 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 atoms 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, okay, and let's just assume that they're 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 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. This 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 are standing 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 flow. 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. 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, right? Because this is in amps per unit area. This is in Newtons per Coulomb. So whatever this thing is, 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. And so if we write that down over here, so here at the microscopic level, P equals rho J, 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, okay, what's the definition of the voltage, right? Voltage is equal to work done by an applied force per unit charge, right? Okay, and this is the work is, if we do one over two here, the work is the integral of the force times the displacement, okay? Which I can write as one over q e, let's see, q e dot dx. So my q's cancel, 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. So in the end, this integral is just going to give me e times L. That's it, okay? 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 is work from the applied force divided by charge, okay? 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 meters squared, okay? 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 it's 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, okay? 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 gonna 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, okay? 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, 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, okay? So there's an art to engineering a nice short path for current in your house so that it's not looping 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 light 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 this is why you need to be very careful with driving more than, say, 15 amps or 20 amps through a house wiring that's rated for 15 or 20. So for instance, the shimmy 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 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 houses on fire. And that's also why you have breakers in your house. They're 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 raw 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 some problem with the material in one part. It broke down, the resistance changed, and suddenly the power dissipating through it increased. 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 is going to do. And the breaker trip. And it probably saved our house from burning down. We still have 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 to a ton of acts. 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. So you have to tear the walls down to get at it. And that means massive amounts of damage to your home. OK, 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 cause 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. OK, 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 it 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's 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. 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. 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. 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. OK, 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, OK, I wanted to highlight these. So sea water and drinking water. So you'll notice that they have very different resistivity. Sorry, yeah, resistivity is here. And as a result of that, that's because in sea water, 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 sea water. You're kind of similar to sea water in terms of your blood conductivity and things like that. What makes you a worse conductor, a better resistor than sea water 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 sea water 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 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 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. Yeah, well, 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 does it all function together? Nobody really understands that. It's a huge area of opportunity for discovery. And money, no doubt. OK, so back to this boring picture by comparison. Power supply, battery, all right? So let me make a comment on 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 a resistor. So there's some current that's driven by the battery and it's clockwise. So 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. 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, it's everything. It 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 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 until later they made the connection to electricity and magnetism and so on. So it's an old term, but it's convenient because whenever you see that little curly E, 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 a circuit. We'll look into that situation later on. 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, 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. You can do the same thing with resistors. You can, if 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. Capacitors store charge. And so at some point, when they oppose the voltage that's put across them by storing of 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, I'm sure let me just show it to you, all right? All right, so here's series. All right, so 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 that 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 gonna 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 nine volts, 12 volts, 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, a 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. 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 and we need to use energy conservation and charge conservation as our guiding principles. Let's 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. 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, 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, 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 you just substitute R's for C's, that's it, okay? 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, okay? I don't know, it's up to the buyer, right? 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 gonna 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, and V over R total equals V1 over R1 plus V2 over R2. And now I get to use the last equation, this one, that 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 resistance. So if I had three in parallel here, it would be one over R1 plus one over R2, plus one over R3 plus 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 just a light bulb. And a light bulb is really nothing more than a resistor. And what we're gonna do is we're gonna exercise resistors in parallel and resistors in series and owns law like mad. And we're gonna look at light bulbs and see, I know, this is so exciting, right? And we're gonna 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 gonna do is set this up so that it's capable of measuring a potential difference of up to 200 volts. All right, and 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 gonna do now is I'm gonna just hook this into the wall. Yes, I have this on voltage. Last thing I wanna do is 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 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 wanna die. All right, so now what we'll do is we'll start simply by hooking up 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. Da-da, 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 gonna 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 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, 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 people tend to think the 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 gonna use them as the manufacturer intended, we're gonna hook it up singly across a single potential difference. V wall is 125 volts, approximately, okay? 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 ohms law? Any ideas? What's power equals? Any ideas? What's power equal to? Anybody remember? IV, yeah, okay, I heard another one in there. So power is current times voltage and we can substitute it with ohms law again. I squared R, all right? So if we put in ohms law, we can get I squared R. Okay, so if we knew the current going through the bulbs, we could figure out the resistance by knowing the manufacturer's 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 squared over R and these are just substitutions of ohms law in for either I 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 and you get V squared over R, okay? Get used to exercising these, 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 P one and R two is P squared over P two. 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 in here? 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, excellent, 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 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 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, and 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 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 gonna 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, close enough for government work probably. Okay, 124.5. All right, and you know, you're gonna start to see a little bit of pattern here as we go 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. 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. We 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 a bulb, 40 watt bulb, bulb number two, so resistor number two or it can go through the 100 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.