Capacitors - A Level Physics

hello today we're continuing in our a level physics revision series looking at the subject top capacitors capacitance measures the amount of charge stored per volt and therefore its units are charge which is Coulomb's per volt but we actually give that the name farad and use a capital F now a farad of capacitance is very large so in most cases you will probably see references to a micro farad and how does a capacitor work well you have a capacitor which is essentially just two parallel plates connected in a circuit with a battery and what actually happens the electrons from this side of the battery will flow to this side of the capacitance and give it a negative charge that negative charge which is building up on this side of the capacitance will repel the negative charge on this side and that will go towards the positive side of the battery and if it repels the negative charge on this side it will leave behind a positive charge on that side it's important to notice that no current flows across the gap it's simply that the electrons flow up to this side and accumulate on that plate making it negatively charged that negative charge repels electrons from this side that then go back to the positive side of the battery leaving a positive charge on this side of the plate and at that point the capacitor is said to be charged how do you make a capacitor well as I've said they are two parallel plates usually of some kind of metal foil which are close together and they mustn't touch obviously there has to be a gap between them so what is usually done is some kind of insulator we call it a dielectric but an insulator that doesn't carry we'll go between them sandwiched between them and then you roll the whole thing up rather like a Swiss roll and that is then packaged as a capacitor now let's do a little experiment here's the capacitor and we're going to put a switch in the circuit we have a battery we're going to have a variable resistance we're going to have an ammeter which will measure the current flowing and we're also going to put in a voltmeter which will measure the potential difference across the capacitor now ordinarily if this resistance were not here and you close the switch then the electrons flow around the circuit onto this side of the capacitor just as they did up here and they will repel electrons from that side and you will get plus and minus minus that side plus on that side but of course initially a lot of electrons will flow but as the electrons build up on this plate so they will also have the effect of repelling any more electrons coming this way so what you find is that when you first switch the current when you first switch the circuit on there's a high current as electrons flow onto the plate but that current will gradually reduce as the current as the electrons get more and more accumulated on this side of the plate because they then repel any more electrons from coming and make it harder than for them to come so the current will vary and what we can do therefore with this variable resistor is to change it so that as the capacitor is charging the current remains the same so you need to start off with a high resistance to have a low current and then as the current would naturally fall you change the resistance and make it smaller so that you get effectively the same current being maintained in the circuit until the capacitor is fully charged and if you do that then you can measure the current against the voltage and because we have set it so that the current remains at a constant level sorry I'm measuring the wrong thing here what we should be measuring here is time so let me just recap as the capital capacitor is being charged you measure the voltage across the capacitor at regular time intervals and what you find is that the capacitor will increase its voltage as time increases so it's time against voltage but we know that charge is current times time because current is effectively the amount of charge use moved in a unit time so if the current is held steady which is the whole point of this variable resistor then we can say that this is also a measure of charge because charge equals I T and I is constant so T is effectively proportional directly to the charge so this is not only a measure of time it's also a measure of charge because the current we've kept constant and since we started off this whole thing with a definition of capacitance as charge divided by voltage so he's saying capacitance equals charge divided by voltage that means that charge is the voltage times the capacitance and that means that if you plot Q against V Q against B the gradient of that line will be C so that's the way you can measure the capacitance now let's consider two capacitors in parallel here's a battery he is our first capacitor and he is our second capacitor and they are in parallel it's a bit like in the electricity series of videos where we had resistor in parallel and what we want to know is what is the effective resistance of those two in parallel suppose they were inside a box and you didn't know what was inside the box except you knew there was some kind of capacitance what would be the effective capacitance of that box that you couldn't see inside well we know that charge is voltage times capacitance so for the individual capacitors let's call this C 1 and this C 2 and the total voltage across them is the voltage of the battery that's V then we can say that the total charge on these two capacitors is V times C 1 plus V times C 2 and that equals the total charge on the effective capacitor which is essentially V times C effective now of course the V's just cancel out and you find that the effective capacitance equals C 1 plus C 2 and that's the effect of two capacitors in parallel you simply add the two capacitors values together contrast that with resistances in parallel where you would have had 1 over R effective equals 1 over R 1 plus 1 over R 2 well what about the other one what happens if they are in series here is the battery here is capacitor one here is capacitor 2 and once again we're imagining that this is a box and we know there's some kind of capacitance arrangement in it what is the effective capacitance once again the battery has a value V and that voltage will be dropped across the first capacitor value V 1 and across the second capacitor value V 2 and we know that V equals V 1 plus V 2 but we also know that q equals V over C which means V is sorry Q equals VC which means that V is Q over C so V equals Q over C so this V here is sorry is Q over C effective where that is the effective capacitor of the capacitance of these two combined equals V 1 well what's V 1 that is Q over C 1 what is V 2 that is Q over C 2 and now we can divide by Q and we get that 1 over C effective equals 1 over C 1 plus 1 over C 2 and that is the way you calculate the effective capacitance for two capacitors in series now we want to talk about the amount of energy that's stored in a capacitor let's take a circuit where we have a couple of batteries this is a little light bulb here is our capacitor C and here is a switch which we can switch either way to start with we're going to switch it that way so that effectively the capacitor will charge and of course what you will find is that that side of the capacitor will be negative and that side of the capacitor will be positively charged now if you now break that switch the capacitor remains charged there is no circuit nothing no current can flow but it has as it were potential energy in it because it's got a charge of electrons on one side and as it were assertive asset electrons on the other side and if you now connect the switch that way so that this circuit is now complete that capacitor will discharge through that light bulb and the light bulb will flash it's one of the principles of the way that a flash bulb works in a camera so just to recap when the switch is in this direction then the battery charges the capacitor when the switch is in this direction then the capacitor discharges through the bulb and produces a flash of light and that means that the capacitor had stored energy because that energy was what causes the light bulb to flash what is the energy that is stored well we know by definition that capacitance is charge divided by voltage so if you've got a fixed capacitor with a fixed capacitor with a fixed capacitance if that is going to be standard because it's a single capacitor what it means is that as the charge increases or rather as the voltage increases so the charge on the capacitor increases if you were to plot the voltage against the charge you'd find that was a straight line and let's suppose that the voltage of the battery was here that means that the maximum charge on the capacitor will be here what is the energy that is stored on that capacitor the answer is it is the area here and what is that area there well that is simply the area of a triangle which is half the base times the height so the energy is half the base which is Q times the height which is V and so now we have a formula for the energy stored in a capacitor it's half the base times the height half QV but since C equals Q over V you can rearrange this formula so that the energy is equal to 1/2 C V squared because we simply say Q is CV and if you put CV in here you get CV times V which is CV squared and is also equal to Q squared over 2 C because if you substitute for V V equals Q over C so Q times Q over C is Q squared over C and then the half Q squared over 2 C what determines the magnitude of the capacitance well the first thing is the distance between the two plates D and the capacitance increases as the distance between them decreases that is to say that C is proportional to 1 over D if you make the distance between the plates smaller the capacitance increases the capacitance is also a function of the area of overlap of the two plates if you don't overlap them completely then it's only that area that determines the capacitance so the capacitance is also proportional to the area of overlap and finally the capacitance is dependent on the dielectric you remember I said that there was an insulator that we put as effectively the sandwich between the two plates and we say that the capacitance is proportional to the dielectric which we represent with the symbol Epsilon so we can now write our formula for a capacitor capacitor equals epsilon a over D representing the three dependencies for the value of a capacitor now let's just think again about how we charge a capacitor here we have a couple of batteries we put a resistance in the circuit here is our capacitor here we have an ammeter to read the current that's flowing in the circuit and here we have a voltmeter to measure the voltage across the capacitor now as the capacitor charges of course the electrons will flow round here and this side will be negative this side will be positive but as we've already discussed what happens is that initially there's a high current but as the electrons accumulate on this side of the capacitor they will tend to deter further electrons from coming and so the current will actually go smaller as time progresses and what that means is that the charge building up on the capacitor initially builds up at a high rate but then tapers off so that if we plot charge against time what we will find is that it will start by going up very quickly charge is VC and so if this is a measure of charge then since the capacitance is constant this is also a measure of voltage so that's you could also a measured voltage along there and that gives you an indication of the charge so in other words this voltmeter will show a sudden increase in voltage but then it will it will ease off until it's level and that will be the level of the voltage of the battery you cannot get a higher voltage than the voltage of the battery and at that point no more charge will flow on the other hand if you discharge the capacitor then let's bring that down so you can see what's happening if you look at charge against time when you discharge then it discharges like this and once again since Q equals VC this is also a proxy as it were for voltage so let's now look at a capacitor that is being both charged and discharged first of all we're going to plot voltage which course is also a measure of charge against time we remember that V equals QC and therefore for a constant capacitance as the voltage increases so the charge increases so this is both a measure voltage or charge and we said that as the capacitor charges up the voltage or the charge goes up according to in time in this way until it is completely charged and that's when it is at the voltage of the battery and then we said if you discharge it it discharges like that and what you find is that the formula that governs this part of the process is that the voltage equals one minus V naught where V naught is the voltage of the battery times e to the minus T over RC where you remember we had a resistance in the circuit and a capacitor well T over RC is the time constant so the voltage that is measured across the capacitor as it is being charged equals 1 minus the voltage of the battery times e to the minus T where T is the time here divided by RC where R and C are the resistance and the capacitance in the circuit by contrast the formula that dictates that for is that the voltage measured across the capacitor as it is discharging is equal to V naught which is the voltage of the battery times e to the minus T over RC again it's the time divided by the resistance and the capacitance in the circuit and we call the time constant which is often given Tam as our C now what happens if T in our formula here equals tau which means it equals RC well in the case of the dish you can see that V would equal V naught times e to the well minus T over T is minus 1 because RC is T so minus T over T is minus 1 which means that V divided by V naught equals 1 over e and that's approximately naught point 3 7 what that means is that in tau seconds where tau of course is value RC in tau seconds V Falls to 37% of the value of v-0 and because V equals QC if V Falls to 37% of the value then since C is constant Q also Falls to 37% of the value of Q 0 which is the total charge when the capacity the capacitance is fully charged you can also show that tau is the time that it takes for the capacitor to charge to 63 percent of its total charge in other words tau is the time it will take for this process to get to charging to 63 percent 63 percent of course is just 100% minus 37% of total charge that it's going to get when it is at its maximum charge now you can see that since tau equals RC and that C is obviously a constant because it's the capacitor we're looking at if you increase R you will increase tau and that will make it take it longer for the capacitor to charge but it will also it will take longer for the capacitor to charge but it will also take longer for the capacitor to discharge and that means that you can get a current flowing for a longer time albeit of course it will be a much smaller because you've got a higher resistance in the circuit but you can get a current flowing for a longer time if you have a higher resistance in the circuit

Channel: DrPhysicsA

Views: 273,735

Rating: 4.9029694 out of 5

Keywords: Capacitor, capacitance, farad, energy

Id: A4WL48koxpM

Length: 22min 17sec (1337 seconds)

Published: Mon Mar 12 2012