Ohm's law and circuits with resistors
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Resistors in series
Let's make our circle a little bit more complicated now. So let's say I have a battery again, and let me do it in a different color just for variety. That's the positive terminal, that's the negative terminal. Let's say I have this perfect conductor, and let's say I have one resistor and I have another resistor. I don't know, just for fun, let's throw in a third resistor. And we know, of course, that the convention is that the current flows from positive to negative, that that's the flow of the current. And remember, current is just the charge that flows per unit of time or the speed of the charge flow. But we know, of course, that in reality what is happening, if there's any such thing as reality, is that we have a bunch of electrons here that, because of this voltage across the battery terminals, these electrons want to really badly get to the positive terminal. And the higher the voltage, the more they really want to get to this positive terminal. So what's going to happen in this circuit? Actually, let me label everything. So let's call this R1, let's call this R2, let's call this R3. The first thing I want you to realize is that between elements that the voltage is always constant. And why is that? Well, we assume that this is a perfect conductor-- let's say this little segment right here, right? And so it's a perfect conductor. Well, let's look at it at this end. So you have all these electrons. This is a perfect conductor, so there's nothing stopping these electrons from just distributing themselves over this wire. Before you encounter an element in the circuit or device or whatever you want to call that, you can view this ideal conducting wire just from a schematic point of view as an extension of the negative terminal. And similarly, you can view this wire right here, this part of the wire, as an extension of the positive terminal. And the reason why I want to say that is because it actually turns out that it doesn't matter if you measure the voltage here. So let's say if I take a measure of the voltage across those two terminals using what we call a voltmeter. And I'll later do a whole video on how voltmeters work, but remember, when we measure voltage, we have to measure it at two points. And why is that? Because voltage is a potential difference. It's not some kind of absolute number. It's a difference between essentially how bad do electrons want to get from here to here. So if we measure the voltage between those two points, it would be the exact same thing as if we measured the voltage between these two points. Theoretically. As we know, no wires really have no resistivity. All wires have a little bit, but when we draw these schematics, we assume that the wires are perfect conductors and all the resistance takes place in the resistor. So that's the first thing I want you to realize, and it makes things very-- so, for example, everywhere along this wire, this part of the wire, the voltage is constant. Everywhere along this wire, the voltage is constant. Let me erase some of this, because I don't want this to get too messy. That's a big important realization when you later become an electrical engineer and have much harder problems to solve. Let me erase all of this. Let me erase all of that. Let me redraw that, because we can't have that gap there, because if there was that gap, current wouldn't flow. That's actually-- well, I'll draw later how you can draw a switch, but a switch is essentially a gap. It looks like a gap in the circuit that you can open or close, right? Because if you open it, no current will flow. If you close it, current will flow. OK, so you now know that the voltage between devices is constant. The other thing I want to convince you is that the current through this entire circuit is constant, and that applies to any circuit in series. Now, what do I mean by series? Series just means that everything in the circuit is after one another, right? If we take the convention and we say current flows in this direction, it'll hit this resistor, then the next resistor, then the next resistor. At no point does the circuit branch off and have to choose whether I want to go down path A or path B. So this circuit is completely in series, and there's a couple ways I can convince you that the current-- let's call the current here I1. Let's call this current here I2. Let's call this current here I3. I could draw another one here, I3. So there's a couple of ways I can convince you that I1 equals I2, I3. One is I could just say if you experimentally tried it out using an ammeter, which measures current, you would see that they are identical. But the other way to think about it, and this time I'm going to actually talk about the electrons, so let's talk about things going in this direction, is-- so these electrons, through this wire, they can go as fast as they want to go, right? The speed of light or close to the speed of light since they have very, very, very low mass. And we'll go into relativity one day. But once they get to this resistor, they start bumping into things, and they slow down. This resistor is a bit of a bottleneck, right? So as fast as they're traveling here, they have to slow down here. And if they slow down here, they have to slow down here, because if they kept going superfast here and then they slowed down here, then they would start building up here, and that just doesn't make sense, because we know that they're evenly spread out, et cetera. And similarly, they might exit this resistor at a certain speed and then slow down even further as they bump into resistors here, but if they're going even slower at this point, then there would be a bottleneck here, so essentially, they would have to go at that rate throughout the whole thing. And another way to think about it is the resistance is kind of a probabilistic thing. I know when you think on a macro level, you say, oh, it has this resistance. It just slows it down. But the longer there's a resistor, it increases the probability that some of the electrons are going to bump into something and create a little bit of heat, et cetera, et cetera. So when you put resistors in series, what you're actually doing is increasing the probability that more electrons will bump into more things, right? Say there's an electron that travels-- say, somehow through freak luck, it doesn't bump into anything as it goes through here's because it's going really fast, but then it bumps into something here, right? It only increases the probability that something bumps into it. So there's a bunch of ways you can think about it, and I encourage you to let me know if there's other ways that help you. But the current through this entire series circuit is constant. Now if we say that, what else can we say? Well, if the current here-- let's say the current through here is I1. If the current through here is I1, what is going to be the voltage if I measured it from here to here? What is this voltage here? I measured it with a voltmeter. Well, V1 is going to be equal to I1 times R1. I don't know why I put an R. That's a 1, not an I. I1 times R1, right? And similarly, if I measured the voltage from here to here, that voltage is going to be equal to I2 times R2. Let's say this is where I3 is. So the voltage, if I were to measure it from here to here-- But anyway, if we look at the voltage from here to here, it's going to be I3 times R3. So what we see is that the voltage across the entire circuit, which I can write as V-total, is going to be equal to the potential drops, the total potential drop across each of these devices. So the way to think about it is that-- well, let's think about the electrons. The electrons here, they really want to get here. But after they've bumped around a little bit and they get here, they've experienced some potential drop. So the electrons here actually are a little bit less eager to get here. And then once they've gone through here, maybe they're just tired of bumping around so much. And once they're here, they're a little bit less eager to get here. So there's a voltage drop across each device, right? So the total voltage is equal to the voltage drop across each of the devices. And now let's go back to the convention, and we'll say that the current is going in that direction. The total voltage drop is equal to V1 plus V2 plus V3, so the total voltage drop is equal to I1 R1 plus I2 R2 plus I3 R3. And what's the total voltage drop? Well, that's equal to the total current through the whole system. I-total, or we just call it I, times the total resistance is equal to I1 R1, plus I2 R2 plus I3 R3. Well, we know that all the I's are the same. Hopefully, you can take it as, just conceptually it makes sense to you that the current through the entire circuit will be the same. So all these I's are the same, so we can just cancel them out. Divide both sides by that I. We assume it's non-zero, so I, I, I, I, and then we have that the total resistance of the circuit is equal to R1 plus R2 plus R3. So when you have resistors in series like this, the total resistance, their combined resistance, is just equal to their sum. And that was just a very long-winded way of explaining something very simple, and I'll do an example. Let's say that this voltage is-- I don't know. Let's say it's 20 volts. Let's say resistor 1 is 2 ohms. Let's say resistor 2 is 3 ohms, and let's say resistor 3 is 5 ohms. So what is the total resistance through this circuit? Well, the total resistance is 2 ohms plus 3 ohms plus 5 ohms, so it's equal to 10 ohms. So total resistance is equal to 10 ohms. So if I were to ask you what is the current going through this circuit? Well, the total resistance is 10 ohms. We know Ohm's law: voltage is equal to current times resistance. The voltage is just equal to 20. 20 is equal to the current times 10 ohms, right? We just added the resistances. Divide both sides by 10. You get the current is equal to 2 amps or 2 coulombs per second. So what seemed like a very long-winded explanation actually results in something that's very, very, very easy to apply. When resistors are in series, we just add them up. I will see you in the next video.