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Parallel resistors (derivation)

Discover the fascinating world of parallel resistors in electrical circuits! Learn how to identify parallel resistors, understand the concept of shared nodes and voltage, and apply Ohm's law to calculate equivalent resistance. Master the formula for parallel resistors and enhance your circuit analysis skills with real-life examples. Created by Willy McAllister.

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Video transcript

- [Voiceover] In this video, we're gonna look at another familiar pattern of resistors called parallel resistors. And I've shown here two resistors that are in parallel. This resistor is in parallel with this resistor. And the reason is it shares nodes. These two resistors share the same nodes. And that means they have the same voltage. And they are called parallel resistors. So if you share a node. Share the same node. Then you share the same voltage. And you are in parallel. That's what that word means. Now, if we go look closer here, we'll see some interesting things. It's hooked up, we have a battery here, some voltage v. And because there's a path, a complete path around here, we're gonna have a current. We're gonna have a current i flowing in this circuit. Let's label these resistors. Let's call this one R one and this one R two. Those are our parallel resistors. When the current reaches this point here, when a current reaches this node, it's gonna split. It's gonna split into two different currents. That current and that current. We'll call that one i one, because it goes through R one. And we'll call this one i two. And that goes through R two. Now, we know any current that goes into a resistor comes out the other side. Otherwise it would collect inside the resistor, and we know that doesn't happen. This one comes here. And they rejoin when they get to this node, and flow back to the battery. So the current down here is again i, the same one as up here. Now, what I want to do is I want to replace these two resistors with an equivalent resistor, one that does the same thing. And by "the same thing," we mean causes the same current to flow in the main branch. And so that's what's drawn over here. Here's a resistor here. We'll call this v again. And we'll call this R parallel. R P. And this resistor causes the same current i to flow here. And now we're gonna work out an expression for that. We want to figure out how do we calculate R P in terms of the two parallel resistors here. Okay, so let's go at it. What we know, let's see what we know about this over here. What we know is the voltages on the two resistors are the same. We know there's two different currents, assuming that these are two different-valued resistors. And now, with just that information, we can apply Ohm's law. And we use our favorite thing, Ohm's law. Which says that voltage on a resistor equals the current in the resistor times its resistance. So let's write down Ohm's law for R one and R two. Okay, we know the voltage, we'll just call the voltage v. This is for R one. v equals i one times R one. And for R two, we can write a similar equation, which is v, same v, equals i two R two. Now there's one more fact that we know, and that is that i one and i two add up to i. And these are the three facts that we know about this circuit. What I'm gonna do now is come up with an expression for i one and i two based on these expressions, plug them into this equation. Okay, I can rewrite this equation as i one equals v over R one. I can write this one as i two equals v over R two. And now I'm gonna plug these two guys into here. Let's do that. i equals i one, which is v over R one plus v over R two. Let me move the screen up a little bit. Okay, now we're gonna continue here. I just want to rewrite this a little bit. i equals v times one over R one plus one over R two. Okay, so here we have an expression. It actually sort of looks like Ohm's law. It has an i term, a v term, and this R term here. Let me go back up here. Here's our original Ohm's law. I'm gonna write this, I'm gonna solve for i in terms of v, just to make it look a little more obvious. I can say i equals v over R. And what I hope you see here is the similarity between this equation and this one down here. So I have this R here. And what's happening is this term is playing the role of that, resistance. So I'm gonna bring this equation down here, and write it right down here. Times one over R. I'm gonna call this R P. Because what I want is for this expression and this expression, I'm gonna set those equal. Same i, same v. These guys are equal. I can write it all, I'll just write it over here. One over R P equals one over R one plus one over R two. This says we have a resistor, we're gonna call it R P or R parallel, that acts like the parallel combination of R one and R two. So this is the expression for a parallel resistor. If you want to calculate a replacement for R one and R two in parallel, you do this computation and you get R P. So let's do one of these for real. Here's an example. Here's an example where I've actually filled in some numbers for us. So I have a 20-ohm resistor in parallel with a 60-ohm resistor, driven by a three-volt battery. And what I want to do is I want to combine these two parallel resistors and find out what is the current right here. Find out what is the current, that's my unknown thing here. I know everything else about this. So let's use our equation. We said that one over R P was equal to one over R one plus one over R two. And let's just fill in the numbers. One over R P equals one over 20 plus one over 60. That equals, let's just make 60 the common denominator. So I have to multiply this one by three. Three over 60 plus one over 60. And that equals four over 60. And so now I'm gonna take the reciprocal here. R P equals 60 over four or R P equals 15 ohms. So what this is telling us is if we have two resistors in parallel, 20 ohms and 60 ohms, that is, for the purposes of calculating the current here, that's the same as 15 ohms. It took the three volts. Just like that. Let's check what the current is. The current is i equals v over R equals three volts over 15 ohms. That's equal to 0.2 amps. Or you can say it's the same as 200 milliamps. So we actually have now simplified our circuit from two resistors to one resistor. And we were able to compute the current here, which is 0.2 amps. And I would invite you to check this by going back and computing this current up here to make sure it's the same. And the way you would do that is you would calculate the voltage. The voltage here is three volts. Three volts across 20, three volts across 60. You'll get i one and i two. And if you add those together, you'll get the total i, and it should come out the same as this. And I think that's a good exercise for you to do, to prove that the expression for a parallel resistor, one over R parallel can be computed from one over R one plus one over R two.