26. Capacitors & Dielectrics
Solving Capacitor Circuits
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All right, guys. So for this video, we're going to be working on how you solve problems with capacitor circuits. Let's go ahead and check it out. Because we saw that in circuit problems, we could figure out what the equivalent capacitance is of the entire circuit. But now you're gonna be asked things like what are the charges and voltages across some combinations of capacitors is. You're gonna need to know how to figure that out. So we saw from the last video that if you have series connections, you can figure out what the equivalent capacitance is by using these rules. We have the for series connections, we have the inverse sum rule, where you basically take the inverse of the equivalent capacitance, and it's equal to the inverse subs sums of all of the additional of all the individual capacitors. Right? So if you have all of these capacitors that are in a line like this, remember that these capacitors mean that there are some charges that are building up on the plates between them. There's negative and positive charges, and they're separated by some distance. What ends up happening is that one of the capacitors builds up a charge and it transfers that charge to the next one in line, and that one transfers the one next one in line as well. So basically what happens is that capacitors in series will share charge with each other. So that means if you have a capacitor that has 3 coulombs of charge built up between the plates, then that means that there's 3 coulombs on the next plate and also on the next capacitor. So all of those charges basically pick up 3 coulombs each. And it also means that when you condense this down to one equivalent capacitance, CEQ, then that will also have a 3 coulomb charge. So in other words, these share charge with the equivalent capacitance. So that's how series connections work. They always share the same charge, and that's always going to be true. Now we also saw that pair parallel connections, we could figure out what the equivalent capacitance is, and it's basically just the sums of all of those things. So if you have parallel connections, the equivalent is just is just summing each of the individual capacitors. Now what happens is if you have this wire and you have it approach this junction and the wire splits off into these three parts, what happens is you can you can imagine these charges sort of branching off in different directions. So what ends up happening is that the charge is not conserved and the charge is not flowing through each one of these things equally. It has to split up. But what ends up happening is that through these points right here, from this point to this point and this point to this point, the potential difference is, in other words, those v's are equal to each other. So unlike for series connections, that capacitors in parallel don't share charge with each other, they share voltage with each other. So whatever the voltage is across one of the capacitors, so I'm just gonna make up a number. If this is 2 volts, then that means that that voltage is the same across all 3 capacitors. They don't share the same charge. So what that also means is that when you condense this down to an equivalent capacitor, CEQ, then that means that voltage is also has to be, 2 volts. So you can actually work forwards and backwards from this kind of thinking. You can collapse all of these down to an equivalent capacitor and know what the voltage is, or you can take an equivalent capacitance and expand it back out into its, individual capacitors, and you know what the voltage across each of those is. So you can sort of work forwards and backwards. Now with these rules, we basically have everything we need to solve capacitor circuits problems. And we have a nice step by step process in order to solve those things. Let's go ahead and check it out. So we've been working with the first step a lot. We've found what the a single equivalent capacitor is for for a circuit. And, now what we have to do is questions will ask you things like, what is the voltage across a this 3 farad capacitor? What is the charge against the across this combination of capacitors? So what we need to do is once we find these single equivalent capacitance out of the entire system, now what we need to do is use the relationships between q, c, and v to figure out what the voltage and charge are for the entire circuit. And now what you can do is you can work backwards and sort of expand out again that circuit to until you figure out the voltages and charges for each of the capacitors that you're interested in. That's gonna be the step by step process. We're gonna reduce everything down to a single capacitor, figure out what the voltage and charges across this whole circuit, and then we're gonna move outwards again, sort of expanding out all of these capacitors until we find the variables that we're interested. Alright. To see how all of this stuff works, we're gonna go ahead and do this example together. So what's the charge and voltage of each of these capacitors in the following circuit? So we've got 3 capacitors. We've got a mixture of parallel and series. So we're gonna go ahead and and first things first, we're going to to find out what the equivalent capacitance is of the entire circuit. Now, we've done this before. We have these, capacitors right here that are in series, but if you look a little bit more closely, we have this 1 and 2 farad capacitor that are in parallel. So we need to work from the inside out. So what happens is this system right here, this circuit can be thought of as this more simplified circuit where you have the equivalent capacitance over here, and then you have the 6 farad capacitor hooked up to the battery right up here. So we have this 10 volts. Now, in order to figure out what the equivalent capacitance is, CEQ, all we have to do is just add them because they are in parallel. So in other words, the these are in parallel, so the equivalent capacitance is just gonna be 2 plus 1, which is 3 farads. And we also have the original 6 farad capacitor right here. So now what happens is we need to figure out the equivalent capacitance. So that's gonna be when we do these capacitors in series, when we add these things in series together. So this is the same exact things if I had one equivalent capacitor like this. And this equivalent capacitance, CEQ, is gonna be well, I have 2 formulas in order to solve it. I have the inverse sum formula or I can use the shortcut equation, in which I multiply each of those two things together on the numerator, and then divide by their sum. So that's gonna be 6 times 3 divided by 6 plus 3. That's gonna be 18 over 9, which is equal to 2 farads. So that's the equivalent capacitance. That's basically what all of these combinations of capacitors reduce down to as a single capacitor. And we also know that the voltage across this battery is equal to 10 volts. So that's step 1. It's done. We figure out what the equivalent capacitance is. Now in step 2, step 2 says we need to figure out what the equivalent charge and the equivalent voltage. We already know what the voltage is, so that's okay. We know what the voltage for the whole entire circuit is. So in order to figure out what the charge is, the in other words, the q equivalent, we just have to use our relationships between q, c, and v. We know q is equal to c times v, so that means that q equivalent is gonna be c equivalent times v. So what's the c equivalent? It's 2 farads, and then the equivalent voltage or sorry. The voltage is just 10. So that means that the entire circuit has 20 coulombs of charge that's flowing through it. But that's not our answer, because we need to figure out what the charge and voltages across each capacitor in the following circuit. So what we need to do is we need to work backwards from these charges, these voltages, and these equivalent capacitances to figure out what the charge and voltages is on all three of these capacitors. Okay? So let's go ahead and draw out a little table because we're gonna have a lot of different variables floating around. It's, it's definitely gonna be worthwhile to get organized, just so you don't lose track of numbers, things like that. Okay? So, let's see. I've got this 2 farads. This is gonna be 1 farad, the 6 farad, and then the equivalent capacitance right here. So I'm gonna be looking for the voltage and the charge, and I know what the equivalent capacitance is. Right? So I've got my little table. Now we said for the equivalent capacitance of the entire system, we know what the voltage is. The total voltage is 10 volts, and the total charge is 20 coulombs. So that's done. So what we can do is we need to work backwards to figure out what the charge and voltage is across the 6 farad capacitor. Okay? So we said that this equivalent capacitor right here of 2 farads is the same as these 2 capacitors right here in series. What's common about capacitors in series? Remember that they share the same charge. So if we can figure out what the charge is on this capacitance, which by the way is just the charge of the entire circuit, the 20 coulombs, then that means that each one of these capacitors here has a charge of 20 coulombs. So that already answers our first question. So this charge right here is 20 coulombs across the the capacitors right here. Because again, if this one capacitor here is the equivalent capacity of the entire sir of the entire circuit, and it has a charge q e q of equal equal to 20 coulombs, then that means that because these capacitors are in series, they have to share the same the same charge as the equivalent capacitor. Okay? So these are 20 coulombs each. Now, how do we figure out the voltage? Well, the voltage across the 6 coulombs, or sorry, the 6 ferric capacitor is gonna be using q equals c v. It's just gonna be q divided by c. So in other words, the voltage is gonna be 20 over, and that's 6. So in other words, that's 3.33 volts. So we're gonna fill that into our table, 3.33 volts. Okay. Now, what we have to do is we have to get the voltage and charges for each one of 2 and 1 farad capacitors. Now in order to do that, I need to look at what the equivalent capacitance is of this 3 farad capacitor, because then I can sort of expand it back out into the 2 and the 1 again. So, let's see. If I if I know that these are going to be in parallel, what's going to be the same about them? The capacitors in parallel share the same voltage, not the same charge. So what I need to do is I need to figure out what the voltage is across this equivalent capacitor. So in other words, v 3 is gonna be q over c, so 20 over the equivalent capacitance, which is 3. So that's gonna be 6.67, and that's gonna be in volts. Now, remember, if this is 6.67 volts, and this capacitor is the same thing as these two things that are in in parallel, then that means that the voltage across each of them has to be the same. So both of these have to be 6.67. So we have 6.67 volts, and we have 6.67 volts. Now, the charges are not the same because they're not in series with each other. So in order to figure out what the charges across each one of these things are, have What I wrote down here is that v 3 is equal to 6.67, so v 2, in other words, the voltage across this one was also 6 point 67. The voltage across this one was also 6.67. Now, how do we get the charge? Well, remember, if we wanted the charge across the 2 farad capacitor, we just have to do c times v. Now, the capacitance is just the So let's see. We have the 2 farad capacitor. Now, the voltage across this one was 6.67. So that means that the q 2 is just gonna equal I've got 13.3. So I've got the charge is equal to 13.3, and that's coulombs. And now for the other one, I've got q of the 1 farad capacitor is equal to the 1 farad times the voltage, 6.67. So that's just gonna be 6.67, and that's coulombs. Alright? So those are all of our numbers right here. We have the charges and voltage for all of these things. So you basically just have to work backwards and figure out all the target variables. You've got this, you've got these charges, you've got these voltages. Anyway, let me know if you guys have any questions with this, and, let's go ahead and do some more practice.
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