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Analyzing a resistor circuit with two batteries

Video transcript

- [Voiceover] Hi, my name's Willy. I'm the Khan Academy Electrical Engineering Content Fellow, and we got a Tweet in from Jane, and Jane asks, "Could you take a video, "could you make a video on solving complex circuit "problems in physics? "For example, something similar to this question." So, Jane sent us a circuit here, here it is over here, and it's an interesting looking circuit. We're going to take a shot at solving it here, and I'll tell you how I would approach this. Let's take a quick look at it. We have two batteries in it, which is a bit unusual. There's two resistors here. They happen to have the same value, 1.4 ohms. There's a third resistor connected up, and the question we're being asked is to find, what's the current in this resistor right here? So that's what we're looking for. So the way I approach these kind of questions, first, I sort of just live with this schematic a little bit. I look at it and see if I can imagine how the currents are flowing and what's going on, and then a really good way to do that is to, I'm going to draw this schematic myself, I'm just going to draw it over and let me own hand kind of learn this circuit as we go, so I'll draw the battery. Here's the two batteries. And I see that they're connected together. Okay, that's interesting. Each battery then goes to a resistor, this goes to a resistor, and those are connected, let me check, yeah, those are connected together, and then they go to a third resistor like this. And now if you notice what I'm doing is I'm taking all the funny corners and dots and curves and things like that and I'm just going to draw a square schematic that I can understand. Now, I'm going to label it just to make sure I got everything right, so, this is 11 volts. And I know from the battery symbol, that the short line is the minus side, and the long line is the plus side, and same here. So, these two batteries are hooked up exactly the same way in the same direction. This one's also 11 volts. And that's going to prove out to be interesting. We'll call this one R1 because the problem had i1 here. We'll call this one R2. And we'll call this one R3, of course. And the current that we're looking for is this current right here, what's that? Okay. So, before I start doing mathematics on this, what I'm going to do is see if I can simplify the diagram, and what I'm going to notice here, this is kind of a trick that was done here. Let's look at these batteries. These two ends of the batteries are connected together, so they are at the same voltage. We don't know what that voltage is, but we know it's the same, and we know that they're both 11 volts and that means that this node here, this node here, and this node here are at exactly the same voltage with respect to this node here. This one is 11 volts lower in voltage and this node is 11 volts lower in voltage, so what I'm going to imagine in my head, if I connected a wire between these two nodes, this end would be at 11 volts, this end would be at 11 volts. So no current would flow through that wire. It would have no effect on the circuit. It wouldn't, there would be no current like diverted from here over to here. I'm actually going to draw that wire in. I'm going to draw that wire in, and I didn't change the circuit when I did that. The same current goes through this wire as before because zero goes this way, the same current goes through this wire because zero current goes this way because I know they're both at 11, at the same potential above here. Right, so this is a good time to redraw this again. If you look here, I have two batteries that are hooked up, their inputs, their positive side is hooked up together and their negative side is hooked up together, so they're actually just acting like one, big battery, so let me draw that. I'm going to draw the circuit again so it looks like this. Here's my combined big battery. And it goes to... Relabel these again so we don't get mixed up. Okay. This is R1. This is R2. And this is R3. So this circuit looks a little simpler, and I'm gonna look at it again, see if I can do any more simplification, so what I recognize right here, right in this area right here, R1 and R2 are in parallel. They have the same voltage on their terminals. That means they're in parallel. I know how to simplify parallel resistors. We'll just use the parallel resistor equation that I have in my head, and that looks like this, let's go to this color here. Okay, so parallel resistors, R1 in parallel with R2. I made up this symbol, two vertical lines, that means they're in parallel, and the formula for two parallel resistors is R1 times R2, over R1 plus R2. Now I'll plug in the values. We know that R1 and R2 are actually, if we look over here at our schematic, they're the same value, and that has a special thing when in parallel resistors so it's actually R R over 2R. Because those resistors are the same. And you can see I can cancel that and I can cancel that and two parallel resistors, if the resistors are equal, is equal to half the resistance. And let's plug in the real values, 1.4 ohms over 2 equals 0.7 ohms. That's the equivalent resistance of these two resistors in parallel. So this is a good time to redraw this circuit again. Let's do it again. Here's our battery. This time I'm going to draw the equivalent resistance. Then we have R3, we'll put in R3 down here. Nothing's changed there, and that goes back to our voltage source. Again, do some labels, so this is R1 in parallel with R2. This is R3. This is the current we want to know, right there, and this is 11 volts, that's the plus side and that's the minus side. Okay, we got a pretty simple circuit here now. Let's fill, well, let me fill in the values for our resistor here. R1 in parallel with R2 is 0.7 ohms. R3, let's go look that up, remind ourselves, 2.7 ohms. Now I can almost do this in my head. These are both now just two series resistors. So, Ohm's law equals V equals iR. This is the form of it I always memorized. If I want to know i, I'm solving for a current, so i equals V over R. We can start filling in values, i equals 11 volts over, what's the resistance? It's the series resistance here, so it's the sum of these two resistors, which is 0.7 plus 2.7. If I work that out, it equals 11 over 3.4 equals, okay let me actually work that out. It's 11 volts divided by 3.4 ohms and that equals 3.23 blah-blah-blah something something something. And that will be in Amps. So, this right here is our answer. Right there. So find the current i, i equals 3.2 Amperes. And now I'm going to check, actually can check to see if that's the answer. Let's see what Jane sent. She actually sent in the answer. So thank you, Jane, for doing that, and, there it is. So that's it. Thank you very much for listening.