 Now that we've seen how to describe, at a macroscopic level, the movement of charges through closed systems, circuits, let's begin to explore what happens as we add a bit more complexity to those circuit designs. Such complexities exist naturally in the world around us, whether they're engineered or emerge in biological or chemical systems. It's crucial to begin to build a toolkit for describing, but then also using those descriptions to calculate the behaviors of such circuit systems. So in the microscopic world, what's going on is sort of pictured in this cartoon right here is you have some material made of atoms, and for a conductor, some of the electrons in each atom, one or more of them, have to be loosely bound to their parent atom. So that under the influence of even a modest electric field, they can be removed and motivated to accelerate in a particular direction. But what makes materials interesting is collisions. The electrons might accelerate a short distance, much like the ball bearing in this model here, but they'll hit something as they are accelerated by gravity. They'll get up to some instantaneous speed that's actually quite high, but then they'll bounce off of an atom and they'll have to be re-accelerated by the gravitational field in this case. And the situation is similar for a material with an electric field. So if one puts an electric field across the material, and let's just assume that they're a uniform for now to keep this nice and simple, we'll just deal with uniform fields for these cases. One can induce, for instance, an electron to drift, remember if the electric field points that way, electrons will drift that way. So correspondingly, it can be said that the positive charge is moving this way with the opposite direction, but equal magnitude feed drift. Everything is done in terms of positive charge when you're defining your directions in a circuit. Where does the positive charge move? So microscopically, if we wanted to imagine for a moment that the electrons were spinning still, but the atoms are moving, that's kind of what Ben Franklin has stuck us with at this point. So it's really the electrons that are moving. But they're moving opposite the direction of the positive current flows. Positive current flows that way if electrons are flowing that way. I know this gets a little backward, but again, these are conventions that we are hindered with by the history of our field. So positive charge is moving this way and this basically induces a, if I can make that J better, a current density. Now current density is a vector and it points in the same direction that the positive V drift points. So let's call this negative V drift to distinguish it from the velocity with which the positive charge is moving to the right. And then this current density will point in the same direction as V drift. So the direction that positive charge is moving, that is the direction of positive current density. That's the only thing you need to define the sign of that vector as positive or negative. And that should point in the same direction as the net electric field on the system. So the positive charge is going to go in the direction that the net electric field is pointing. So that's what I've drawn here in this cartoon. And so we kind of get a feeling that there must be some relationship between electric field and the current density that's set up, and that that current density will depend somehow on the properties of the material. If the material offers a lot of resistance to the motion of charge, then you can put an electric field on it, but you won't get a very big J. And if you put more electric field on that same material, you can increase J. Correspondingly, you could instead change your material. You could put the same electric field on a material that offers less resistance to electric current, and that would allow you to set up a bigger current density. So there must be some relationship, which we can write as the electric field equals some constant of proportionality that takes care of all the units, because this is in amps per unit area. This is in Newtons per Coulomb. So whatever this thing is, and we'd have to figure it out for each material, whatever this thing is, it will have units that correctly map one onto the other. Yeah, so this is the Greek letter Rho. It is known as the resistivity of the material. So at the microscopic level, there's some relationship between the current density that one can set up and the electric field that you place on the material to establish that current density in the first place. So if we write that down over here, so here at the microscopic level, P equals Rho J work. And then at the macroscopic scale, we have the expression of Ohm's law, which is that a potential difference will establish a current, and that current will depend on the amount of resistance of the material. And we can attempt to relate the two of them. We want to find the connection between this Rho, this constant of proportionality, and R, the resistance of the material. So that constant of proportionality summarizes atomic physics, what's going on with collisions down inside the material. And this summarizes the net effect of all of those collisions on the relationship between the voltage one applies to a wire, for instance, and the current that one is able to drive through the wire. Well, OK, what's the definition of the voltage? Voltage is equal to work done by an applied force per unit charge. And this is the work, if we do 1 over Q here. The work is the integral of the force times the displacement, which I can write as 1 over Q e, let's see, Q e dot dx. So my Q's canceled, and I just have the integral of E dot dx. Well, I've given a nice simple case. I've got a uniform electric field, and it's a length of material L. And so in the end, this integral is just going to give me E times L. That's it. So voltage is E times L. Nice, simple equation. And this is maybe a demonstration of how you can go back to the basic definition of voltage's work from the applied force divided by charge, work through all your stuff. OK, well, let's remember the basic definition of current density. This is the current divided by the area of the material through which it's passing. So if we were to slice this, if this was some kind of cylinder, we'd have some area A in meter squared that this current is passing through. That's the definition of current density current per unit area. And now we can take that equation up there, E equals rho j, and we can rewrite it. So we want to solve for rho. We want to figure out what this thing is. So rho is just going to be E divided by j. Well, I have things I can substitute in. E is equal to V over L. And j is i over A. So j is in the denominator. So that flips this relationship over. OK? And finally, I can try to use Ohm's law to figure out the relationship between R and rho. So let's see if we can get this into something that looks like Ohm's law. Let's leave V on this side, but let's move current area and length to the other side. So we're going to get L over A rho i equals V. And in order to recover Ohm's law, V equals iR, I now only have to make an identity between that thing multiplying i and resistance. So R is equal to rho L over A. And let's think about that formula for a moment, OK? That's the identity that we can solve for from this little exercise. Whatever this rho is, what we see is as it goes up, resistance goes up. And so that's why this thing is referred to as resistivity. So as resistivity increases, so increases the resistance of the material. If I put more material in the path of the charge, if I lengthen the amount of material that it has to go through, I offer the opportunity for more collision. And correspondingly, R goes up. So if I take a material that's a very good conductor and has very good conduct, it has very low resistivity, I can make it a really terrible conductor by just adding more of it. So the longer your house wiring, for instance, the more resistance there is in the wall to the flow of current in your house, OK? So there's an arts to engineering a nice short path for current in your house so that it's not moving wildly all over the place. That wastes energy basically because the more copper you put in your wall, even though copper is an excellent conductor, it has some resistivity and that will add up as you add more and more and more lengths of copper. So you want a shorter path between where, for instance, the electric potential comes into your house and your blender or your lights as possible. If you put too much copper in, you basically just create a big resistor in your wall and now you're wasting energy in the form of heat from collisions in that material. And then finally, if one decreases the area of the material, basically you're squeezing the charge now to travel for a much smaller area and that will increase the number of collisions as well. And so that increases resistance as overall. So if you want to figure out how to go from the atomic properties of a material, the length the charge has to travel, the area through which it has to travel, and the inherent structure of the material which is summarized by its resistivity, you have this equation. Right, and so this is why if you're driving a lot of current, your wire thickness goes up. Because if you have a small thickness and a high current, you'll generate a huge resistance and that can heat up the wire to the point of melting. So, you know, this is why you need to be very careful with driving more than say 15 amps or 20 amps through house wiring that's rated for 15 or 20. So for instance, the shipping volt that my spouse and I drive, it can be charged at a higher current, but the company has explicitly limited the rate at which that car will charge to no more than 12 amps. And that's for a couple of reasons. One, that's safely below the limit of the house wiring in any standard house that gets built today. Most of the house wiring is rated for 15 to 20 amps. There are special circuits in the house, usually for even higher current appliances like washers and drivers. Okay, that can draw a lot more current. The other reason is that the company, Chevy, they don't know whether you're plugging your volt into a new house or a 40-year-old house with 40-year-old wiring, which may be decaying and decrepit and actually have a higher resistance as a result of its lack of input condition. And so, they don't want to put it at 14 amps or 15 amps the limit of the house wiring now. They put it safely below that is for a margin of error. Because Chevy doesn't want to get sued for setting house at this point, right? And that's also why you have breakers in your house. There are little switches. When they overheat, they trip and break the circuit. And that's on purpose. That's so you can't accidentally put 25 amps of draw on a 15 amp circuit and set your house on fire. So the breakers are there to save you from yourself, essentially, or from a malfunctioning appliance. For instance, we had a heating element in an oven and it exploded one day. And why did it explode? It exploded because the resistance of the material in the heating element suddenly changed for a catastrophic reason. We'll never know why. Could be that there was just a cheap, you know, some problem with the material in one part. It broke down, the resistance changed, and suddenly the power dissipating through an increase. But it exploded with a bright flash and a pop and a trip to breaker. Because for just a brief moment, as the resistance of the material failed, the current going through that part of the house suddenly exceeded 15 or 20 amps, whatever the stove was limited to. And the breaker trip. And it probably saved our house from burning down, okay? We still had to deal with smoke and sparks and so forth inside the oven. But at least nothing else bad happened behind the walls where we couldn't get through without an axe. All right, that's what the fire brigade would have done if they had rolled into town to save our house from a house fire. And electrical house fires, particularly nasty, because it usually starts in the wall. And so you have to tear the walls down to get out of it. And that means massive amounts of damage to your home. Okay, so things to keep in mind. If you're goofing around with too much stuff plugged into one wall socket, think about it carefully. Many of the wall sockets, like this one may be on a different breaker than the one down here that I'm plugging my laptop into. And that's why you should distribute your appliances across multiple plugs so they don't draw too much current. Because your wiring can only handle so much current before you start wasting energy in the form of lots of heat. And you can ignite the wall on fire or melt material or worse. Okay, so that's the connection between the macroscopic and microscopic worlds. And we're going to start exercising now Ohm's Law. We're going to exercise Ohm's Law. We're going to exercise in conjunction with conservation of energy and conservation of charge and start thinking about circuits. So you can look up resistivity of materials on the web. Wikipedia has a great table. Here's copper. For instance, resistivity is measured in Ohms times meters. And you have to do it at a particular temperature. We'll come back to that later. For instance, copper has a resistivity of 1.68 times 10 to the minus 8 Ohm meters. And that is exceedingly small resistivity. Another convenient way of quoting resistivity or rewriting it is by taking one over the resistivity. And this is known as the conductivity. So if you have a big resistivity, you have a low conductivity and vice versa. If you have a big conductivity, you have a low resistivity. Okay, so this is something that depending on what table of numbers you're looking at, if they give you conductivity, you can get resistivity from it just by inverting conductivity. It's a very simple bit of math. Okay. All right, so copper is pretty good. Graphene, which got a Nobel Prize a few years ago now. This is a very special material that's showing up more and more now in electronics. Yeah, so on this list here, the lowest resistivity material is actually graphene. And so you're going to see a lot more. It's also really cheap to make the way to discover it was they found a mono layer of graphite on a piece of scotch tape. And when they analyzed it, it was this amazing one atom thick layer of repeating carbon atoms. And it turned out to have these incredible electrical properties. Okay, and then you go down the list. Platinum has a resistivity that's a bit bigger actually than copper. And so forth, let's see here. And then, okay, I wanted to highlight these. So seawater and drinking water. So you'll notice that they have very different resistivity. Sorry, yeah, resistivity is here. And as a result of that's because in seawater, you have a lot of salt. So there's a lot of free ions available to be moved. In drinking water, it's been desalinated. It tends to have a lower mineral content. There's less free charge to move the water dipole really holds onto itself. In order to rip that apart, you basically have to make hydrogen and oxygen out of water to get those charges to separate. So they tend to have a much higher resistivity as a result than seawater. You're kind of similar to seawater in terms of your blood conductivity and things like that. What makes you a worse conductor, a better resistor than seawater is the fact that your skin is often dry and if you're not sweating, you're in good shape. But if you sweat, you're basically putting seawater on your skin and any current that it wants to flow across you will flow across you very easily. So we're going to start to build up a toolkit to deal with situations that involve a source of voltage, conductor, and resistance. And then eventually we're going to add capacitors into this. Now a preview of why this is important to do is something that I mentioned earlier, and I'll show a picture of it, probably not this lecture, but maybe next one. And that is, for instance, if one wanted to try to model the body as a large group of electric circuits, you could start thinking about the smallest pieces of that circuitry and how it functions. And so one of the things that actually has, you know, had a long time ago, I believe it won a Nobel Prize in Physiology and Medicine, was the description of the firing of the neuron as a series of voltages, resistors, and the capacitors. And with that model in mind, one could grossly reproduce the features of the action potential that occurs in every neuron in your body. And by changing the potential across the neuron transmits information, the aggregate storage of electrochemical information is possible through the way that voltages and chemicals are controlled in the brain. And this is something which is still not completely understood today. Yeah, well, I mean, all this is fascinating, right? I mean, neuroscience is really a, it seems like it's advanced, but it's really quite in its infancy. And that's why, for instance, there's a huge attempt to inject lots of federal dollars into this to try to have like a shoot the moon moment with the human brain. Europe is doing this too. They have a huge brain initiative to simply try to answer basic questions about the brain. And that requires synthesizing chemistry, biology, physics, mathematics, computation, and a lot of other areas. Engineering, because I mean, we're just wetware, right? We're just a large computer with some software loaded into it. Where does all that come from? How's it all functioned together? Nobody really understands that. It's a huge area of opportunity for discoveries and money, no doubt. So, okay. So back to this boring picture by comparison. Power supply, battery, all right? So let me make a comment about batteries here. So far, we've been assuming that a battery is an ideal provider of an electric potential difference. And one does have to be a little bit careful, right? This is the simplest circuit you could construct. Basically, you take some conductor and here for the purposes of the schematics, it's assumed that this conductor carries no resistance. And all the resistance of whatever's in this circuit is summarized by this squiggly symbol here. Which is the universal circuit symbol for resistor, okay? So there's some current that's driven by the battery and it's clockwise, all right? So current is emitted, the positive charge is emitted out the positive end of the battery. It flows through the resistive material and then back down here. And it's re-upped by the chemical reaction inside the battery, which won't last forever, but we'll ignore that for now. And then this cycle just continues. Current just keeps flowing. And we know from looking at the drift velocity last time that this is a very slow moving thing. I mean, in a typical house wire, you can walk faster than electrons are drifting through this wire. But nonetheless, because the electric field is established immediately in the system by the battery, all the electrons over here start to move, even if it's only a little, and they start doing work in the resistor by colliding with atoms. And so that, for instance, will give you light from a light bulb almost instantaneously. All right, so this is our basic picture. Battery, so a voltage source. Current driven by the battery. Resistor with resistance to motion of current in the circuit. And we're going to play with this archetype as we go forward. All right, now there's a symbol here that is introduced in the chapter. It's an old term. It's a vestigial term left over, but nonetheless, since everything contains it when you look at circuits, you're sort of stuck with having to adopt this bit of terminology. But we have a special symbol for the electric potential difference that's established by the battery. And it's this curly E, which is short for electromotive force or just EMF. It's an old term before we understood what batteries were actually doing. So, you know, when people were playing around with these cells back in the 1800s, they knew that they had some electromotive force in them that would drive charge through the system. It would do work somehow, but they didn't know what the force was. And so later they made the connection to electricity and magnetism and so forth. Okay, so it's an old term, but it's convenient because whenever you see that little curly E, all right, that's a prettier one than the one I'm drawing. But that means the battery voltage or a battery voltage. There could be multiple batteries in the circuit. We'll look into that situation later on. Okay. All right, so one of the things that we'll have to deal with right away is the organization of resistance in a circuit. Just like capacitors can be next to each other with each end at the same potential, that's parallel. Or just like capacitors can be one after another so that any charge that passes through one has to pass through the other before getting back to the battery. So called series, okay. You can do the same thing with resistors. You can, we have a battery here. You can put two of them in the system such that, right. So here's our electromotive force. Here's R1. Here's R2. You could put two resistors in so that they are parallel to one another. That is that the voltage across them, if I were to measure this here, the voltage is the battery voltage. They both are at the same electric potential difference. But the current that's flowing out of the battery, I, has a branch. It can go through resistor one or it can go through resistor two. And what you have to figure out is how the current is distributed through the resistors. And your guiding principles on this as always are energy conservation and charge conservation. Any current that enters a branch in a circuit, no matter how it branches, the sum of the branches must equal what went in and will equal what comes out. There's no place in these circuits where charge can build up yet. We'll get to that later when we add capacitors, okay. Capacitors store charge. And so at some point when they oppose the voltage that's put across them by storing charge with a separation, current will cease to flow through a capacitor. And that is something that we have to analyze by adding time into the situation. Right now we're ignoring time, all right. So, yearly current is established, it flows through the resistors, it comes out the other side, goes back to the battery, and so forth, all right. So here you have a branch and any place you have a branch, you can apply conservation of charge, okay. Now in the case of series, which we'll get to in a moment, you will have the current, well let me just show it to you, all right. So here's series, that's why I drew this picture so I wouldn't have to repeat it. All right, so in series, you have the voltage from the battery. It is driving a current, so here the plus side is on the right. All right, so the current is going counterclockwise in this circuit. Okay, and it has to flow through R2 before it flows through R1. There is no way the current can skip R2 and go to R1. There are no branch points in this circuit, so whatever current goes into R2, goes into R1 and comes out again, all right. So whatever the current is in the whole circuit coming out of the battery, it's equal to the current going through R2, and that's also equal to the current going through R1, all right. And that's written down here, current is not split anywhere, and by conservation of charge, that means that the current must be the same literally everywhere in this circuit. All right, and we're going to check that assumption in a moment, all right, with a real example. Okay, so the other thing to keep in mind here is that if I were to, I know what the potential difference is coming from the battery, this could be 9 volts, 12 volts, you know, whatever the problem gives you. The potential difference, however, is split across R2 and R1. If I were to make a measurement with a voltmeter or potentiometer here and here, I would get the battery voltage because there's a path that connects directly back to the battery with no resistor in between. But if I make a measurement of the potential here, okay, I am measuring the potential across that resistor, and if I measure the potential here, I'm measuring the potential across that resistor. And I know from Ohm's law that V equals IR, so if I know R and I know I, I can figure out the V across each of those resistors. All right, so what will be true here is not that the electric potential differences are the same across these two, but rather that the sum of the electric potential differences on R1 and R2 will equal that of the battery, okay, over here. That's conservation of energy. Any energy changes that happen through here have to be matched by the energy changes in the battery. This is a closed system. There's no place for energy to go in or go out, all right. So with those two principles in mind, we can do the same trick that we try to figure, we try to do with capacitors. And that is to find the equivalent resistance of these two resistors. So we sketch here for a second. What we would prefer is a simple circuit with just some total resistance, but we've got that picture. So to reduce that picture into that simpler circuit, we have to do a little algebra. We need to use energy conservation and charge conservation as our guiding principles. I sketch that out down there, but I'll just rewrite it up here. So we know from energy conservation that that has to be true. And we have Ohm's Law. For every resistor in the system, Ohm's Law applies, all right. So there is a V1 equals I1 R1 and a V2 equals I2 R2. Just like for every capacitor in a system of multiple capacitors, there is a capacitance equation that applies to that capacitor. Similarly, there's an Ohm's Law for every single one of the resistors in the circuit, all right. So that is also true for this R total that we're trying to figure out. So V total here is just I total, R total. And that's equal to I1 R1 plus I2 R2. Just subbing in with Ohm's Law into this equation, all right. So whatever V is equal to up here, it's equal to the current, total current in the circuit times the total resistance of the circuit. And then similarly, V1 is I1 R1, V2 is I2 R2, we're just plugging in Ohm's Law. Well, the other thing we get to take advantage of now is that the total current in the circuit, all right, so I total is equal to the current going through the first resistor and the current going through the second resistor. There is no place in this circuit where current branches. So in order to conserve current, it has to be the same here, here, here, here, here, everywhere. Current has to be the same everywhere, all right. So that's the last bit of information we need. We now know that I total, I1 and I2 are the same numbers. So they cancel out of both sides of the equation. And you're left with this relatively nice formula that for series, series resistors, the total resistance is just the sum of the individual resistances. That's it. Now, series capacitors was a little different. In series capacitors, the total capacitance was one over the total capacitance was one, you know, the sum of one over the capacitances of the individual capacitors. So if you can remember the capacitor rules, you can figure out the resistor rules by remembering that the rules that apply for series capacitors apply for parallel resistors. And the rules that apply for parallel capacitors apply for series resistors and just substitute R's for C's. That's it. So if you can remember one of them, you can remember the other one just by swapping them. So I find that tip helpful, but it may just be easier to memorize them all. I don't know, it's up to the buyer. Okay, so that's the situation for the series resistance. Now let's look at parallel resistors. So in this case, I have my battery, it's driving a current. I have now a branch in the circuit. So some current will flow through R1, some current will flow through R2, but the sum of those will be equal to the current driven by the battery through the whole circuit. So I have I coming in here, I1 going up through the top branch, I2 going down through the bottom branch, I1 plus I2 meet again over here and I just get I. So whatever goes in comes out, conservation of charge. Now I also have this set up so that the resistors are at the same electric potential difference. These sides of the resistors are both hooked into the same side of the battery. These sides of the resistors are both hooked into the other side of the battery. So whatever the electric potential difference is across R2, it's the same as R1 and the only one that's in the system is the battery. So in this case, V equals V1 equals V2 and I just have to use the fact that I will be equal to the sum of I1 and I2. Whatever the total current is driven by the battery, it will be equal to the sum of the currents in the branch points. So to sketch this out is I want to simplify this picture so that I have just some total resistance. I have some total current and one resistor but I'm stuck with that picture to begin with. So I'm going to try to relate the two of them and to do that, I'm going to start with the current conservation equation. I1 plus I2 so this is I total and again there's an Ohm's law for every one of these resistors. Write that down. All right so I can make substitutions. There's also V total equals I total R total. So I can make substitutions. I can rewrite this as I1 is V1 over R1. I2 is V2 over R2 and I total equals V over R total. All right I will now sub that into the conservation equation. So I have V1 sorry V over again V over R total equals V1 over R1 plus V2 over R2 and now I get to use the last equation, this one, that the each resistor is at the same potential difference and that potential difference is given entirely in this case by the battery. So V is equal to V1 is equal to V2. They all cancel out of both sides and I'm just left with a very familiar looking equation and this is the parallel resistor case that one over the total resistance is equal to the sum of one over the individual resistances. So if I had three in parallel here would be one over R1 plus one over R2 plus one over R3 is one over R total. So wherever you see resistors in parallel in a complex circuit merge them together using the parallel rule wherever you see them in series merge them together using the series rule your goal with any circuit picture is to get it down to as much something resembling this as possible maybe one voltage supply one resistor do the best you can. What I'd like to do right now is take the last 20 minutes of this lecture and talk about the light bulb game okay because something happened when we did this last week that maybe was a bit perplexing because you attempted to use intuition to answer the question right so what had I done I'll reproduce what I had done. So each of these is a just a light bulb and a light bulb is really nothing more than a resistor and what we're going to do is we're going to exercise resistors in parallel and resistors in series and Ohm's law like mad and we're going to look at light bulbs and see I know this is so exciting right and we're going to see what happens if we can make predictions about what will happen in this circuit okay and these are great I mean these are getting harder to find but these are great demonstrations of resistors okay so to illustrate what we did last time what I'll start by doing is just hooking up the I've plugged nothing in I've just plugged in this really sketchy circuit to wall voltage all right all right so what I'm going to do is set this up so that it's capable of measuring a potential difference of up to 200 volts all right you'll see right now it's measuring zero all right so that way that's terrible zero there we go so what I'm going to do now is I'm going to just hook this into the wall yes I have this on voltage last thing I want to do short the building out okay so okay we'll let that kind of average up for a second and how much voltage do we observe coming out of the wall yeah about 125 volts so that's a 125 volt potential difference that's delivered by that socket okay doing nothing right now because it's not hooked up to anything all it's doing is making my my screen display the potential difference so it's doing some work to power this thing now okay we'll take that out unplug that because I don't want to die all right so now what we'll do is we'll start simply by hooking up the the bulbs the way the manufacturer sort of intended this to happen so one end will go black to black one end will go red to red so what I've done now is I've hooked up a single bulb this is a simple circuit I just showed you it's this you have a voltage difference you have a single resistor that thing okay it's a 40 watt light bulb all right so what I will do is now plug that in that uh there was light okay so nothing exciting happens there you go nice decently soft bulb you put a shade over that and it actually wouldn't be too bad to have in a room okay now let's hook up the other bulb so this is a 100 watt bulb exactly as the manufacturer intended it I'm going to put 125 volts across just it just like I just did the 40 watt bulb by itself a second ago and even brighter can't even look at that thing right now I've got my retinas seared like tuna steaks as Archer would say okay so there all right no one no one gets the Archer reference no okay retinas seared like tuna steaks no nothing you should watch more TV I'm ordering you as a doctor okay all right so that was pretty bright and let me just ask how many people think that the 100 watt bulb is the bigger resistor raise your hand if you think the 100 watt bulb is the bigger resistor okay raise your hand high I mean really commit okay thank you all right all right so good the 40 watt bulb this how many people think the 40 watt bulb is the bigger resistor okay and some people aren't committing at all you have commitment issues you should get over that at some point all right that would be helpful all right so okay so fine all right so more more people tend to think 100 watt bulb is the bigger resistor than the 40 watt bulb let's just keep that in mind all right that's unplugged I'm not about to kill myself good now let's do the following let us start so I want everyone to start keeping numbers here okay we are going to figure out what the resistance is of these different bulbs so we're going to use them as the manufacturer intended we're going to hook it up singly across a single potential difference the wall is 125 volts approximately okay give or take it's good enough for what we want to do the manufacturer says that the power of what I'll call bulb one is 100 watts and that the power of bulb two is 40 watts how can I calculate the resistance of the resistor using power and ohm's law any ideas what's power equal to anybody remember IV yeah okay I heard another one in there so power is uh current current times voltage and we can substitute with ohm's law again i squared r all right so if we put in put in ohm's law we can get i squared r okay so if we use the current going through the bulbs we can figure out the resistance by knowing the manufacturer is rating for the power but we don't know the current what do we know what do we know in this situation when I hook up one of those bulbs to the wall we've already measured it voltage right so is there another equation that involves just voltage in r v square to over r and these are just substitutions of ohm's law in for either i or or v inadequately okay so if you take v equals ir and if you want to get rid of i because you don't know it you can write i equals v over r plug it in here you get v squared over r okay get used to exercising all right that's why we're doing this okay great so we know that the power is equal to v squared over r so we can solve for resistance of bulb one right that's just going to be equal to v squared over p1 and r2 is v squared over p2 so calculate them we'll get some numbers here but what do you already notice collectively the rest of you not the two of you who are calculating right now what do you already notice about the relative resistances of the 100 watt bulb and the 40 watt bulb if i put 100 in here is that going to give me a bigger or smaller resistance than if i put 40 smaller so the brighter bulb offers less resistance apparently very good okay and then i'll just check because i did this exercise before class as well and i get 156 and 39 okay great so let's start playing around with this okay because we can actually do measurements now we can compare them to predictions all right so for instance what if i hook these up in parallel all right we saw what happened last time let me do that all right so i will hook them up in parallel what i will do is i will attach the ends of the these bulbs here both to the same side of the wall potential and these ends to the same wall potential okay and what happens if i plug this in do we get them both kind of at their expected brightness or is or is one really faint and one really brighter they both don't come on what happened last time they both turn on yeah so let's just verify that yep and they're both pretty freaking bright okay that one's nicer to look at than that one all right that is the situation with a power strip if you buy go to walmart whatever and buy a power strip that's got like six or seven or eight sockets on it you can plug a bunch of things into it they in all the things you plug into it are resistors and you're plugging them in in parallel and that's why it is that your appliances can all function normally you know you can have a hairdryer plugged into a power strip with a lamp and when you turn the hairdryer on the lamp doesn't dim suddenly and it's because you're putting them all all those devices at exactly the rated electric potential difference that the manufacturer intended about 110 125 volts depending on your your wall outlets okay so they both lit up but let's see if what we think is going on is actually going on so for instance we could measure the potential difference i'm going to have to turn these on so you know don't stare too much all right so what i can do is i can measure potential differences across the bulbs what do i expect them to be in parallel the same and what value will it be 125 pretty close yeah closer for government work probably okay 124.5 all right and you know you're going to start to see a little bit of pattern here as we go uh that some of the numbers might not be quite what we predict but this is good because if we fail we will learn and we'll see if we can fail in this exercise the failure is very important all right but that's basically 125 okay let's try the other one now average up 124.5 okay so they're at the same potential it's a little bit lower you have five volts lower than wall potential we'll come back to that later that's an interesting observation maybe it's just a glitch maybe you know maybe it's just that's the error in the instrument or something like that we'll come back to that in a bit but yeah they're at the same potential that much is clear even if the potential is not quite 125 all right now we can do another thing too here oh man all right we can also measure the current flowing through each of the resistors now this is a parallel circuit situation all right so let me sketch this here what I like about circuits is that with a few basic things ohms law energy conservation current conservation you can feel very powerful you can suddenly start to understand complicated systems by breaking them down surprise surprise pieces at a time and understanding each piece and adding it all up okay the same trick we've been doing the whole time in the class all right so I have created this situation so the battery in this case the wall socket is driving some current I counterclockwise in the circuit what does it do when it gets to this branch point that I've now created right because current can flow in through the red wire and then it gets to this point here and it can either do what it can go through bulb 40 watt bulb bulb number two so resistor number two or it can go through the hundred watt bulb resistor number one all right and what's the relationship again between the currents going through resistor one and resistor two they add to the total conservation of current okay that's a prediction let's see if it's true