 So, for lesson four, what we're going to be looking at is specific types of circuits. Series circuits and parallel circuits. A series circuit is a circuit where the resistors are connected in a row with no junctions between them. In my ski hill analogy, we say two resistors are in series if the same skiers have to go through both of them and have no choice. So which of these shows a circuit or a resistor in a simple series? We say these two resistors here are in parallel because skiers don't have to go through both. They can go through one or the other. This circuit here is in series because the same skiers that go through this resistor must go through this resistor, must go through this resistor. Although instead of skiers, what was skiers in the circuit really? Current. So what I'm really saying is the same current has to go through those resistors. They are in series. Example two says find an expression for the total resistance of a series circuit. And you know, we could derive it, but I'm going to keep it very, very simple. If your resistors are in series, they're the easiest type of circuit. If you wanted to find out the total resistance of this circuit, it would be resistance of one plus resistance of two plus resistance of three. If resistors are in series and you want to combine them mathematically as one to make your math easier, you add them up. And that part we probably don't need our calculator for. We will later on. You will have to memorize that for resistors in series. You can add them up. It's not on your formula sheet, although we'll be doing so many of them. I think you'll just naturally memorize it. So in the box on the next page, if you want to find RT, the total resistance, resistance total, and they are in series, it's add the first one plus add the second one plus add the third one if there is one, et cetera. So example three says, select the best answer for the voltage drop across the 20 ohm resistor. We would like to find the voltage here. We said to analyze a circuit, what we would really like to find, last day I said the first, do you remember what I said the first thing I always try and find was? Total current. Now this circuit has no current listed on it. No problem. I can tell you total resistance. What's the total resistance of this circuit? Well because it's in series, the total resistance is 25. And I do know the total voltage of this circuit because the total voltage is the height of the mountain and the height of the mountain must be the chair lift, the battery. The total voltage in this circuit has to be 100. Oh and let's remember it's ohms and volts. And then I have an equation, this is the equation that we used the most yesterday, ohms law. What was ohms law? V equals I times R, except I'm going to just tweak it a bit. Total voltage equals total current times total resistance, which makes sense to me. Oh, how could we find total current? Which was our goal, we said whenever I want to, I want to find total current. How would I rewrite this to get total current by itself? Total current equals total voltage divided by total resistance. In this case it's 100 divided by 4. Divided by 4 Mr. Dewey? How about 100 divided by 25 and the answer is 4 Mr. Dewey. What's the total current running through here? Four amps. How many amps are going through this resistor then? Four amps. How many amps are going through this resistor then? Four amps. Why is that so nice? This question wants the voltage in this resistor. Voltage is what times what? I times R. Do I know I the current? Yes. Do I know R the resistance? Yes. I can tell you exactly what the voltage drop is in this particular resistor. What is it? It's 80. What's the voltage drop here? 20. And you will notice 100 high, lose 20, lose 80, back to zero, the circuit works. The circuit works. This is where we're going to be using Ohm's Law, an awful lot. The, unlike yesterday, rarely Justin will they give you the current. But if you can find the total resistance, almost always they will give you the battery, the total voltage, and you can find the total current. And once you find the total current, as I said yesterday, then the question falls apart. So oh, more than 50 volts, B equals I times R, the current in the 20 Ohm resistor is equal to 4 amps, the resistance is 20, therefore the voltage is exactly 80 volts. So if we have a circuit in series and they ask us to analyze it, again I said our first goal is we'd like to find total current. So this question says find total resistance in current and then find the current. Do I know the total current? No. Are these in series? Yes. So what's the total resistor mathematically? What's the total resistance of this circuit? Sorry, 25. The total voltage is 100. The total current is going to be voltage divided by resistance, total voltage divided by total resistance. The total current here is also 4 amps. Oh, it's always 4 amps. But no, we're just making up numbers and these happen to give you the same current. So now that I know the current, what's the current right here at this section of wire? 4 amps. What's the current in this resistor? 4 amps. Try drawing it 4 properly, Mr. Dewey. What's the current in this resistor? 4 amps. 4 amps. What's the voltage drop when we go through this resistor? 60 volts, 20 volts, 20 volts and conveniently those do add up to 100 so we do lose all of our voltage going through the circuit which is a check to make sure, Sally, you haven't missed anything. Series circuits are pretty easy. I don't know. Parallel circuits are where we're going to need our calculator. A parallel circuit is a circuit where resistors are starting and ending. It says our start and end should be our starting and ending at the same height, same voltage. I.e., the resistors are directly connected at the top and the bottom with no resistors in between. Which circuit shows resistors in simple parallel, the top one or the bottom one? The top one. Now this bottom one has these two in parallel with this one so it's not a simple parallel circuit. We have to do some. What we would do is we would, because these two are in series, the same skiers go through here is here, we would just add those two together mathematically and make it mathematically one resistor and then mathematically we've turned it into a simple parallel circuit. Preview of coming attractions. Example six says find an expression for the total resistance of a parallel circuit and I would love to derive it because the math nerd within me thinks it's way cool but I'm just going to go straight to the punch line. If you want to find the total resistance, you can't, all you can find is one over the total resistance and then once you've done that, Kellen, once you have an answer to get from one over to just the total resistance, you have to take the reciprocal of your final answer but the total resistance is, sorry, one over the total resistance is one over the first resistor plus one over the second resistor plus one over the third resistor dot dot dot. It's a lovely proof but I'm going to pass on the proof and that's what we're going to write in the little box over here where it says parallel resistance. Can't find it directly. Instead we're going to go one over parallel resistance and usually instead of our total because I'm doing parallel, I'll often put our parallel right there, my little symbols for parallel lines. It's going to be one over the first resistance plus one over the second resistance plus one over the third resistance dot dot dot and the actual total resistance is take the reciprocal of your, take the reciprocal of this. Take reciprocal of answer and I'll show you what I mean. Says find the total resistance, the total current and then V and I for everything. Solve this circuit. Find everything. Are these three in parallel with each other? Yep. How do I know? Because we start at the same voltage and we end at the same voltage but we don't go through the same ski hill. That's our definition of parallel. So what I'd like to find is the total resistance, one over the total resistance is going to be one over ten plus one over fifteen plus one over twenty. And I do this all on my calculator. Forget trying to do common denominators and blah blah blah. I go like this. One divided by ten plus one divided by twenty, sorry fifteen, let's go in order to do it. Plus one divided by twenty and then I just hit enter and then all of your calculators Tyler have a reciprocal button and one of the reasons they all have that is because of parallel resistors. My reciprocal button is the little x to the negative one right there. I just hit that when I'm done and it takes the reciprocal of my answer. It's a little easier than going one divided by. So find your little x to the negative one button on your scientific calculator. It's there somewhere and it looks like the total resistance is four point six one five three eight four six one five ohms actually it repeats for a while. And you're going to find a lot of these don't work out evenly. Or anybody who cannot find their reciprocal button this is your chance to ask help you find it. Yep, so we're back. I found I helped people find their reciprocal button because apparently it wasn't very obvious on their calculator. And as I said to Dylan, I would ask you to show me this line of work. It gets you part marks. Oh, and then I go like this. I don't do an equal sign because this is one over the total resistance. I go like this and I say our total I draw an arrow equals. And I'm going to go to four point six one five. Most of these won't work out evenly because when you add a bunch of reciprocals together sometimes if we do a lot of work ahead of time we can give you a nice answer. Often you'll have yucky repeating decimals live with it. So I'm going to say our total is four point six one five ohms. Now if they said find our total I would go to two or three sig figs. Okay, they did. I would say four point six two. But I'm going to use four point six one five for the purposes of calculation because now I can find total current. What was total current? What was the equation? Oh, and you'll notice I'm rewriting Ohm's law in my head. So V equals I times R, what's I? What did we do on the previous pages? Yeah, this one also you'll probably end up memorizing because you use it so often. It's total voltage over total resistance. Total voltage was 60 divided by total resistance was four point six one five. I'll use this number on my calculator, 60 divided by that answer. And I get a total oh the voltage worked out sorry the current worked out evenly. The total current is 13 amps. Okay, so how many amps are right here 13? Do I know how they split up? Not yet. Oh, oh, how many volts are right here 60? So how high am I right here? If I go like this down this ski hill, can I get to the bottom of the chair lift? So how many volts do I have to lose going down that one ski hill? So this is 60 volts. If I know two, I know three remember? What's the voltage? Sorry. What's the current in this resistor? Oh, I guess current is voltage divided by resistance. Voltage divided by resistance sorry six amps. What's the voltage in this resistor? Well if I leave 60 volts high, can I ski down this hill and only hit that hill and get back down to the bottom? So how many volts must there be in this resistor? 60 volts better lose 60 volts. Oh, and while we're at it, how many volts are there in this resistor? In fact, if resistors are in parallel, what can you tell me about their voltage drops? They are the same. They have to be because we shook hands, we met up, we must have gone through the same drop in height. Oh, and what's the current in this particular resistor? Four amps. What's the current in this particular resistor? Three amps. Now you'll notice we never actually used this 13 yet, but I can double check six plus four plus three. Oh, I confirmed that 13, and what I've also done because we never used this yet is I've showed you that this rule with those currents works. That's the derivation of the one over r total equals one over the first resistor plus one over the second plus one over the third. Parallel. A little bit trickier, sometimes decimals, yuckier numbers, but not too bad still. It says, what can be said about the voltage drops across the resistors in the circuit below? Let's see if we can fill in as much as we can. Do I know total current? Yeah? You know what? That's worth writing down somewhere in my DFIC approach. Total current is five amps. Now these two resistors I find interesting because they are the same. What do you think if I have, well how many amps do I have right here? Total current, right? So I have five amps right there. How many are in each of these resistors if they're the same? 2.5 amps. Oh, and how many volts is in each of these resistors? How many volts do I lose? What's the voltage drop? Remembering that V equals I times R. How many volts? Sorry? 30? So at a minimum how many volts does this battery have to be? Not 60. At least 30. I'm not adding these together because from here to here at least 30 high. Let's see what else we can figure out here. What's the ratio between these two particular resistors? 4 to 1. And we said yesterday that bigger resistors have less current, smaller resistors have more current, but the ratio will be the same, 4 to 1. How would I break five amps up into a 4 to 1 ratio? Oh, I think actually 4 to 1. I think this is going to be 4 amps. This is going to be 1 amp. Oh, how many volts in this particular resistor? What's our voltage drop? 20 volts. And these two are in parallel. It's also going to be 20 volts. And I could have gone I times R and convinced myself of that as well. Right now that means that this battery has to have at least 50 volts. Okay. What's the ratio between these two resistors? Well, I started out saying, well, it's sort of 1.5 to 1. And that's yucky. I like what Pat did. Pat said that's the same as 3 to 2, which it is. Okay. If this is a 3 to 2 relationship and the smaller resistor has the bigger current, the ratio for the current is also going to be 3 to 2. How would I break 5 into a 3 to 2 ratio? Oh, 3 and 2 conveniently. I think this is going to be 3 amps in the smaller resistor, 2 amps in the bigger resistor. Oh, how many volts? 30? How many volts? 30 volts. How many volts must this battery have? Let's see. If I walk down any ski hill and get to the bottom, it doesn't matter which of the many paths I take, but that's one path without lifting my pencil, always going downhill. I go through 30, 20, 50, 30, 80. Oh, you know how many volts this battery must have been? 80 volts. Now that's one way to do this, and that's the cleanest way. If you have trouble with that ratio idea, and I know it's a bit tricky, what you could have done is gone combine these two parallels as one resistor, 1 over 10 plus 1 over 15 equals 1 over r parallel. Combine these two as one resistor, combine these two as one resistor, and then you would have three series resistors and added those up, and then gone r total, i total, v total and worked your way backwards. We'll look at both approaches. But this got us there. And oh, even though I gave you a whole page, we did almost all of our work right here. Example 8 illustrates an important point. It shows us that the voltage drop across parallel resistors is the same as the drop across the equivalent resistor in the equivalent circuit. Let's go back. You guys have all this room here, right? I'm going to put a little red 1 right there. You can just put a 1 and circle it. I'm going to put a 2 right here. I'm going to put a 3 right there. And I want to find mathematically what the combined resistance of this is. What is the combined resistance of this? Well, 1 over r1 is going to be 1 over 10 plus 1 over 15. Don't forget, take the reciprocal when you're done. What's that those top two resistors, if I replace them with a single one, what's it the same as? 1 over 10 plus 1 over 15 reciprocal. What do you get? 06 ohms. And if I find 1 over r2, which is going to be 1 over 20 plus 1 over 5, what's the second group of resistors the same as? If I wanted to replace this red circle with a single resistor, what's that the same as? 1 over 20 plus 1 over 5 reciprocal. 4. And if I went 1 over r3, which is going to be 1 over 12 plus 1 over 12, what's r3 the same as? Also 6. So this big ugly circuit is the equivalent, and we're going to draw our first circuit together of this battery. What are those top two resistors the same as if I just make them a single resistor? 6 ohms. And what are these two resistors the same as if I make them a single resistor? 4, not amps, Mr. Dewick, 4 ohms. And what are the bottom two resistors the same as if I make them a single resistor? 6 ohms. The black circuit that I've just drawn, mathematically Gordon, is equivalent to the one at the top of the page. Oh, what was the total current? He told me that. What was it? I scrolled down. 5 amps. So I have 5 amps. So if I have 5 amps here, 5 amps here, 5 amps here, what's the total voltage? What's the voltage in this top one? 30, which is what we had above earlier, was it not? What's the voltage here? 20, which is what we had above earlier. What's the voltage here? 30, and that forces us to have an 80-volt battery. So those are the two ways, Kellan, of solving these. You can do it with ratios. And I'll be honest, I find the ratio method tough. Or take one second to do an equivalent circuit. Quickly sketch it, and now it falls apart. Because in series, oh, by the way, what's the total resistance of this circuit? These are in series. Series is the easy one. How do I find series resistors? Add them up. The total resistance of all those parallel ones up there is 16 ohms. That's the total resistance. This circuit is actually equivalent to this. One battery and one 16 ohm resistor with an 80-volt battery and a 5 amp current. Mathematically, all three of these are the same. That's what this statement is saying. Voltage drop across parallel resistors is the same as the drop across the equivalent resistor and the equivalent circuit, the mathematical equivalent. In a household circuit, by the way, plugs and lights are wired in parallel to 120 volts. Each circuit has a circuit breaker in the circuit. The circuit breaker is an automatic switch that opens when the current exceeds a set value, usually 15 amps. When that switch is opened, no more current flows through that circuit. So let's suppose that, oh, sure, Tyler is doing his hair. He's just finished. He's gotten out of the bathtub. And he's blow-drying his hair, getting those golden locks. Or you know what? He's shampooed it four or five times because he's trying to get the last trace of the rooster red out of his hair. He's desperately trying to get the last trace of it out. So he's done four or five shampoos in a row. Lab their rinse, repeat, repeat, repeat, repeat. He's blow-drying his hair. But Tyler is not too electrically savvy. You know, his feet are cold. And so he figured he'd stand in the nice, warm tub water, keeping his feet warm while he's blow-drying his hair. And he drops the blow-dryer into the tub. Most bathrooms now have what's called a ground fault interrupter. What that means is as soon as it senses or as soon as the current starts to spike, 15 amps, let's say, boom, the circuit breaker would blow right here. That means current can't flow here, but it can still flow to the TV. So just because Tyler is a meathead in the tub, his sister can still watch TV quite nicely without losing her days of our lives or whatever she's watching. You can also blow a circuit by just overloading it. So here's what this question is asking. If we connect a 1,200-watt hair-dryer and a 500-watt TV to the same circuit, will we cause the 15 amp breaker to trip? OK. I'm going to redraw this circuit in our circuit diagram. So here we have, well, wait a minute. OK, I can't use a battery symbol because battery is direct current. The symbol for alternating current is a circle with a little sine wave in it like that. But there's our voltage source. And we have a resistor here. There's the blow dryer. And we have a resistor here. There's the television. Good gosh, Mr. DeWitt, let's draw this a little better. OK. My voltage source is 120 volts AC. Did they tell me the resistance of this? What did they tell me? Power equals 1,200. And power equals 500. Oh, hey, not only does v equals i times r, we had an equation with power in it as well called it Joule's Law. What was power equal to? Vi. OK, that's good. Can a skier go like this? How many volts must be going through this resistor then? Got to be 120. Now it's alternating. My skier analogy, because the whole downhill thing doesn't work as well with alternating current. But it's still got to be 120 volts. This has got to be 120 volts. Oh, and this has got to be 120 volts. Because they're in parallel. So oh, do I know the voltage? Yes. Do I know the power? Yes. What's the current in either of these? What's the current flowing through this particular section of the circuit? How many amps are going through the TV? 0.4? 2.4 amps? Enough to kill you. But not enough to be dangerous to the wires of the house. The wires of the house, most of them are designed to take about 30 amps. 15 amps is the safety cutoff. Because if wires heat up, fire, bad, right? How many amps going through here? How many amps going through there? Anyone? I think it's 10, yes? Oh, what's the total current in this circuit? See it? 12.4. Note to self. Don't go to that corner for information. What is it over here? 4.2 amps. OK. And 10 amps. So what's the total current? 14.2. Are we going to trip the breaker? No. Hey, what if this was a 1,500 watt blow dryer? What if then? Probably. So this is how your breakers get tripped. Oh, what if I want to use a 1,500 watt without tripping the breaker? Turn off the TV! And you'll notice that often you'll blow a breaker when you flip that one last appliance on. Where I live my kitchen, there's one plug. It must be on a different circuit. I haven't quite puzzled it out. I can't plug my microwave into this one plug. Because anytime I do, inevitably it blows the breaker, which is bad because it's the best place for my microwave on my counter. So I have a long little extension cord running elsewhere, which is not necessarily the best way to go. Homework, number one. Number two, 3, 4. For the circuit in question four, complete the equations. I, 3, the current in the, oh, eh, nah. I'm not into doing an abstract algebraic. I am into having you solve stuff though. 7, 8, 6, and 7. Sorry, good gosh. I'm reading the next question as I'm circling the previous one. Clearly you know where my mind is. 8 is good. I don't know how much more is there here. OK, oh, 9 is cool, but it's a lot of work. I think I'm going to go with 10, though. So when it says what's the maximum number of the following in the house circuit, what causes a house circuit to blow if the current gets bigger than 1? And what's the voltage source in all of your house circuits? And we're ignoring the special plugs. I know your dryer and your stove has a 220-volt plug. We're not using those. We're plugging into the regular plugs, which are all 120. Technically, 114 points. Ah, 120 is the nice number to do the math with, OK? 13 is a good one to think about. And 14. Wow, a whole half hour. I almost want to start the next lesson. But I won't right now. It's going to come back to haunt me.