 Thank you. So they're real and they are part of imaging technology. So they will allow you to amplify an electric signal by accelerating electrons and smashing them into the copper. So today we're going to start to get into some of the – well, okay, I think this is, for me, this is the most fun part of the course. It kind of started last time with capacitors. Avery asked a really good question, which is why do we have to know this stuff? And I actually love that question because there's actually good answers to that, although I suspect many physics professors will just stay shut up and calculate, which is technically the phase of this course that we're entering, but I promise you I'll continue to justify why this stuff is useful. And today we're going to do some basic terminology with electric current and all of the stuff related to electric current. So we're going to talk about the macroscopic picture of current flow through a material. We're going to look at the microscopic picture of current flow through a material. What are the charges actually doing inside of a material? But the big summary picture is illustrated by this cartoon that I ended the last class with. And that is you have some material, maybe it's copper. This is kind of a copper-colored loop of material. You hook a battery into it so that there's a plus end here. There's a minus end here. This side can emit positive charge. That's what the plus side of a battery does. And you can set up a current flow through the loop that is constantly maintained by the battery's constant electric potential. Now, as we'll see, it's very bad to hook the positive end of a battery, essentially straight to its negative end, without something in between it that resists the flow of current. So a material that has resistance to the flow of current. There are materials in nature. They're called superconductors. They exist. They're real. Humans have learned to control their properties to some degree, but they're not cheap enough to mass produce yet. Although they're used in medical applications, superconductors are used in every MRI machine in the world. Superconductors are used in high-speed rail systems in some places in the world, like in Japan. I think China is trying to build one of these now as well. Superconductor is at the heart of all of the accelerator technology that's used in my field. Without it, we would waste so much power in accelerating current through material that we would easily use more electricity than most major cities to study the fundamental properties of the universe. And at some point, you can't get away with that anymore. You have to be smarter. You have to build better materials so that you're not wasting electricity. Incidentally, when the Large Hadron Collider is in operation, it's using about a third of the energy during peak energy usage times that the city of Geneva, Switzerland, is using. And it would be a lot worse if we didn't have control over conduction and conductivity of materials. So this is our basic cartoon, a battery of source of constant electric potential difference, a material through which charge can be pushed like water through a pipe. We'll call this our conductor, so the thing that allows the conduction of charge. And a few definitions are needed. So we have this symbol here, I. Okay, I. And this is electric current. And I is simply the amount of charge that you push through a certain point in this picture, in this cartoon, in some amount of time. So this is in coulombs. This is in seconds. And this gets its own name. This is known as the ampere, or just the amp. Okay, so we'll get to ampere later. He's important primarily in magnetism, although he did other things. But he was so important, he actually has his name on one of the fundamental laws of nature that he gets a unit named after him. Okay, so ampere, or just amps. So amps, if you've ever heard of house current. This house has a wiring in it that can handle a maximum current of 20 amps. So that's what they're talking about. They're talking about the maximum number of coulombs that can be driven through the wire every second. And we'll exercise this concept as we go through the lecture today. Now, this is something that involves charge, coulombs. And just as all things involving charge, there are basic concepts that you can apply to current that are fundamental, that just work for current, like they work for the more fundamental things. So since charge is neither created nor destroyed, it's simply moved around. Okay, it can be separated, it can be recombined. Charge is conserved, so current is conserved. So if you push a certain amount of current, certain number of coulombs per second into a device, or it comes out the other end taking account of all the branches in the device, must be what you put in, that's it. Okay, so if you imagine in that cartoon, if I go back a slide, if I had taken this copper, and I cut it with a razor and pulled these two pieces apart so they're not, the two halves are not physically touching anymore, I would create a branch in the circuit that eventually recombines later in the copper. And then you can ask, well, what's the current in each of those branches? Well, whatever it is, the total, the sum of them has to equal the current being pushed in. No current is lost anywhere in the circuit. And so as a result of that current, like charge, must be conserved. So anytime you have a branch, the one thing you know right away is that whatever the current is in the top branch, like I1 in this picture, and if you know I2 and you know I, you can figure out what I1 is from this simple conservation equation, that the total current in is the sum of the currents in the branches. And if you have three branches, there could be three of them and so forth. Yeah, Kat. There's like currents. We haven't gotten there yet. Okay. You're getting ahead of things again. So right now the basic concept you wanna keep in mind is if you have current going into a branch, if you just forget anything else that's going on in the circuit, okay, in the device. If you have a branch point and you have current coming in and you look just past the branch point in the two branches, the sum of the current in those branches must equal what went in, independent of anything else that's going on. And that just boils down to charge conservation. That's it. Now it may be true that more current will go through one branch than the other because of the devices that are in place in each branch. Just like we can use capacitors to divide voltages, we can use other things like resistors, which we'll get to in a bit, to divide current. But nonetheless, the sum of the current in the two branches must equal the current that went in. That's it. And so that's good because that grounds any analysis of a circuit in a fundamental conservation law. And you can always fall back on it. Like conservation of energy, conservation of charge will come in and save you. If you're not sure what's going on in a circuit, try looking at the circuit in a way that you think about conservation of energy or conservation of charge and see if you can solve it, unravel it that way. Thank you. And if anyone else would like to look at this at all more, just let me know. I gotta return it to Professor Strunowski's office that we'll notice it's missing. I'm up for tenure this year. I want a nice letter from him, okay? So they make the mistake of giving us all faculty keys and we can all open each other's doors. I don't know why they thought that was a good idea. All right, so current density. Is this thing recording? Hang on. So a better concept, one that's independent, for instance, you'll see why this matters in a bit, but the area of the conductor can have a big influence on its ability to move current, to carry current, given a certain electric potential difference. And often, conductors are not simple in shape and we're gonna deal with things that are quite simple in shape over all of it. That should calculate things very easily, but there are structures in nature that push current through them, but they have a very irregular shape. And so it's often much more convenient not to talk about the current that is the charge moving past a point in space at a single, through some time, but rather the density of current. That is the current per unit area of the conductor. All right, so this is just another definition. We're gonna define this thing J and it's simply the current divided by the area through which it's moving. All right, so if I take a very simple picture of like a cylinder of conductive material, okay? And I have some charge Q that starts out over here and ends over here on the right-hand side. And this thing has some cross-sectional area A, which is pi times its radius squared. Okay, if I know how much time it takes to make this trip, then I can just take Q over delta T and I can divide that by the area and that's the current density. Okay, so Q over T is just current and then that winds up in the numerator of the next part. So current divided by area is current density J. So it's amps per meter squared. That's it, amps per meter squared. So it's the current density, but it's a two-dimensional density. It's the current moving through a slice of your conductor. So if I were to slice this in like a circle right here and just consider what's the current moving through that little slice through the conductor that is the constant of current density. And why is this useful? Well, let's go back to this picture again. Okay, you saw it on the first exam. You have very complex structures in nature, like the cell membrane. And as a physicist, I look at this and I see charges. I see charges that can be moved, that can be pumped by electric fields from one side of the membrane to the other. And the pumping action can be used to maintain a certain gradient of charge carriers inside and outside the cell to maintain an overall balance of negative charge here and positive charge here, like a capacitor. But you've got to move that charge and to move it, you have to pump it through the cell membrane. And this is not exactly a simple cylinder of copper that we're talking about here. We're talking about a very complex and irregular structure over, spread over a wide area in three dimensions out of which or into which charges are being constantly pumped by the cell. And so here it's far more convenient, for instance, to make a measurement of the number of potassium atoms per second that come out of the membrane or go into the membrane and simply divide by the surface area of the cell to get the current density. That's a far easier thing to do than to try to put a little ammeter or a little current probe on the cell wall and at one point, try to measure the current going through. It doesn't work that way. It's too small. We don't have devices that can do that. So in this case, this is a great example of why current density is a far easier thing to deal with. And then once you know the area involved, if you can measure the current density by some other means, you can get the current itself. So if you can find one piece, you can solve for others, just like everything in algebra, okay? All right, so those are some very basic concepts to kick off resistance, all right? So let's get into that subject. And I'm gonna start with the microscopic picture. Actually, I'll start with the macroscopic picture and then I'll move on to something else, to the microscopic picture. So, where are you? Where are you on? Ah, excellent. And I can't stand seeing myself on camera. So, oh, for the love of you. Yay, all right. Okay, so let me get this camera aimed over here. Now, really, technically, I don't need to put a camera on this. You're gonna see this just fine. But what I have here is a dangerous toy. I have a very shoddily constructed light that I can use. You never, ever operate something that's poorly constructed in your house, or at least you shouldn't. Although, having lived in a college dorm room, I'm pretty sure I had at least one lamp that looked exactly like this, all right? It's pretty simple. I'm gonna take advantage using this copper wire. It's got two plugs. Okay, one goes into one side of the wall socket. One goes into the other. And these two plugs represent basically a physical potential difference. I can put an electric potential difference across the wire. One side is at one potential. One side is at the other. Now, they're wrapped up in this sort of rubber or plastic insulator so the copper wires don't touch. But, you know, they're small. They're, you know, maybe two millimeters in diameter each, something like that. And it's a braided cable, all right? So it's very sturdy. One side of the potential difference goes in here over on the red wires. And the other side goes in here over on the black wires. And I can hook things into them, okay? I can hook stuff into that potential difference. And that's what I'm gonna do right now. This is a standard 100 watt light bulb incandescent. It's getting harder and harder to find these as they're phased out in favor of compact fluorescent bulbs, which are really like little computers because they have circuits built into them. This is pretty simple. It's just a filament probably made from tungsten or something like that inside of an evacuated glass bulb. And you don't wanna break the bulb. That would be very bad. And it will heat up as I put in electric potential difference across it. You'll see that and that heat will generate light. All right, so we will have, there we go. Very bright, all right? Yep, super bright. Don't wanna look at that too long. They're gonna my retinas. We'll study those at the end of the semester. All right, I have another bulb over here. I got a 40 watt bulb. And often what you wanna do in a house is you wanna have like a power strip, right? You got one socket. It's the only one that your stereo TV, your laptop, your phone charger isn't plugged into. And so that's the one you use for plugging in light bulbs. But you've only got one and you wanna multiply it. So you can buy these power strips, right? They come with multiple plugs on them. And that's exactly what I have here. I have a really poorly manufactured power strip that again, no electrician would ever in their right mind allow you to have in their house. What I can do is I can wire this up just like the ones in, you can buy in the store that are nicely made. They pass the underwriters limited UL rating and so forth. So then I can set your house on fire. I'm like this one. I can plug in two lamps to this thing and we'll get to this later when we do circuit analysis but what I've done is I've hooked them in in what's called parallel. So just like the capacitors, right? Where each side, the top side of each capacitor is at the same potential and the bottom side of the capacitor is at the same potential. I've done the same thing here. The red side represents one potential. The black side represents the other potential. They're on separate sides of the light bulb. And when I plug this in, I get two different lamps. So I could put a nice soft 40 watt light in one corner for reading, maybe for entertaining or for dinners. I want a bright light over the dinner table and I can do that, okay? There's a reason why they hooked them up in parallel. Let's instead be really stupid and we'll build a power strip where everything's in series. That is current has to go through one light bulb and then it has to go through the second light bulb and so forth. And so every time you plug a lamp in, you're putting it later in the chain of current. So first current has to go through one bulb and then it has to go through another and then so forth. And you'll see why this is a terrible way to design a power strip. And again, we'll come back to this later, but I want to demonstrate the macroscopic business here of resistance, okay? Resistance to the flow of current will radically alter your expectations of what will happen in the system. The 100 watt bulb has a different resistance to the flow of current than the 40 watt bulb, all right? This one gives off more power than that one when it's plugged in by itself to an electric potential difference of about 110 volts. So let's just think about this a second. When I had them hooked up so that they were each, each at the same potential across the bulb, I got a dimmer light and a bright light, okay? So 100 watts of power output, 40 watts of power output. Let's stop and pause for a second and think about what's gonna happen if I hook this up differently so that I force any electron that's coming in this side has to go through this bulb first and then it comes out this side and then I hook it in so that it goes into the other bulb and then comes out its other side and then finally that goes back to the wall potential. Okay, so let's make some predictions about what's gonna happen. What do you think's gonna happen when I plug this in, all right? So we can kind of do a poll here on this totally damaged board. All right, A, both will light just the same as before. B, the 100 watt will light but the 40 watt will be super dim and C, the 100 watt will be super dim and the 40 watt will light and then we could do D, neither will light. Okay, so let's do show of hands on this one, right? So now I've got it so that the electric current being driven by the potential difference from the wall must first pass through the 100 watt bulb and then it can pass through the 40 watt bulb and then it can go back to the wall. So what's gonna happen? How many people think A, both will simply light exactly the same as before? I really primed you for that one, okay, good. B, the 100 watt bulb will light and the 40 watt bulb will be super dim. Okay, so one, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, what a bunch of followers, 15, 16, okay, 16 should make it, make you all close your eyes when you do this so you're not influenced by your peers. Looking at you back there, all right. All right, C, 100 watt will be super dim and the 40 watt bulb will light. Anyone for that, raise your hand high. Okay, and then D, neither of them are going to light. There's like two of you that haven't voted. Who hasn't voted? Raise your hand if you haven't voted. Okay, fine, then there's absent people today, no problem. All right, so basically everybody who was willing to vote and some of you who are sheeple voted for B, okay? So let's see what happens. Damn it, right? Who bet on this? So actually, with somebody maybe, Emily, why don't you come up here? And would you do me a favor and just look into the 100 watt bulb? Is it not lit at all or can you see the filament glowing? I can see it glowing. Glowing, yeah, so it's super dim, right? But this one, I mean, this one's not as bright as it was before, but it's still lit up, right? And this is why it's important to learn about resistance apart from all the other reasons I could argue is because otherwise you'd walk into building appliances like this and you'd make really bad decisions or you won't understand why it is that energy output from a circuit behaves a certain way. Energy is really important in everything in chemistry and biology and physics and engineering. So all of this is very basic, right? It's a simple problem. What if you built a power strip so that the current has to go through one appliance then the next appliance then the next appliance and finally at the end. Do you think it'll make any difference if I swap the bulbs? Should we do that? Okay, yeah, some nods. Cheaple, man. I feel like I'm on a PBS Kids show. Although they be far kinder. Their hosts are really nice. I'm kind of a jerk. All right, let's see what happens. Same outcome. So 40 watts over here now, 100 watts over there. It's still barely glowing, but it's glowing a little bit. All right, and we'll come back to this. This is what I call the light bulb game and it's important for you to try to look at a problem intuitively without a deeper understanding of what's going on and then fail. Failure is extremely important in science. It's how we make progress. If you never risk intuiting an answer and then risk failure as a result, you'll never learn anything about the world. Failure is how we as human beings have made progress for hundreds of thousands of years. We tried something, somebody died and other people decided not to repeat it. Or we tried something and it didn't work out. No one had to die, but we'll never do that again and maybe we'll find a better way of doing, trying to get the same outcome with another means. This is how you improve technology. This is how you improve all kinds of things in life. Yeah, Rachel. Do you think it was, I'm sorry, one thing, my physics teacher- Sorry, you got a very emotional helpful message. No, very. My physics teacher in high school had a lot of hairlicals signed on his belt or so, but he was early fell off and I don't have a smiling picture of him so it was hilarious. You had to be there, but it was- No, no, it's true. Yeah, no, and it's true. There's also, my favorite motto actually is, I forget who said it, but in Latin, it translates from Latin as I know that I don't know. And that's an important thing to admit right away, right? The first time I did this demonstration, I didn't actually sit down and do the math and I got totally the wrong intuitive answer as to which light bulb was gonna light and I spent then an hour sitting down and going, wait, why did that just happen? And I sat down with the equations, I sat down with Ohm's Law, which I'll get to in a second and I thought about resistance and I worked through the math and I came to the answer. Oh, okay, I should have expected the 40 watt bulb actually to still be pretty bright and the 100 watt to not. But that's the tricky business in this. Okay, Jen, do you have a question? No. Oh, okay, all right. I had like a part two of my question. Okay, part two, sorry, yes? Or comment, whatever. So the 100 watt bulb is more wanted. Why didn't more of the current or whatever goes through it and make it light up? We'll come to that. Yeah, we'll get to that. I'll come to come back to this. This isn't even the last time you're gonna see this, I promise, yeah. We're supposed to get out of that. What you're supposed to get out of that is that if you just start making guesses about what's going to happen in a circuit without understanding resistance, power, current, and voltage and how they all relate to one another, you will get wrong answers. Okay, because you think one thing is going on, but something else entirely is actually going on. And we'll kind of look at this a little bit microscopically in a second, okay? The higher the wattage, the lower it is. Well, we'll get to that, yeah. So yeah, so there's, we'll get there. So actually the higher the wattage, the higher the resistance. And that's because to take the same voltage and get more power output, you have to drive the resistance. Okay, we'll get back to it. We'll get back to it, yeah. Okay, so let me set that aside for a second and come back to this. Okay, so let's take a look at the microscopic picture of this and the graph is just crashed. All right, we're gonna come back to the microscopic picture in a second once I can get this back up again. What's going on inside of a material? Well, you've got atoms. And in a conductor, some of the electrons, maybe one, maybe two, three, something like that, out the outermost shell of the electron, of the atom is free to move. It can be easily ripped off of its parent atom by even a little electric field and moved away from the atom. That electron is gonna travel through the material but it's not just gonna travel and never touch anything. There are going to be other atoms in its path as it's moved by the electric field. And so the issue here is that in materials, there's a very complex set of phenomena going on that have to do with motion, collision, speeding up, slowing down, being accelerated, being decelerated. And that's the microscopic picture which we're going to look at now. Okay, now that recovery actually went well. All right, so let me kill the lights a little bit here. So imagine we have some conductive material which I've represented by this sort of hazy, shaded area up here. And in a conductive material with no electric potential difference applied to it, that is no electric fields, no net electric fields in the material or on the material. If you imagine these atoms or the conduction electrons, they can even be knocked out of a copper, for instance, you can knock the conduction electrons out just by the vibration of the atoms alone. And so these electrons are very loosely held by their parent atoms. And so they will execute little random motions. Maybe this one's zipping up into the right, maybe this one's moving down, that one's moving up, that one's moving down into the left. Then one of these will bounce off another atom and then it'll deflect and it'll go in some other direction. So you have all this sort of random motion of atoms or charges inside of material. And the net effect of this is that they go nowhere. On average, the net velocity, if you sum over all the velocities of all the individual things is zero. And that's good news because if I left a cord just sitting on a table with no electric fields, no forces of any other kind acting on it, I really don't expect it to, I don't expect to come back in an hour having found it moved a centimeter over to the right because its atoms are all drifting in one direction, net, as time goes on. That's not happening in materials. They are, each atom is moving, but the net effect of all the motion is that they all kind of cancel each other out and all the random motions lead to no net motion whatsoever. All right, so this is just thermal motion. If you are sitting in a room, you are being bombarded by air molecules, moving quite fast, roughly at bullet speeds. And those air molecules are smacking into the atoms on the surface of your skin. They're transferring kinetic energy into the atoms in your skin. And that's causing heat to be added to you. So on a hot day, when you've got solar radiation, convective air currents outside, you walk outside, you're absorbing a whole lot of heat. And basically that's from motion of particles bombarding the surface of your skin and your clothes. That heat has to be then absorbed and dissipated someplace. So blood is very good at pulling heat away from your brain. For instance, if it didn't do that, you'd suffer brain death every time you walked outside. That heat has to go someplace. And in our case, it's removed from the body by perspiration, for instance. So it takes a lot of energy to evaporate water. So your body secretes water and then it uses the fact that you can dump a lot of energy into water and then it evaporates away from the body taking the kinetic energy away again. All of that is needed to thermally regulate a system. But all that thermal motion is essentially random inside of a material. There's no purpose to it. It's not going anywhere. Nothing's directing it. But we can direct it. If these are charges that we're talking about here, we could, for instance, apply an electric potential difference. So we could now apply a high potential on one side of the material, a low potential on the other side of the material, and suddenly an electron that was moving down only will start to move also to the right as it's accelerated. So think about the two components of motion when you fire a projectile. If you shoot a cannon ball off the surface of the earth, it will rise up and it will fall down on a parabolic trajectory. That parabola shape is governed by the fact that in the horizontal direction, if we neglect air resistance for a second, there's no force acting on the cannon ball. But in the vertical direction, gravity is slowing it down and then re-accelerating it back down to the surface of the earth. And so similarly here, we have an electric field, for instance, that points to the left. These, let's say these are electrons. So they're gonna move to the right under this potential difference. So the electric field must point to the left. And they're gonna get a little bit of net velocity along the horizontal direction. So they'll still have a component that points off the horizontal direction. But they'll gain a little bit of net velocity in the direction that the electric field is pushing them. All right, so now what happens is that what was once random motion becomes much more organized and these conduction electrons will begin to drift to the right in the material. They'll still collide with atoms. So they'll begin to drift, but then they'll smack into an atom and deflect. But then they'll re-accelerate in the electric field and then they'll smack into an atom and deflect and then they'll be re-accelerating the electric field and smack into an atom and deflect and this will go on and on and I'll demonstrate this in a minute. The net effect of the electrons moving in response to this now established electric potential is known as the drift velocity. And the drift velocity is simply the, you can compute it very easily. It's the distance that an electron, for instance, has to travel, L, divided by the time that it takes to travel that horizontal distance, T. So drift velocity is fairly simple. It's just like any other velocity. V drift, its magnitude, is the distance you have to travel divided by the time it takes you to travel that distance. Okay? So let's take this picture and let's, there we go, this thing's useless to me. Let's now think about drift velocity and current density. Let's come back to I over area. So current over area and think about how we can relate this, for instance, to drift velocity, right? So we'll take the pole away here. This is going to be a very interesting exercise at this point and I'm going to leave that just because it's going to be handy in a second. So we have this equation that the drift velocity is just the distance that, let's say an electron has to go and the time that it takes it to get there. That's it. That's V drift. And absent an electric field, there will be no net drift velocity but when you apply an electric field and electric potential difference to a material, you can induce a drift velocity. So you're moving some charge through a conductor. Take this picture here. All right? So we're going to take some charge Q and we're going to move it through the conductor along a length L in some time T. Okay? Now materials are macroscopic things. It's not just one electron that's moving. It's a whole lot of electrons that are moving. And so the amount of charge that's actually being moved in a copper wire, like this one here, it can be quite large, right? I mean, there's sort of Avogadro's number worth of atoms in this thing and copper atoms. Does anybody know how many conduction electrons it has per copper atom? None of you remember the atomic structure of copper? Just like off the top of your head? Weird, okay. Well, good news is we have this thing called Wikipedia and we can go look it up. So let's do that. So the nice thing about Wikipedia is there's like tons of information available on it and my generation invented what we know is the web, but you guys are supposed to know how to use it. Okay? Your web 2.0 is. So per shell, all right? So if we look at the shells of the atom, you have two in the inner shell, that's a filled shell. Eight in the next shell, that's a filled shell. Eighteen, remember these are all the magic numbers in atomic physics of filled shells. Eighteen is a filled shell, one in the last one. That is not a filled shell in that last outermost shell. So you have this one electron way out from it's what, 28 other electron cousins that are stuck in the atom. Those cousins are all screening the electric field from the nucleus of the atom. That guy's barely bound. This is the oldest child wants to get out of the house. Nothing's holding it there anymore. All right, so that one electron is the conduction electron and it is free to move under the influence of an electric field. That thing can be very easily removed from a parent copper atom. So for copper we have one conduction electron for every atom. And so you can do some gymnastics and you can calculate for instance, well what we wanna know is, okay, we have some N. This is number of conduction electrons, E minuses per unit volume. All right, so given some volume of material like a cubic meter of copper, if you knew the number density that is the number of conduction electrons per cubic meter, you could calculate the number of electrons in that cubic meter that can be moved by an electric potential difference. If you knew the width of the copper, you could figure out how far they had to go. And if you could somehow measure the time that was required for them to make that trip, that's the tricky part, then you could figure out things like drift velocity very easily. But we'll get to that, all right? So this is a number that can be calculated. I'll show you an example of how you could do it given other information like the density of copper. You can actually figure it out, this number, number density from the things like the density of copper. You have the volume. And I apologize, but I'm gonna write this as a V but with these little hats on top so you know I'm not talking about voltage. We have only so many symbols in the English alphabet, the Arabic alphabet, okay? So I'm gonna modify my V a little bit for volume and put these little hats on top of its ears, all right? So that we have some volume of material and if we have a nice simple picture like this, it has an area A, it has a length L, so the volume is just the area times the length, okay? You can figure out the volume of a cylinder by its cross-sectional area times the length. That's just sort of the number of little circles you have to add up to get to the total length of the cylinder, okay? So based on that, we can actually solve for the charges, the amount of charge that's actually being moved through this. It's the number density that is number of conduction electrons per unit volume times the volume, which is the area times the length times the charge of a single electron. So one electron, this is the number of electrons total that are moved, so you're just taking the total number multiplying it by the charge of a single electron and you get the total charge. That's the charge we have to move through some length L to get it to the other side of the potential difference, for instance, okay? So the time that it takes is going to be related to the drift velocity. So v drift, the drift velocity, is equal to the length that has to travel divided by the time, okay? And the time that it's gonna take therefore, we can just solve for that. That's just going to be equal to the length divided by the drift velocity. And you'll see where I'm going with this in a second. What I'm trying to do is I'm trying to relate drift velocity to something else like, for instance, current density or current that we can actually measure. We can measure the number of amps in a circuit, for instance. All right, well, the only thing left to do at this point is to, let's see, that's absolutely incorrect. Why is that in my notes that way? That's fascinating. Ah, yes, okay, good. Now I see what I did wrong. I'm gonna have to reclaim this board. I'll rewrite the useful thing in a second. Okay, so current density, let me write that down. Current density is current per unit area. All right, well, current is charge per unit time. And we've got an expression for charge in terms of the number density, the area and the length of the material and the electric charge. And we've got an expression for time here in terms of the length divided by v drift. And I can rewrite this as n, a, l, e, v drift divided by l. Well, that's nice, because that l cancels with that l. And I'm left with the current is equal to n, a, e, v drift. And this is why current density is so useful. So we don't have to worry about the area of the material if we rewrite this as i over a. So current density is just the number density of conduction electrons times the charge times the drift velocity, okay? So this allows you to, playing this game, and I'll show you an example in a second, playing this game allows you to relate the current density, current per unit area in a material to the number density that is conduction electrons per unit volume times the charge of a single electron times the drift velocity of the electrons in the material. Now what has been observed, and it was sort of demonstrated here, is that if you can measure the electric potential difference across a material, or if you can create a certain electric potential difference across material v, and then you measure the amount of current moving through the material in response to that electric potential difference. All right, so if you set up an electric potential difference v, and you measure the current going through it, that there is, in a certain class of materials, there is a direct linear relationship. If you double v, you double i. If you quadruple v, you quadruple i. And that relationship is known as Ohm's Law, and the constant that relates these two things is resistance. This is the material's tendency to resist the movement of charge through it. So if I put 20 volts on a material that has a low resistance, let's say one Ohm, I'll get to the units in a second, but one Ohm, okay? And I put the same voltage on another material that has four times the resistance, four Ohms. In the first material, I'll get a big current, and in the second material, I'll get a much smaller current. Because that material resists the flow of electric charge through its volume much more effectively than the first material does. So let's see here. We did this. Oh yeah, let me go through this. This is what I wanted to demonstrate to you about why I did this business with this right here, okay? So imagine you wanted to know the drift speed of electrons in a wire, right? So I plug this in. All right, so the minute I plug this in, which I'll do in a second, tell me how fast the light comes on. All right, was it essentially instantaneous? Yeah, okay, it was. It was essentially, it seems like to our eyes, it was certainly less than like a tenth of a second or something like that, okay? So it seems instantaneous that you plug it in and the light bulb comes on. Well, how fast are the electrons actually moving in the copper when I put that potential difference on the copper wire? Well, it's plugged into something like a 110 volt or a 125 volt potential difference. And typically, for a light bulb like this, we're talking about maybe about an amp. So nine tens of an amp moving through the wire. It doesn't take a lot of current to get a light bulb to light. If the conductor is made from copper with a radius of about two millimeters, so that's roughly the size of this, that you can calculate its area very easily, okay? So you get 1.310 to the minus five meters squared. And then you can ask, well, how fast are electrons drifting in the electric field inside the cord? Given that potential difference, that current and that area? Well, where'd it go? All right, so the current is NAE V drift. We have to get N somehow. We've got the area already. We know the charge of the electron. I mean, we can know the current that's going through the cord, it's 0.9 amp. So we wanna solve for V drift, but we need N. And here's an example of how you can get N, all right? So in copper, as we said already, you've got one electron in every atom that can be moved, a conductor, a conduction electron. The density of copper is 89.60 kilograms per meter cubed. Copper's really heavy, okay? A meter cubed of copper, you're not gonna lift that. There are 6.02 times 10 to the 23 atoms per mole. So atom here cancels with atom here, that's nice. And then finally, you can look up the kilograms per mole of copper. And that's 63.54 times 10 to the minus three kilograms for every mole. And if you put them in this format, so electrons per atom times kilograms per meter cubed times atoms per mole, divided by kilograms per mole, you wind up with what you're looking for, the number of electrons per cubic meter of material. And that's a whopping great number, eight and a half, 10 to the 28. Okay, so it's bigger than Avogadro's number. And so if you plug that in, if you plug in that end there, we've got E, we've got A, all right? And so we got I, so we can solve for the drift velocity. And it comes out to be an incredibly unimpressive, 5.1 times 10 to the minus six meters per second. The electrons in that copper wire are moving slower than I can walk the length of the copper wire. All right, if I lay this out and I were to plug it in in two seconds, I can cross the length of the wire roughly on one to two seconds. But they're only moving, let's see, that's about two meters roughly, okay? They're only moving 5.1 times 10 to the minus six meters per second in that electric field. That's terribly slow. So why is it that the light comes on right away? Why doesn't it take years for that light to come on? Any ideas? No, the material that makes it move faster. Yeah, it's all copper tungsten in the light bulb, which has a different resistance, but yeah. Just the concentration, there's so many, like this guy, this guy, just basically the room is warmed up. Yeah, what in the material is established instantaneously when I hook it up to the wall potential? The potential difference creates a what? The electric field. Electric field. Electric fields, it turns out, propagate at the speed of light. So it's actually not instantaneous that the light bulb comes on, but it takes so short a period of time for the electric field to propagate through that copper that to my eye, it appears to come on right away when I plugged it in. There's a small delay, but you can't measure it with human senses. The electric field is established immediately and the density of conductors, of conduction electrons is huge. There's conduction electrons here, here, here, here, here, here, everywhere in the circuit. So the minute I plug it in, the electric field comes on and the electrons just inside of the light bulb will begin to drift and hit other atoms right away. And so they'll immediately start doing work by smashing into other atoms, which will emit light. So the electric field is established immediately. The drift is not that fast, but the density of conduction electrons is so huge that they just start smacking into things right away. Yeah. Yes, the drift velocity is only going to exist as something other than zero when a voltage is placed across the conductor. And that's because if you don't put an electric potential difference, if you don't put a voltage on the material, there's no electric field that causes electrons to wanna go in one direction over any other. So there's no preferred direction that the electrons will travel and they'll just bounce off atoms randomly. Well, the field is pushing them and it tries to, I mean, you guys have calculated like the acceleration of a little bit of charge in an electric field, that acceleration can be like 10 to the 13 meters per second squared. That's a big number, right? Some of you even got worried about that on the exam, that number's too big. No, accelerations can be huge, but the problem is that there's so many atoms in copper that it might accelerate and then one atom later, bam, smack into an atom and slow down and then re-accelerate and then bam, smack into another atom and re-accelerate. And I'm actually gonna demonstrate that right now in fact, so why don't I do that? One last bit of information here. So Ohm's law is just that there is a direct proportional relationship between voltage and current and that the resistance of the material isn't dependent on those things, okay? So Ohm's law is basically that R is not dependent on either I or V, the voltage. And so you get to use this relationship to figure them out. So this per se is not Ohm's law. This is what is a consequence of Ohm's law. That is Ohm's law, that the resistance is not dependent on current and voltage. Now, when a material fails that statement, it is non-Ohmic. So Ohmic materials obey this rule, non-Ohmic materials do not. There are plenty of non-Ohmic materials in nature and you can take a perfectly good Ohmic material, expose it to too much electric field, damage it and turn it into something non-Ohmic by doing that, okay? All right, so the unit for resistance is the Ohm. All right, so resistance, you can write it as V divided by I, so it's volts per amp, okay? Or V over A, okay? You can work that out in kilogram meters in seconds if you really want, but it has a symbol. It's symbol is the Greek letter Omega, okay? So that's Ohm's, you can treat it like a chant. This is George Simon Ohm, okay? A very serious looking German fellow. He was a German physicist. He began his studies using at the time what was known as the newly developed electrochemical cell. We call them batteries now, okay? But back then they were pretty nasty buckets full of all kinds of horrible chemicals with metal sticking out of them, not very fancy. That was invented by the Italian Count Alessandro Volta. See where all this is coming from now, okay? So he published in 1827, his discovery of this relationship between current and voltage in a certain class of materials which we now call Ohm's Law. So yeah, Catherine. Well, for instance, it could be that as you apply an electric field to the material, you can alter the spacing between atoms as you put more electric field into it and that would cause the drift velocity to change with electric field. And so that would induce, that would mean the time between collisions would change and it would be much harder for electrons to move through the material if you compress them into an electric field. So a material that can respond by its structure changing in response to the electric field, maybe compressing or expanding or something like that, that might render it non-omic. More electric field, more voltage would then change the structure more and would actually alter the relationship to the amount of current but basically by altering the drift velocity, okay? So resistor's resistance is everywhere. These funny little colored doohickeys here appear in circuits, that's what we call a resistor in a circuit. It has a symbol, just like the capacitor has a symbol with two parallel plates next to each other, the symbol for a resistor is this little squiggle, this little zigzag, it looks like Charlie Brown's T-shirt design, okay? Do any of you know who Charlie Brown is? Am I that old? Okay, great, excellent. All right, so yeah, so that little squiggle represents in a circuit diagram the presence of a resistor. It is normally assumed in a circuit diagram that all the little lines that connect the components are perfect conductors and that any resistance that needs to be represented in the circuit would be represented by a squiggle like this. So all the resistance of a copper wire, which is not zero, would be represented by one little squiggle in the wire, for instance, okay? Light bulbs are resistors, people are resistors. Pretty good ones, as it turns out, but not great ones, all right? So I've said this already, the unit of resistance is the ohm. I've got some conversions here on the slides. We'll come back to the microscopic picture in a second and how you relate microscopic properties to other things, but let me show you this model here first, okay? So how can we think about resistance? So I have here this giant pin board. Let me get this unlatched. All right, and get this aimed at it. Okay, so, yeah, I'll see about it. I have the ability to drop charges. So little ball bearings into this. This pin board will be an analogy. So this pin board, it's got a bunch of little nails in it in here, okay? And plastic covering the front so the little ball bearings can't escape. This is our material and the pins represent the locations of atoms in the material and those locations won't change. They'll simply remain fixed and the conduction electrons have to get around them. The gravitational field in the room is our analogy to an electric field being established in the material, all right? So if I have a ball bearing like this, and this is gonna fly someplace, if I just drop it, I would expect this thing to accelerate freely under the influence of the gravitational field. In this mechanical picture, this is an analogy to the electric field accelerating the ball bearing, okay? So nothing exciting there. If we neglect air resistance, it falls pretty fast. It takes maybe less than a second to make the trip from the top of the pin board to the bottom of the pin board, okay? So on the other hand, if I were to put a bunch of stuff in this ball bearings path, what do I expect to happen to the time it takes for it to make that same trip from here to here? Increase, right. And that's because it's gonna suffer collisions. So let's watch. Some of them are quite extreme, right? The first one kicked it way over here and take another one, right? And imagine this is a one coulomb charge, for instance, okay? And it has to travel, what is that, about a foot and a half, okay? So this is like one coulomb. It's gotta go a foot and a half. If there's nothing in the way, if we were in a superconductor, we put an electric field on it, that's how fast it would take to make the trip, maybe about a second. But instead, if we put a bunch of atoms in its way, so it's jittering around, it's smacking into the atoms, and every time it's released from smacking into one, it's re-accelerated by the gravitational field down, it hits another, it bounces, then it's re-accelerated again, it hits another, it bounces, and it executes this sort of little jittery motion in the horizontal direction, but in the vertical direction, it's sort of moving down very slowly. And what took a second before takes one 1,000, two 1,000, three 1,000, four 1,000, about four, four and a half seconds, something like that. So it took about four times longer to make this trip with all these atoms in the way. And this is an analogy, okay? This is an analogy to what's going on inside of a material. And imagine now if we were to take a whole bunch of these things, let's see if this is gonna work, all right? Now, what did you notice? What was different about this? And putting a whole bunch of them through. What changed? They hit each other. They hit each other, yep. In a material, it's possible for electrons to hit each other, but they're so tiny, it's unlikely. They're much more likely to hit atoms. This is not a good proportional representation of the size of an electron to the size of an atom, all right, or even the size of an electron to another electron, okay? But what else? What macroscopic thing intensified as we pushed more electrons, more charges through our material here? You mean just downward? Movement downward, yeah? Well, let's put these all in at about the same time and see how long it takes, all right? So one 1,000, two 1,000, three 1,000, four 1,000. About four, four and a half seconds again. So the drift velocity was still about the same, okay? So it took about four, four and a half seconds to go roughly 18 inches or so. But what intensified? Something got bigger. Sound. Sound, yeah. What does sound represent in this model? Giving off energy. And that's exactly what electrons are doing when they smack into atoms. They smack into them. They cause the atom to vibrate in response. That atom doesn't just sit there and take it, it may vibrate in response. An electron might have so much energy when it strikes another atom, it might free another electron. Or much more likely, not free it, which takes a lot of energy, but it might take an electron in the outermost valence band and excite it for a moment up to a higher energy state before it then drops back down. And what happens when you excite and then de-excite an electron in an atom? It gives off energy. It gives off energy in the form of light photons. Yeah, so what sound actually is, is the vibration of atoms against one another. And then that's transmitted to other material and you get air that's displaced by the vibration of atoms. And we interpret that as sound. In a light bulb, there's no sound, but there is light and there's heat. And the heat comes from the fact that the atoms are vibrating and they're hitting the few gas molecules that are inside the bulb. And then that's then, we've also got the light that's striking the bulb glass and causing vibrations in that as well. And you've got the light itself. And so the sound that you hear in this analogy, that's an analogy to the light coming out of our light bulb, which is a resistor. So this is a resistive material, so is the light bulb. Here you emit sound with every collision. There, because we're talking about atomic phenomena, you're exciting and de-exciting electrons in the atom. The excitation comes from the collision. The de-excitation happens on its own and has to conserve energy, so it emits light. And that's where the light from a light bulb comes from. It's the electronic equivalent of this. And that's all that's going on. So the light that you see there is a macroscopic effect. It's due to microscopic collisions of electrons, the actual things that are drifting inside of the electric field, with the atoms that are essentially fixed in place in what's known as a crystal lattice. So copper is a crystal, believe it or not. It doesn't look very pretty, but all a crystal is is a regular, predictable arrangement of atoms. Quartz, glass, these are also crystals. Glass is a more amorphous crystal. Quartz is a very regular crystal. And it's quite beautiful when you look at it. Diamond is an outstanding example of a crystal. It's a very regular crystal lattice. It's properties, it's transparency, it's brilliance. Those are all driven by the structure of the lattice itself, where the atoms are placed in relation to one another. You can, this is how you can grow man-made diamonds. This is how you can alter the properties of natural diamonds by altering their structure, okay? So all the macroscopic properties we take for granted, the fact that copper is kind of orangey colored. The fact that gold is a bit more yellow than copper. All of that is a byproduct of the atomic structure of the material and how electrons can move in that material. That to me was a revelation the first time I learned it. Well, I took a class in college and I actually learned why gold is gold colored. And it's because of the atomic structure of gold. The fact that it gives off, it's able to reflect light up to yellow, but then no further than yellow gives it its golden color. And that's entirely because of the atomic structure of gold atoms. If those structures were altered a little bit, the color would change. So it's all kind of remarkably well tied together and the microscopic picture has to add up to the macroscopic picture. And there I am again, so let's get rid of me. All right, so what I wanna wrap up with today is just a last bit that you can get from Ohm's Law and then I'm gonna come back and talk about the microscopic picture one more time. Because I wanna give you something useful right here at the end with Ohm's Law. I'll get rid of this. And that is power, okay? Power is just energy per unit time. In this model that I just showed you, the energy that was being dissipated per unit time was coming out in the form of sound waves. And if we were, for instance, to make very precise measurements of the sound wave properties coming off of that material, we could actually figure out the kinetic energy that the ball bearings had when they collided with a nail just by knowing the frequency and amplitude of the sound waves that come off of the nail when it vibrates in response to being plinked by the ball bearing, okay? In analogy to that, any power dissipated in an electronic circuit by light or by heat, if we can count that up, we can figure out exactly how much power was being put into the material in the first place through the acceleration of electrons. So we should be able to relate this to things like current and voltage. So our goal is to relate power to the things going into the circuit, the voltage put across it, the current moving through it. Because energy is conserved, there must be a relationship between those things and the power that's output from the circuit in the form of light, heat, and so forth, okay? Well, power is just going to be the work per unit time, okay? So work done by an applied force, for instance, per unit time. In this case, the applied force is the resistance of the material. It's ability to get in the way of conduction electrons. Well, we know that that is just equal to change in potential energy per unit time, okay? Work done by an applied force is just the change in potential energy. Again, it's just energy conservation. Now we can do the physicist trick of inserting the clever one. What we want is voltage and current, but we have changes in potential energy and time, but we can get voltage and current by putting charge in because voltage is the change in potential energy per unit charge and current is the charge per unit time. So if we're moving a certain amount of charge through a potential energy difference, the voltage is just the potential energy difference divided by the charge we move through it. The charge we move through it in a certain amount of time is the current and so a clever one is in order. So what I'll do is I'll divide by q over here and I'll multiply by q up here and I get delta u over q times q over t, which is just vi. The power dissipated in a resistor of any kind is merely equal to the voltage applied across the resistor times the current being driven through the resistor. It's a really nice simple equation. This is why this part of the course is kind of fun. If you can just remember some of these basic equations, v equals IR, p equals IV, you can do a whole lot of stuff pretty easily, okay? So, I think I'll save resistance and resistivity for next time. We'll come back to that because I wanna show a video today. So the other thing we can do is, we can substitute in, right? So if we know the resistance, all right? We know Ohm's law, if we know that v is equal to IR, we can also get resistance into this equation. I guess I didn't get this. Basically that analogy I just gave you, be very careful with it. It's a great way of picturing, sort of using gravity ball bearings and nails. What's going on inside of a material, but don't let that be your only guide about a material. Sometimes you just have to appeal to mathematics to understand things, okay? So that was an analogy with the pin board. Use it wisely. Every analogy breaks down at some point. Electrons are not ball bearings. Atoms are not nails. Gravity is not the same as electricity, okay? So just, if you keep that all in mind, you'll be fine. But it's not a bad place to think about what's atomically going on in the material. All right, so we have p equals IV and v equals IR. So let's say we're given the resistance and the current. We can then do things like substitute for v. So if we're given i and r, we can figure out p by just substituting in for v and that will give us i squared r. If v and r are given, okay, or you can figure out v or r in some way, then p will be equal to v squared over r. Okay, so you can do some very simple Ohm's Law stuff with power and very quickly, if you're not given one thing, you can substitute for using Ohm's Law and figure out in terms of what you are given in a problem, okay? All right, so to close out today, I'll come back to this material in the next lecture. What I'd like to do is, here we go, human beings. We're pretty good resistors, but you do wanna be a little bit careful. So has anybody ever stuck their fingers in an electric socket before a wall socket? No one? Okay, thank you for admitting it. I've done it like four times. It is not a good feeling, okay? So let's take a look at that situation using what we've learned, basically, okay? You accidentally stick one finger from each hand, you know, like your fingers and bam, right into the socket. Like, this sounds like a great idea. I'm gonna do this, okay? Okay, good, we have sound. How much current goes through your body under perfectly dry conditions? Let's say, okay, maybe I don't lick my fingers. Got chalk on my hands, my hands aren't sweaty. I shoved my fingers in. Turns out the resistance of the human body under dry conditions is about 10 to the five ohm. That's a big number, all right? Copper, in comparison, is tiny fractions of an ohm in resistance, so that's pretty big. But 110 volts is also a pretty big potential difference, all right? If you then calculate the current going through your body by taking that voltage and dividing by that resistance of the human body, you come to find out that the current going through your body is 1.1 times 10 to the minus three amp, 1.1 milliamp. So, will you feel this? Is it enough to kill? James, are you gonna feel this? From experience, go ahead and answer that question. Yeah, you're gonna feel it, all right? What happens when you shove your fingers into a wall socket? What happens physically to your body besides pain? You feel kind of a clenching, like opening, closing feeling, right? And that's because wall sockets are alternating currents, so the current direction is reversing 120 times a second, so you feel your muscles kind of clenching, unclenching, clenching, unclenching. But what's really, what's going on? The clenching and unclenching are muscle, what? For the physicians out there. Spasms, contractions, yeah. So you're basically spasming your muscles. You can't control your muscles. In fact, if you are in a situation where you are really grasping firmly onto something and it comes into contact with a potential difference at that level, you will be unable to release it. So it's very important if you're ever working in a situation where there's high voltage. And as a doctor, you may, if you're in the field, for instance, and there's been a downed power line or something like that, okay, where there's exposed wiring and a lot of water around, a lot of minerals in the water, you're gonna wanna have a buddy nearby, have a partner, somebody who in the event that you accidentally come in contact with a significant potential difference can knock you, literally run into you and knock you out of contact with it. It will potentially save your life, okay? So let's look at human biology and electric current. Point five to two milliamps is the threshold of sensation, okay? 10 to 15 milliamps and you'll go into involuntary muscle contractions, you can no longer let go at all, all right? So wall current probably won't do that to you. But again, if you're wet and your resistance changes, your resistance goes down if you have sweat on your hands, you can drive a bigger current through your body. Yeah, Ariana. Yeah, yeah, if you're able to feel it, it's somewhere between point five and two. If you can no longer control your muscles, it's between 10 and 15, okay? So there's a gray area in here, right? Between threshold of sensation and where you hit involuntary muscle contractions. These are rough rules of thumb. Don't play too closely with the boundaries of this. 15 to 100 milliamps, you get severe shock. You completely lose control of your muscles and breathing becomes difficult. It will now interfere with your respiratory system. You can no longer execute routine predictable muscle contractions. Fibrillation of the heart occurs between 100 and 200 milliamps. Your heart can now no longer execute a routine build up and release of charge on a rhythmic cycle. So you can no longer pump blood through your system. Death will now occur within minutes at this level. Greater than 200 milliamps, you will immediately induce cardiac arrest. Your breathing will completely stop and you will no doubt suffer severe burns at the site of contact with the potential voltage, okay? So with that in mind, humans have invented devices that are designed to disable our fellow human under conditions of perceived threat. Tasers, these are stun guns. They're electric guns. They involve the firing of a pair of barbed probes into the perpetrator. You aim for the chest. The goal is to create an electric potential difference across the chest, although you can really get it anywhere, neck, chest, legs, whatever, okay? Your goal is to now create a potential difference. The barbed probes separate when they're fired. They will stick into the person. They're like fish hooks. You can't pull them out easily. If you do, you'll do damage, okay? But you can't pull them out. In the immediate, the moment that they make contact with the skin, they deliver about 1500 volts potential difference to the human body, okay? So that's about 15 milliamps of current if you do the math. And that's in the range to paralyze a human being but not stop their breathing. So we'll close class with this, okay? So they're demonstrating a stun gun here. Normally there's sound with this, but whatever. So this is Rick Sanchez. He was a reporter for CNN and he's about to have a stun gun demonstrated on himself voluntarily. Now you'll notice there are people behind him who are going to hold him when he's shot because he's going to lose control of his body. And boom, he's shot. He can now no longer, okay? So they turn off the stun gun, all right? So he's completely out of control of his muscles at the time when the thing hits. Now depending on the conditions of a human being if they have a weak heart, if that's what they're talking about here, if they have a weak heart or something like that, it's actually possible to cause cardiac arrest. You can never know when you're shooting somebody for instance, whether or not they have a weak heart or some heart arrhythmia or condition. So there's been a lot of debate and discussion about these devices and when it's okay to use them. That's a social and a values and a moral and an ethical discussion. But now you can use Ohm's Law and understand why it is that human beings respond so poorly to being shot in the chest with an electric potential difference of 1500 volts, okay? So be careful. We'll pick this up on Tuesday.