance D effectively it's the same sort of thing but I want to write it a little bit different I think it'll make it a little bit clearer if you were to look at this top down then we would have an inner guy and you would have an outer guy right and so let's say you know the inner guy might be positive charge let's say and the outer guy could be negative charge there could be backwards or if it's an oscillating circuit it reverses back and forth positive negative the inside gets positive and negative every cycle and so on but that's basically what you would have and then so let me switch colors here the radius of this inner guy we're gonna call a and the radius of this outer guy we're gonna call B and we're gonna say that this inner guy has a charge Q and this outer guy has a charge of negative Q and we're gonna say that this capacitor has a total length of L so that sort of describes the geometry just like for the plates all you needed was the distance in the area here you need to know sort of the inner radius the outer radius and the length to fully describe the geometry and once you have that the capacitance that you get is going to be 2 pi times the permittivity epsilon times the length of the capacitor divided by the natural log of B over a so you know the distance is B divided by a take a natural log the length divided by that quantity times these numbers that is gonna give you the capacitance in farad's again it looks complicated but if you were to set up a Gaussian surface around and do exactly what we did before this logarithm would pop out as part of the integration that happens there and you would see it there so that's don't be surprised like why is there a logarithm here it's as a result of the derivative the integrals that you end up doing when you do that derivation but I'm not going to derive it for you because number one you're really not going to be using this in real life too much you're not gonna be really deriving capacitance too much anyway in real life and the second thing is I think it's more instructive just to sort of know where it comes from and sort of that you can at least say that you know how its derived and then if you really wanted to dig into those details I'm confident that with Gauss's law you can figure that out and it's a better use of our time just to continue moving on so another kind of capacitor you'll probably see in your book is a spherical capacitor it's a spherical capacitor and it actually looks exactly the same because you know but it's not the same but it looks the same if you draw it on a piece of paper because if you have two concentric spheres instead of a cylinder where they're nested inside let's say you had a central sphere and then an outer sphere again you have a separation between two surface areas is really all you need to make a capacitor and let's say that you know this inner radius was again a and this outer radius was again B there's no length to it because these are spheres and let's say this is you know charged up to positive Q and let's say this on the outside is negative Q so the capacitance in that case would be equal to 4 pi times the permittivity times a times B over B minus a that's the capacitance of a spherical capacitor again it's just the geometry a times B divided by B minus a times these constants out in front it kind of makes sense that you have a 4pi here because spheres involves you know you know 4 PI R squared for instance and then over here 2 pi kind of makes sense because you have some spherical cylindrical stuff going on there and you would see all that fall out of your derivation now the final one that we're going to talk about is an isolated sphere now this one's a little bit interesting to think about if you have just an isolated sphere you you know for capacitors you have to have two plates but it turns out that if you have like an isolated sphere here if it's not connected to anything you can charge it up and it'll hold the charge you know that can be discharged at a later time so it sort of it behaves as like a capacitor and the easiest way to derive it is just to sort of taking this result and make the second dimension B go out to infinity so you can sort of think of as an isolated sphere as having this central sphere where the outer sphere is so far away that it really isn't even relevant so if you start off if you start off with what we had before the capacitance is 4 pi times the permittivity times a B divided by B minus a and if you take B going to infinity then what are you gonna have as a result what are you gonna have as a result you'll want to basically take the limit as B goes to infinity but if you do that infinity is going to be on the top and then infinity is going to be on the bottom and so it's going to be a little bit hard to do so the easiest way to do is to rewrite this capacitance as the following four pi times epsilon not if you divide B on top and B on the bottom that will you'd be left with is a over and then what you would have on the bottom is one minus a over the right because if I divide the top might be then I'm just gonna be left with a and if I take this denominator and divide it by B which is totally legal to do then I'm gonna have one here and I'm gonna have a over B here this is sort of a trick that you learn when you solve simple limits so then if I take the limit as B goes to infinity and then this term is going to drop to 0 1 minus 0 is just simply 1 and so the capacitance that we're seeking here is 4 PI epsilon naught a which is the basically the distance from the center of this guy out to the edge and so generally when you're talking about a sphere you want to talk about the radius so you just relabeled at 4 PI epsilon naught R you know you can just use these formulas in the book it's fine it's just to try to give you a little bit of an idea where they're coming from if you take a sphere a spherical capacitor and take that second you know the dimension B as the second part of the capacitor way off at infinity and just do the simple limit right here then what you'll get is this capacitance of an isolated sphere so if you know the radius of a sphere and how you know that basically the geometry of it from the radius then you can calculate its capacitance and it can behave like a capacitor you can hook it up and charge it up and it'll it'll maintain charge and you can bleed charge off like that so this is sort of the the end of the section here we're gonna work a lot of problems in the next section we're gonna get you a very comfortable dealing with capacitance and calculating how many farad's a capacitor is and understanding what it really means more than just sort of the theoretical things that we've talked about here but a parallel plate capacitor is basically going to be related to the surface area and in the distance between the plates the capacitor of a cylindrical capacitor you see this a lot in real circuits because if you look at a lot of those capacitors they look like cylinders and again that's gonna be related to the length and also the dimensions of the inner and outer same thing here it's gonna the capacitance is going to be related to the dimensions of the inner and the outer sphere and here is going to be related to the dimensions of our of the sphere that you have so what I'm trying to say here my pattern is that the capacitance of any object or any kind of arrangement is always just going to be related to the Gion three of how you've arranged it the capacitance of parallel plates it's only related to geometry this is just a constant the capacitance here it's only related to the geometry the the in other words the length the width the height how it's all set up it doesn't have anything to do with anything else other than how you've constructed it and the same thing here so you can calculate them all with Gauss's law you can look in your book for that detailed derivation my advice to you is just understand how this one was calculated to know basically how you would go about calculating the rest of these by choosing your Gaussian surfaces but I would spend most of my time understanding the fundamentals of capacitance and and what it means and then on into the next section of this DVD course where we're going to work a lot of problems to really teach you you know how to use capacitance and how to calculate and put some numbers to some of these equations I'm Jason I hope you've learned something from this section let's go on to the next section and work some problems dealing with the calculation of capacitance

Channel: Math and Science

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Keywords: capacitor, physics, energy storage capacitor, dielectric, physics electricity, physics capacitor

Id: uVvh18owfZk

Length: 68min 50sec (4130 seconds)

Published: Thu Feb 04 2016