 In this video, we're gonna explore one more application of using linear systems to investigate conservation of flow throughout a network. This time we're gonna look at electrical circuits. So we just have a simple electrical circuit flowing through the diagram you see here on the screen. And this is gonna follow the linear systems, the conservation law, I should say. It just really comes from what's known as Ohm's law, R I equals V. So what do these symbols mean if you're not familiar with the electrical laws here? V stands here for voltage. R here stands for resistance. The I is gonna represent the amps, the flow, the current flow of the system right here. So Ohm's law tells us that resistance times amps is equal to volts. That's what we get from that. The units for voltage, of course, is volts here. The flow is amps. I already mentioned that one. R here, the unit for the resistance is typically measured in Ohms. And you see that they actually use the capital Greek letter Omega to represent the resistance there. So you could have like a power source of some kind, like maybe there's a battery in the system, some power source. And the stuff is gonna flow out of the power source. You see that right here. So in my diagram V1, that's our first voltage. It has 12 volts of power coming out. 12 volts of electricity, I should say, coming out of the power source right there. Then the circuits gonna have some type of resistors. They often draw them as these zigzags in the diagram here. Now, this could be like, oh, the electricity flows through a light bulb or flows through a motor, right? And then it kind of gobbles up some of the electricity as it's used to power the light and things like that. So there's that resistance. So we'll just focus on this idea of conservation, R i equals V when we look at these diagrams. Now to set up a system of linear equations, we're really gonna use Kirchhoff's law, his Kirchhoff's voltage law, which tells us that if we take the algebraic sum of all of the R i's in the system, this is gonna be balanced with the sum of all the voltages in the system as well. And so what that means for us when we have this electrical circuit, you wanna look for loops in the system. So you look for something like this. There's one loop here, there's one loop here, there's one loop here, there's one loop here, there's another loop here, there's a loop right here. And so we wanna measure what's the flow of each and every one of these loops? Cause we know the voltage, we know the resistance, so what is the current's flow? Can we measure that in amps? Using Ohm's law, the flow is measured with i. So for each of these loops, I'm then gonna have some unknown flow. So we have like i1, i2, i3, i4, i5, and i6. Now, when you label all of these flows in the loops here, you should give a direction to them. Notice how it oriented everything in a clockwise manner. You could do that counterclockwise or you could mix and match if you want to. It doesn't really matter. You need to specify what that orientation is, so you understand your answer when you're done, but the direction doesn't really matter. Just label it and go with it. Because in the end, if you get a positive answer for i, that means you're gonna get a positive answer for i, that means you have the correct orientation. If you get a negative amp for the answer, that actually means you just label it backwards so you can correct it later on, not such a big deal. So using Kirchhoff's law, how can we set up a system of equations for this electrical system here? And the idea is each loop gives us an equation which conservation on each loop, right? So we look at the first loop for i, what's in play there? So we have the resistance times i equal to voltage. So the first equation, the first loop, that's the only place, whoops, that's the only place where there's any power source whatsoever, so we get the 12 volts over there. The left-hand side's gonna look like a bunch of RIs. So as we flow through, we hit this resistor, right? So what you're gonna do is you're gonna get 10 amps times i1. Then as we continue to flow, we hit the second resistor. So that's gonna be 20 ohms for i1, but it's also as part of the i2 loop. So it actually runs backward with respect to i here. And so what that does for us, meaning we're going to get 20 times i1 minus i2. Should've given myself some more space here. And then, so we also have to consider this one right here. So you're flowing through. This is going to go backward to this one, right? Because this is the flow of i3 going that way. So they're going in opposite directions. So you're then would label this as 30i1 minus i2, and then you'll get the 12 volts like we did before, okay? So then we look at the second loop. The second loop this time, there's no power source. So the right-hand side's just gonna be zero. But if we go through all the same resistors again, we're going to get 20i2 minus i1 minus i2. So you'll notice here that we're flowing with respect to the second loop this time. That runs counter to how i2 runs. So we get that negative i1. If they were flowing in the same direction, we would say plus, but it's negative in that case. So then continuing on, we have this resistor right here. Right, so that gives us a five i2 minus i3. And then next we get a 20. We're gonna get a 20i2 minus i4, and that's equal to zero. All right, looking at the third loop, again, going through all of those, you're gonna get 30 ohms times i3 minus i1. You're then going to get five i3 minus i2. And then lastly, you'll get 45i3 minus i4, and that's equal to zero. Myself a little bit more space there. Looking at the fourth loop, that's our biggest one right here. So looking at all of them, I'm just gonna go in there, just go in the numerical order there. So resistor five, you get 20 times i4 minus i2 in that situation. Then we're gonna get 45i4 minus i3. And again, check the directions here. It's gonna flow through it here, but with the other one, i3 flows backwards. So you get a negative sign right there. Next, we're gonna get then looking at r7, 20 times i4 minus i5. And then lastly, we get 35 ohms times i4 minus i6. This is equal to zero. So the next one's gonna get a little bit crowded because the thing I'm trying to look at is kind of running off the screen right here. So if you look at that one, resistor seven, you get 20 times i5 minus i4. Then resistor nine, you're gonna get 10 times i5 minus i6. And then you'll notice this very last one there right, r10. Only i5 interacts with that one. So you're just gonna get a plus, just a plus 20 ohms times i5. And that's equal to zero. And then the very last one, we'll squeeze it in there. You get 35 times i6 minus i4. You'll get 10 times i6 minus i5. And then lastly, you're gonna get 15 times i6. And that gives you zero again. So I'm gonna zoom out right here. And so then we have this system of linear equations that then models the conservation of Kirchhoff's law in these electrical circuits. So unlike some of the other examples we saw, we actually can see that we have six equations, six unknowns, right? There was an equation for each of the loops and the loops themselves, the flow of the loop is the variable in play here. So this is our square system, right? We can highly anticipate that maybe there's gonna be a solution here. And if you saw this system, you'll actually get that. And that'll then give you the correct flow of the electricity through this electrical system. So that brings us to the end of lecture 17 in our lecture series. Thanks for watching. If you have any questions, feel free to post those questions in the comments you see below. If you liked what you saw, give this video or any of these videos a like. And certainly if you wanna see more videos like this in the future, subscribe to the channel. I'd love to have you all. Bye everyone, see you next time.