Equivalent Capacitance for Capacitors in Series and Parallel Circuits Reduction | Doc Physics

I like capacitors you know that the governing equation for resistors is V as I are a little bit of Ohm's law right but you also know that the governing equation for capacitors is Q is CV and this is a very interesting relationship between these two and we'll see that well V is Q over C in a very similar way like except that R is linear here with V but C is inversely proportional to V so we're going to get some different results here with capacitors as we try combining them I want capacitors in parallel and in series and we'll start with the simpler one seed remember which was simpler with resistors was it so parry to the parallel or series and of course it was series it turns out that in capacitors you're talking about a simpler situation if you put capacitors in parallel so there's a capacitor this is just a battery and here's another capacitor that I'll put here and then I'll give you one more capacitor over here and let's say in principle that these are different capacitors C 1 C 2 and C 3 and I ask you what is the same and this should be obvious because you know a lot about metal what is the same for each capacitor in parallel so this is teaching you physics in a way that you'll be able to figure out why it is what it is so if you go to a desert island you're the only person there and you have to rebuild civilization at least you'll be able to start with physics of the 19th century which would be pretty awesome you could get a lot of civilization going there again I'm not going to tell you this until you tell me what is the same for each of these capacitors is it the capacitance no that's stupid right is it the charge I don't think there's any reason these have to be holding the same charge is it the potential is the potential difference across this similar to the potential across that similar to the potential living across that course they're connected by a wire so this is an equipotential down here that's an equipotential up there so the capacitance is different and the charge is different but the voltage is the same for each which gives us some really cool ability first of all I'm going to draw our equivalent circuit and that is just a single capacitor here that has an equivalent capacitance C equivalent right there plus M minus and we'll put a little bit of color on this slide before we're all finished let's say that the voltage of the battery that's over here V that is the voltage across capacitor one is the voltage across capacitor two is the voltage across capacitor three but charge we're going to investigate qmc and V just like with a resistor we investigated v and I and ara so the charge total that would be all of these charges on the positive plate that's going to be Q total that's just going to be the charge on the positive plate of one plus the charge on the positive plate of two plus the charge on the positive plate of three and you know that each of these Q's is CV so I've got c1 times V battery plus c2 times V battery plus c3 times the battery all right and I can beautifully factor out the V battery so I get C 1 plus C 2 plus C 3 times the battery and this says Q total is something times V battery and I know that Q is CV so if I'm talking about total charge and total voltage this thing that's right here has to be total capacitance or effective capacitive or equivalent capacitance so I'm going to say C equivalent times V battery which gives me an equation for equivalent capacitance for capacitors that are in what are we talking about parallel and that says the equivalent capacitance for parallel is it's just going to be the sum of the individual capacity we'll go from one to the total number of capacitors that are in parallel and we will add up each of the capacitances next we go to series and in series we have a battery and a capacitor and a capacitor and a capacitor and now we have to make some sense out of this thing plus and minus and these capacitors can in principle be different capacitances c1 c2 c3 what is the same for each what is the same for each of these capacitors well is it voltage no I'm not sure it is is it capacitance no there's no reason that we have to use exactly the same capacitor but it must be it must be the charge and I'm going to try to argue why the charge on this capacitor there's a a plus Q here and a minus Q there has to be the same as the charge on this capacitor plus Q and minus Q which again has to be the same charge on this capacitor plus Q and minus Q there and there consider if you will this section of metal now it's kind of weird because I'm thinking about half of one capacitor and half of the other capacitor and the wire that's connecting them but if you look at this chunk of metal in this circuit it's really an isolated chunk of metal and so what it is is it's a chunk of metal in between two plates and as a result of being in an electric field it's in those two plates it well this chunk of metal is polarized so there if the chunk of metal was neutral and in fact there are no wires connecting to it so it has to simply have a net charge of zero so if there's a positive charge here then there must be the same negative charge over there and the same situation is happening in what I'll do as the pink circle in this circle right here this also has just no net charge yet it's got polarization it's got negative charge there and positive charge there so if I've got some charge here that has to be the opposite the charge here and this charge here has to be opposite that charge there which has to be opposite of that charge there and the opposite of that charge and that charge in that charge in that search so of course all the charges are the same but the voltages are different and the capacitances are different so let us go then to a little bit of math I'm going to argue first that the voltage total I'm going to do a little bit of Kirchhoff's loop rule I'm going to say that the total voltage is the voltage of the battery and that's the voltage of all these capacitors and that's going to be well it'll be v1 plus v2 plus v3 the voltage across this capacitor plus the voltage across that capacitor plus the voltage across that capacitor that is the voltage from here to here and it's the voltage of the battery all right and then I want to point out that those voltages because Q is CV I know that V is Q over C so this voltage here voltage one is charge 1 divided by capacitance 1 plus charge 2 divided by capacitance 2 plus charge 3 divided by capacitance 3 but we said the charges are the same so they can factor out I've got Q times 1 over C 1 plus 1 over C 2 plus 1 over C 3 oh here's what I've got this is the charge on each capacitor and it says here V total equals the charge on the capacitor times the equivalent capacitor divided by the equivalent capacitance because Q is CV so I've got myself a funky-looking equation in fact you've seen one of identical form 1 over the equivalent capacitance is well what am I going to get I got to write this in a funny way I'm going to write it as a summation from the first capacitor to the last one that you've got in series we have to add those capacitances up like that so I'm going to be adding the inverse of the capacitance this means that any additional capacitors in series decrease the capacitance of the equivalent capacitor we draw an equivalent capacitor circuit so that you remember this is the circuit we can simplify it to and however in parallel any additional capacitors check this out any additional capacitors over here would increase the capacitance if we add more and more and more we add more in oh and that makes sense because I'm essentially creating one enormous capacitor who cares if these guys aren't connected it's kind of as if they are connected so I'm adding more and more area as they put more and more capacitors here in parallel and so of course that increases the capacitance because capacitance is epsilon naught times area over the separation of the plates in a simple situation and in this one as I add more and more and more of them in series of course I'm decreasing the capacitance because just by the same token capacitance is epsilon over times area over a distance and what essentially what I'm doing is I'm separating the plates by more and more and more distance as I sneaked more and more capacitors in here this plate is really the only plate that's meeting any business and that plate right there so I'm separating them by a greater and greater distance like that distance Plus that distance Plus that distance but the great advantage of this is that each capacitor gets to experience a lower voltage and you know that if the capacitor sees too much voltage that sucker will zap it will short out and chill and probably melt the goo that's inside that the dielectric and will cause a path where the current can always flow and now you don't have a capacitor you have a wire that sucks it might also be a good idea to add more and more capacitors this direction but as you increase this capacitance you don't increase the greatest potential difference that you can withstand in fact this this enormous capacitor effective capacitor they're making by puttin more and more and more of them in parallel will still be limited by the lowest absolute voltage of the most pathetic capacitor in here as soon as that capacitor shorts your whole game is shot unless you pull it out but really if any one of these fails then the whole game is over shoot so they've both got advantages and in a real world situation where you're trying to build a super capacitor that you can shock people or destroy cans or stuff we working on this in physics club you want to probably connect some of them in parallel and then take those parallel groups and connect them in series and then take some of those series parallel groups and connect them in parallel to get yourself an awesome beautiful combination of capacitors that well solving it's going to give everybody a headache but the beautiful thing is it's really simple in each little part you've got this parallel configuration for equivalent capacitance and you've got this series configuration for equivalent capacitances and notice that the series for capacitors looks like the parallel for resistors and the parallel for capacitors looks like the series for resistors because the relationship on voltage is inversely proportional depending on whether you're talking about capacitors or resistors goodbye

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Keywords: AP WGHS, solve, tutor, understand, Electronics, help, problem

Id: xgQt8dRhDnE

Length: 12min 10sec (730 seconds)

Published: Thu Jan 17 2013