1

concept

## Combining Capacitors in Series & Parallel

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Hey, guys. So we saw how When you connect a battery to a capacitor, it becomes a simple circuit. Well, a lot of times, things won't be so simple. You have to know how to deal with situations where you have multiple capacitors. All right, so let's go ahead and check it out in this video. So basically, the idea is the in circuit problems you could take in collapse and combined multiple capacitors into what's called a single equivalents capacitor. So this is the word that you're going to see a lot. What is the equivalent? Capacitance? And basically, the idea is that if you have a circuit that's like this where you have a bunch of capacitors in some random combinations, you can take all of these capacitors with their own individual capacity. So let's say this is hooked up to like a battery like this. Let's make that a little bit better, so if you have a battery that's hooked up to a combination of capacitors, let's say each one of these has their own individual capacitance. So you're gonna start labeling these c one c two c three c four. You can use some rules to basically simplify this circuit down so that it turns into and behaves the same way as one, uh, equivalents capacitor hooked up to one battery so that these two things are basically the same. And you have the ce que here, which is the equivalent capacitance. So all four of these capacitors in this arrangement were behave exactly the same way is if you had one capacitor hooked up to this one battery. Okay, so there's a couple of rules we're gonna follow a couple of situations. The first is when you have a serious connection of capacitors like we have over here with C one and C two. So a Siri's is when you have one capacitor and then just follows straight into the next one. So you have all of these capacities like this. And so the way I like to remember, it's like kind of like how you're watching Netflix. You're watching a syriza just binging one after the other after the other. So the idea is that these three capacitors, each with their own individual capacity C one c two, c three. All behave is if you had a single capacity like this, which has an equivalent capacity Ce que and I'll write. This s here because we're gonna be talking about serious connections. So these air direct connections from one to the next and the equivalent capacity to find that we're gonna use a rule called inverse sums, which is that the want The reciprocal of the equivalent capacitance is gonna be the sum of the individual reciprocal capacitors. So one over C one, C two and C three. And then, of course, if you had more than you would just keep adding them. Now what happens is what trips most students up is you. You'll do all of this stuff in your calculator to do one oversee one whenever C two and C three, and then you'll forget to take the inverse of that when you're finally done, so you'll get the wrong answer. So you have to remember that this is one over the equivalent capacitance. Ah, lot of the times, like nine times out of ten, if you screw this up or if a lot of times you'll get answers wrong, it's because you screwed up this step. So add all of the inverse capacitance is, and then in verse that so you have to do one over that to get your equivalent to capacitance. Okay, so just remember that step right there. Now, the other kind of connection we have is a parallel connection. So this is where you have a wire, and basically it splits off into a little junction, and you have capacitors that are not following directly one after the other. But you have these capacitors that sort of break off into their own individual loops right here. So if all of these parallel connections right here because all these capacitors at basically are in parallel with each other's you see all these parallel lines here, so they're not following directly after one another. They sort of break off into their own loops. So the idea is that this connection right here is the same. And we can collapse this down into equivalent capacitor c e que parallel. And so these things split off and form their own individual loops. And to figure out what this equivalent capacitance is, this is actually the easier rule. All you have to dio is if you have C one, c two and C three, you just add them up together. So C one C two C three and then so on and so forth, depending on how many capacities that you have. Okay, so this is the simpler rule. If they're in parallel, you just add them straight up. If they're in Siris, then you have to take the inverse sums and then in verse, that when you're done with that Okay, so these are the two rules now, sometimes, like I have in this situation right here, we have a circuit with combinations of these things. Some of them are in Siris, some of their in parallel, and so work with so to figure out, how you get to these equivalent capacitance is so we get down toe one single equivalent capacitor, you have to work basically from the inside out. So what I mean by that is we have all these capacitors right here. So if I could figure with the equivalent capacities of this is, then this would be in Siris with these other two. But if you look a little bit more closely first, I have to do the parallel connection first, and then I could do the serious connection is okay, so to get a little more practice with that. Why don't we go ahead and just take a look at some examples? So we've got this first example right here. What's the equivalent capacities of the following capacitors. So I've got a parallel connection right over here, right? Because I've got these wires will split off into their own little junctions. So I've got these are in parallel. But if I look a little bit more closely, what I've got is I've got each of the each of the two capacitors pairs that Aaron parallel. Both of these are in Siris, so I have to work from inside out. What's the smallest arrangement that I have? I have to capacitors in Siris on the top of the bottom. And then those pairs are in parallel with each other. So the way this is gonna work is I want to go from Siri's. I want to do all the Siri's once first and then I want to move over to the parallel. Okay. So for the Siri's, basically, I have these top and bottom capacitors that they're gonna form an equivalent top and bottom capacitance. So I've got the rules for something in Siris. I have one over. See top I'm gonna call. It is equal one over one plus one over three ferrets. I have to add The inverse is of those individual capacitors. Okay, so we've gotta remember a rules for adding fractions with, unlike denominators. So this one over one is the same thing as if I had three over three plus one over three. So this actually turns out to be four over three. But remember, I'm not done yet because I have to do the inverse of my of these of this fraction right here. So that means that see, top is equal to the inverse of four thirds, which is equal to three fourths now. The other sort of shortcut that I can give you guys is that if you're every adding to capacitors in Siris with each other, this Onley works in Siris and Onley works when you're working with two. Then one shortcut to this equation is you're gonna multiply the two capacitors and you're gonna divide by the sum of the two capacitors. We can see how this, um, this shortcut will give us the exact same three fourths. So you have one times three is three and then one plus three is four. So this would equal three fourths for the top one. So you could use either one of those sort of paths or those equations in order to get the equipment. Capacitance is okay. So now for the bottom one Now, we're just gonna go ahead and we can use the same process here, or we can use our shortcut equation. So we've got C one time See, too. So in other words, four times two is equal to eight. And now we've got divided by four plus two, which is six. In other words, we've got three or four over three. So now what happens is we have this four over three or started this three or four and this four over three. And those equivalent capacitors are in parallel with each other. So that means that the total equivalent capacitance that one equivalent capacity of the entire circuit is just gonna be see top plus C bottom, because these things are in parallel. So, in other words, ce que is equal to I've got three fourths plus four thirds. Now again, these things are not the same denominator. So I have to make them the same denominator. This is gonna be nine twelfths. And this is gonna be, uh, this is gonna be sixteen twelfth. So if you work this out, you should get twenty five over twelve, but and yeah, that's it. So, actually, this is our answer. So this is actually our equivalent capacitance in ferrets. Okay, So that would be how you deal with these kinds of situations. Let's do another example. We've got now the equivalent capacity of this circuit. But if you take a look, what happens is we have these equivalent capacitors are in Siris with each other. But if you look a little bit more closely, we have each one of those ones in Siris has a parallel connection. So instead of doing Siri's two parallel now, we actually have to do parallel and then go into the Siri's. So we have to handle those the opposite way. So it all depends on what the innermost arrangement of capacities that you see is now for parallel ones. What I'm what I have is on the left. So see, left. I have just you just go ahead and add those two things together. So that's just gonna be two ferrets plus two ferrets because again, I could just add them straight up because they're just in parallel with each other. So in the words see left, which also just called C l is equal to four ferrets and just similarly See right, I have one favorite plus four ferrets is equal to five ferrets. Right? So I've got that CR. Now what happens is these two capacitors. So this basically boils down to I've got a capacitor that is four ferrets and then plus a capacitor. That's five ferrets. And both of these things are in serious with each other. So now what I have to dio is that the equivalent capacitance is gonna be the inverse. So I have to take the inverse of this and this is gonna be one over four plus one over five. And again, these things are not the same denominator. So you have to make them one way to do that is just by multiplying the denominators. You remember that? So that's just gonna be Let's see, one of four is the same thing as five over twenty, and then one of her five is the same thing as four over twenty. So if you add this up together, you're just gonna get nine over twenty. But we're not done. How last step is to take the reciprocal of that? So that means that ce que is just equal to twenty over nine. And that's in ferrets. So that's how you work with these kinds of problems. All right, so let's go ahead and get some more practice. But they're taking a look at more examples. Let me know in the comments if you have any questions about this.

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example

## Find Equivalent Capacitance #1

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Hey, guys, let's get a little bit more practice with solving equivalent. Capacitance is in complicated circuit problems. Okay, so we've got all these four capacitors right here. All of them are labeled, so we know we have to work from the inside out. If we have combinations, we have some parallels. We have some Siri's. So, basically, what's happening is that if I could collapse all of this down to a single capacitor, then all of these two will basically this one and this one will be in serious with each other. But if I look a little bit more carefully, a little bit more closely than what I have a situation where I have these three capacitors and I have these are in parallel. And if I look even closer, I've got these two capacitors right here are in serious with each other. So we have to do is we have to work from the inside outwards. So that's step one we're gonna be solving with the equivalent capacities is of these two. In other words, when we do that, we're gonna get a circuit that basically looks like this. We're gonna have an equivalent capacities right here. we have the one on the bottom, which we know is three ferrets. And then these two things are gonna be in parallel with each other, and then they're gonna be in Siris with the five Fareed Capacitor. So we need to figure out what is this ce que right here? And that's gonna be the one in red. Okay, so we know that we're dealing with a Siri's, and we have to capacitors, which means we can use our shortcut equation for equivalent capacitance C e Q is just gonna be two times two divided by two plus two. Right, The C one C two divided by C two C one plus C two. Now, both of these happened before, which means that the equivalent capacity is equal to one ferret. Okay, now, what have is I have these two or these So the equivalent capacity right here and it's one of the bottom. So, in other words, this situation right here I have it in parallel. So that means you need to use my parallel equations for ce que and what happens is when I figure out what this equivalent capacity is, this is gonna behave the same way, Because if I had a single capacitor right here, that's gonna be in blue. And then that capacitor was in parallel with the five ferret capacitor. Okay, so these equivalent capacities right here is going to be Well, I have them in parallel. So that means all I have to do is just add these things together. So I have one. Ferid plus three ferrets is just four ferrets, and that's it. So I've got four ferrets right here. Okay. Now, for the last part, the last step, the equivalent capacitance for the entire circuit. Now I have in Siris. So this was parallel, and then this is gonna be in Siris. Now, I have, uh, two capacitors that I have here. So I could use really either one of the equations that I have So I could use the fact that the equivalent that one over e the equivalent capacitance is gonna be 1/4 plus 1/5. And let's see 1/4 plus 1/5. The common denominator is 20. This is gonna be 5/20. This is before over 20. So this is gonna be 9/20. But I have to take the reciprocal once I do this. So that means that my equivalent capacitance C e Q is gonna be 20/9. We could have done that or on alternate way, so I have, Or we could have just done that. The equivalent capacitance for two capacitors is going to be the multiplication of these 24 times five divided by four plus five. And we would have gotten the exact same thing, 20/9 ferrets. So either way, using either one of those approaches, we get the correct equivalent capacitance for the entire circuit. So that means that all of these forced capacitors behave as if you had just had a single capacitor of 20/9 ferrets. And that's the answer. All right, let me know if you guys have any questions. There's a very, very useful step by step process and how to basically work from inside out. Okay, let me know if you guys have any questions

3

Problem

What is the equivalent capacitance of the following capacitors?

A

0.923 F

B

1.08 F

C

9.0 F

Additional resources for Combining Capacitors in Series & Parallel

PRACTICE PROBLEMS AND ACTIVITIES (4)

- In Fig. E24.20 , C1 = 6.00 uF, C2 = 3.00 uF, and C3 = 5.00 uF. The capacitor network is connected to an applie...
- For the system of capacitors shown in Fig. E24.16 , find the equivalent capacitance (a) between b and c.
- Figure E24.14 shows a system of four capacitors, where the potential difference across ab is 50.0 V. (c) How ...
- Figure E24.14 shows a system of four capacitors, where the potential difference across ab is 50.0 V. (b) How ...