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Example: Analyzing a more complex resistor circuit

A hairy resistance problem. Created by Sal Khan.

Video transcript

Let's see if we can apply what we've learned to a particularly hairy problem that I have constructed. So let me see how I can construct this. So let's say in parallel, I have this resistor up here. And I try to make it so the numbers work out reasonably neat. That is 4 ohms. Then I have another resistor right here. That is 8 ohms. Then I have another resistor right here. That is 16 ohms. And then, I have another resistor here, that's ohms. Actually, I'm now making it up on the fly. I think the numbers might work out OK. 16 ohms. And let's say that now here in series, I have a resistor that is 1 ohm, and then in parallel to this whole thing-- now you can see how hairy it's getting-- I have a resistor that is 3 ohms. And let's say I have a resistor here. Let's just make it simple: 1 ohm. And just to make the numbers reasonably easy-- I am doing this on the fly now-- that's the positive terminal, negative terminal. Let's say that the voltage difference is 20 volts. So what I want us to do is, figure out what is the current flowing through the wire at that point? Obviously, that's going to be different than the current at that point, that point, that point, that point, all of these different points, but it's going to be the same as the current flowing at this point. So what is I? So the easiest way to do this is try to figure out the equivalent resistance. Because once we know the equivalent resistance of this big hairball, then we can just use Ohm's law and be done. So first of all, let's just start at, I could argue, the simplest part. Let's see if we could figure out the equivalent resistance of these four resistors in parallel. Well, we know that that resistance is going to be equal to 1/4 plus 1/8 plus 1/16 plus 1/16. So that resistance-- and now it's just adding fractions-- over 16. 1/4 is 4/16 plus 2/16 plus 1 plus 1, so 1/R is equal to 4 plus 2 is equal to 8/16-- the numbers are working out-- is equal to 1/2, so that equivalent resistance is 2. So that, quickly, we just said, well, all of these resistors combined is equal to 2 ohms. So let me erase that and simplify our drawing. Simplify it. So that whole thing could now be simplified as 2 ohms. I lost some wire here. I want to make sure that circuit can still flow. So that easily, I turned that big, hairy mess into something that is a lot less hairy. Well, what is the equivalent resistance of this resistor and this resistor? Well, they're in series, and series resistors, they just add up together, right? So the combined resistance of this 2-ohm resistor and this 1-ohm resistor is just a 3-ohm resistor. So let's erase and simplify. So then we get that combined resistor, right? We had the 2-ohm that we had simplified and then we had a 1-ohm. So we had a 2-ohm and a 1-ohm in series, so those simplify to 3 ohms. Well, now this is getting really simple. So what do these two resistors simplify to? Well, 1 over their combined resistance is equal to 1/3 plus 1/3. It equals what? 2/3. 1/R is equal to 2/3, so R is equal to 3/2, or we could say 1.5, right? So let's erase that and simplify our drawing. So this whole mess, the 3-ohm resistor in parallel with the other 3-ohm resistor is equal to one resistor with a 1.5 resistance. And actually, this is actually a good point to give you a little intuition, right? Because even though these are 3-ohm resistors, we have two of them, so you're kind of increasing the pipe that the electrons can go in by a factor of two, right? So it's actually decreasing the resistance. It's giving more avenues for the electrons to go through. Actually, they're going to be going in that direction. And that's why the combined resistance of both of these in parallel is actually half of either one of these resistances. I encourage you to think about that some more to give you some intuition of what's actually going on with the electrons, although I'll do a whole video on resistivity. OK so we said those two resistors combined-- I want to delete all of that. Those two resistors combined equal to a 1.5-ohm resistor. That's 1.5 ohms. And now all we're left with is two resistors in parallel, so the whole circuit becomes this, which is the very basic one. This is a resistor: 1.5 ohms, 1 ohm in series. Did I say parallel just now? No, they're in series. 1.5 plus 1, that's 2.5 ohms. The voltage is 20 volts across them. So what is the current? Ohm's law. V is equal to IR. Voltage is 20 is equal to current times our equivalent resistance times 2.5 ohms. Or another way to write 2.5 five is 5/2, right? So 20 is equal to I times 5/2. Or I is equal to 2/5 times 20, and what is that? 2/5 is equal to I is equal to 8. 8 amperes. That was not so bad, I don't think. Although when you saw it initially, it probably looked extremely intimidating. Anyway, if you understood that, you can actually solve fairly complicated circuit problems. I will see you in future videos.