 This is just going to be a little exposition to buy time for people to get here from other classes before I hand out the quiz. The quiz will take approximately five to ten minutes. I'll allow ten minutes, but when everyone's done, we'll just stop. So to pick up from last time, I demonstrated electric phenomena last time. To use some of the keywords, I rubbed a piece of paper, one material against a piece of plastic, another material, and induced something called the triboelectric effect. Basically what I'm doing is I'm transferring electrons from one material to the other via friction, and this builds up a separation of charge. This is a conductor. It's originally neutral, but charges are free to move on it. And if I put the charge wand near it, the can can be induced to change direction by the application of the electric force. And if I touch, let's see if I get this to soak up some charge. Okay, it's not soaking up enough. I should be able to get this to repel if I can transfer enough charge to it. Because this will take up the same net charge that's on this, and they will drive each other away. I also demonstrated the triboelectric effect a couple of different times to demonstrate that there are two kinds of charge, okay? So if I rub this glass, and then use this sink with the metal on it to drain the excess charge off me and the pipe, and then rub again, I can get the glass to follow, and I can slow it down from this side and get it to go the other way without ever making any physical contact. And this is the beauty of the electric force. Like gravitation, it's a force that acts at a distance. It doesn't seem to have anything to do with mass. If I were to double the size of this and then repeat my experiments, nothing would really change. So it's independent of mass. It seems to have something to do with a different property of material. And we've labeled that property charge, electric charge to be specific. Now charge distributions can be complicated. I know I've transferred a whole bunch of charge. I don't know how much. But enough that when I put it near this conductor, the charges respond to the electric force from the wand. They move over as close as they can get to the wand on the conductor, and the whole thing starts to roll. And then the charges move back up to the point closest to the wand, and then the whole thing repeats again. So I can get this to roll, essentially doing mechanical work with the electric force. That's a pretty complex distribution of charge. I mean, we've got a plastic cylinder. I rub the whole surface. I mean, who knows how that charge is distributed all over this thing? It's probably chaos down there, okay? This is an example of something called an insulator. Insulators do not allow charge to move freely on their surface. And that's in contrast to conductors like aluminum that do. So aluminum, copper, silver, gold, platinum, these are pretty good conductors. They're often used in applications in electronics manufacturing or in home wiring. Copper is used in home wiring because of their electrical conductive properties. They allow, with very little loss, electrons to move freely through the volume under the influence of the electric force. In contrast with that, insulators don't allow charge to move freely. You can trap charge on an insulator, but once it's there, it's basically stuck unless you mechanically remove it or have some other means to remove the charge. Okay, so this is why if you have a plastic comb and you comb your hair on a dry day, that's a plastic rubbing against a protein, those are dissimilar materials, there's the possibility to transfer charges from one to the other and your hair will frizz up and the comb will gain a charge and you might get a little electric shock if you're not careful. Does anybody know, incidentally, what's the trick to avoiding hair frizz? I mean, besides spraying your hair with some expensive product, anyone know? There's a cheap way to do it. You may not like the smell, but I don't know, it could be okay. Does anyone know how to remove static electric build-up in hair frizz? Olive oil. Olive oil, okay, well, you can do that, but that's smell and texture. So any other ideas that's less liquidy? Baby powder. Baby powder, dryer sheets. Dryer sheets are designed to eliminate static cling and clothe. You're supposed to toss one of those into the dryer if you haven't used fabric softener. There's a good physics reason for that. Chemicals there are designed to prevent the build-up of static electric charge. And so if you rub your hair with a dryer sheet, then comb it, you should reduce greatly the amount of frizz you'll have on a dry winter morning, for instance. Okay? If it's humid, water is very good at soaking up. If there's water in the air, it's very good at soaking up net electric charge that's sitting on a surface. So when you comb or brush your hair in wet conditions, like humid day, the frizz is less likely to happen. Of course, you'll have other problems with your hair, because it doesn't respond well to humidity, but that's a completely different problem, okay? So in principle, these charge distributions can be quite complex. And so if we think about something like glass for a moment, we're in a situation where, you know, who knows how the charges are distributed on this. But the beautiful thing about physics is that it teaches us about energy and matter and space and time, basically everything in the cosmos. And so by doing experiments, we can reveal what the structure is of things like glass or other things that are, you know, material in origin. And what we've learned over a long period of time is that things like glass are made from atoms. And in particular, glass has a lot of silicon in it. Glass is made from silicates, so essentially you can take sand and if you heat it enough, you can turn it into a liquid. And if you're good at glass blowing, you can make some pretty beautiful things out of that, okay? If you were to use a technique to try to look as deeply inside the glass or any other thing made of silicon, like a silicon crystal, like you might find in a computer, you would see something like this. Now, you notice this is quite a blurry image. Each one of these white blobs is the most likely location of a single silicon atom, a single silicon atom. To give you a sense of distance here, this is approximately 1 times 10 to the minus 10 meters or so. So we're talking sub-nanometer in size that you're seeing here. And it's not that humans aren't good at imaging distances this small. Actually, we're exceptional at it. We've gone into the subatomic realm deep inside the nucleus of the atom to see what's kicking around down there. But what you're seeing here are the limits of nature. When you get to something the size of an atom, what is meant by where something is or how fast something is moving loses all the meaning we take for granted in our macroscopic world. If you're driving down 75 to commute here or the Dallas North whole way, you're pretty confident that you're going 60, let's say 65, let's be honest, 75 miles an hour on your way to SMU because that gauge tells you, okay? So in our classical macroscopic world, the world we're raised in, the world we experience every day, we take for granted that I know where you're sitting and I know where you're not sitting. But in the world of the atom and smaller, there is a new, bigger, more general set of rules that takes over. As you scale things up, you record the behavior we take for granted every day, knowing where something is and where it is not, knowing how fast something is moving, exactly using some measurement technique. When it comes to atoms, we have no idea exactly where the atom is at any given moment or how fast it's moving. You trade one piece of information for the other and this is something called quantum mechanics. We won't learn about it in this course, but quantum mechanics underlies everything that we do every day. You take it for granted because its aggregate behavior is to make the world sensible and knowable and measurable. But when you get down to something the size of an atom, you have to accept a new set of rules that's more general than the rules we take for granted every day about motion and time and space. They work everywhere. It's just that you don't need these special rules when you get to something the size of, say, a long chain of proteins or bigger. A cell. Quantum physics doesn't really play a huge role moment by moment in the properties of a cell, but all the structure that underlies that relies on atoms and atoms can only be described by quantum mechanics. So in 1307, you learned essentially what physicists call classical mechanics. You've learned that time is an absolute thing, that everybody experiences the same way, that you can measure position with infinite accuracy and that you can use those to derive velocities. In the quantum world, you have to have a more broad set of rules. You have to be willing to accept that actually time is not experienced by all things the same way. That distances in space are not measured by all observers the same way, depending on their state of motion. This seems spooky at first, but measurement after measurement after measurement has revealed that this is the way it is. You're seeing individual silicon atoms here, not blurred because the camera's bad, but blurred because there's actually no physical way to measure precisely where the atom is at a given moment. And looking inside of an atom, we have to make cartoons. We have a lot of data that tells us what atoms look like, but there are no pictures of stuff inside of atoms. There's no pictures of an electron. There's no pictures of a proton. There are only all kinds of data about smashing stuff into other stuff and looking at what angles it all comes off at and then you reverse engineer what must be going on inside the atom based on that. There's no camera that takes a picture of an atom. So this is the classic picture of an atom that most people see in textbooks. It's utterly wrong. It's very helpful. It's sort of a planetary model. It's formally known as the Bohr model of the atom. It's very formally the Bohr-Rutherford model of the atom. And it was developed from experimental data in the early 1900s. And at its time, it was correct. It's remained because it's a helpful picture. But it gives you a sense of accuracy that you're not going to have in the quantum world, in the atomic world. There is an electron. There is an electron. There is an electron. There is a neutron. There is a proton. And look how close they all are and how similar in size they are. All wrong. There's no way to know exactly where any of those electrons are. Only the probability of finding them at a specific location. And that nucleus of the atom is about five orders of magnitude smaller than the atom itself. So it's essentially to, in a sense, it's sort of the sun in the center of a very big, if you're in the Oort cloud outside of our solar system, you're probably where the electrons are hanging out in an atom. And then the sun is that tiny, dim dot way down in the center that you can barely see. That's about appropriate for scale. So the Oort cloud objects only notice the sun mostly because of gravitation. A light from the sun is not so bright out there. But the Oort cloud gives us comets. And so the sun must have some influence on what happens out there. So this planetary model is very nice. You can put it in textbooks. It makes people think they know what's going on. And it's utterly wrong. This is more accurate. So this is from the Wikipedia article on the hydrogen atom. And what you're seeing here are S wave, P wave, and D wave states of a single hydrogen atom. And all that happens is that the one electron that's present orbiting the one proton in a hydrogen atom changes the probability of its configuration depending on its energy. So all those different shapes you see there. They're blurry. One electron. Not really sure exactly where it is. Only probabilistically where you might find it at any given moment. So the white spots indicate hot spots and probability, most likely. And the black spots down here represent least probable places of finding electron. Not impossible to find it there. Just not really likely. Luckily, because we have this picture of nature built on 200 years of observations, we can take a complicated charge distribution and we can represent it as a sum of point-like particles. As far as we know, the electron's not made of anything else. The proton we know is made of other things, as well as the neutron. It's made of the same things the proton is made from. But they're so tiny that for the purposes of this course, we can pretend like a proton and neutron are very point-like. The proton and the electron have the same magnitude electric charge. It's known as the elementary charge. It's denoted E. You should have read about it. And it's 1.6 times 10 to the minus 19 coulombs. Coulombs is the unit that we'll use for electric charge. It's a measurement of electric charge. And since everything is made from elementary charges, as far as we know, the sum of the charges of electrons and protons should tell you the total charge on an object. Now, most objects are electrically neutral. They have equal numbers of electrons and protons. And because they have opposite sign, they cancel each other out. So the proton, the proton charge, charge of a proton is positive E. Okay, so it's positive 1.6 times 10 to the minus 19 coulombs. And the charge on an electron is negative E. It's actually interesting. We have no idea why that is. That's either a happy coincidence or a fundamental observation of nature that we still have never understood. Yeah? Is the E one electric charge? What is the, oh, this one here? Or this one here? This one? This is known as the elementary charge. It was the charge measured by a man named Milliken using a famous experiment called the Milliken real drop experiment. That's an awful experiment, by the way. It's really messy and it takes a lot of time. But basically, he measured the smallest unit of charge that he could find anywhere in a material, and it was that number. And now we know that this is the charge carried by an electron in magnitude, but it has a negative sign in front of it. It's also the charge carried by the proton, and it's positive. Okay, so it is the building block of charge as far as we know. That's why it's called elementary charge. So, yeah? You could say one electric charge, that's what. Well, when I say one elementary charge, that's what I mean. But one electric charge could be, you know, 10 to the 10 coulombs or 10 to the 5 coulombs, in which case you'd have to divide it by that number to find out how many individual charges it's made from, individual elementary charges it's made from. So think of this as our building block for all charge distributions. And it happily coincides with the charge magnitude carried by the electron or the proton. Okay? All right, so we're going to use point-like charges as our building block for everything in nature. And so what we're going to do today is I'm going to work a problem involving Coulomb's law, which is the law that describes the electric force between two point-like charges. And then I'm going to give you guys a problem, and you can work it in pairs, so you'll partner up with somebody next to you. And when anyone gets stuck, we will stop and we'll talk about where the sticking point is. Okay? So first, let's get the quiz out of the way. I'm going to hand these out now. So if you put your, you know, note pads and stuff away. You know, thank you. As far as we know, everything is built from the ground up from point-like charges. Everything that we know is built from the ground up from elementary point-like charges In fact, as far as we know today, the electron can be best described as a perfect point in space. We don't know that it has any physical size whatsoever. Coulomb's law, which we're going to explore today by doing, is the law that describes the electric force between two point charges, and it will be our archetype for solving problems. Okay, so a couple of announcements for all of you. Your next assignment, I won't always do this. I'll expect that you'll look at the course of the last material page and then look ahead to see what your assignments are. But for next Tuesday, I want you to read chapter 22.1 to 22.2. So just the first two sections of chapter 22 on electric fields. There's an accompanying video. So it's up to you. You can read first and then watch the video, watch the video, and then read both at the same time and ignore both activities simultaneously. It's up to you. Okay, I will quiz you on material from these next Tuesday. Homework one, I actually still have to assign it. I forgot to click that button this morning, but it'll be due next Thursday by 9.30 a.m. I doubt any of you were looking at the homework right now anyway. If you haven't already signed up for Wiley Plus for this course, there's a link from the class webpage. I'll send another email around today reminding everybody where that is. If you have a code from last semester, it should still be good. If it's not working, contact the Wiley Plus, help people and see if they can get it working for you. Your numerical answers to questions will go in Wiley Plus. That's how the homework will be assigned. You're also expected to write up solutions to each and every problem and hand those in. I'll staple together and the teaching assistant will randomly pick one and grade it on process. If you're not sure what I mean by that, read the syllabus and you'll see what I mean. If you get a wrong answer here, it won't be counted against you here unless your process was just totally hosed. Okay? Any questions? Yeah. Well, you should be able to find me through Wiley from Southern Methodist University. That's the institution that you should search for. Southern Methodist University, it's not SMU. It won't show up as SMU. But either will work. And I'll send an email with the links later today. Okay? Okay, so answers to the quiz questions. Let's talk about these. So, very exciting. And now it's going to crash. So, I'll just do it verbally. And then I'll get that back up again. All right, so the first question. Which of these is true by electric charge? One, it is conserved. Whatever total charge you start with in a system you must end with. Two, it is not conserved. Whatever total charge you start with in a system can be changed. Three, there are three kinds of electric charge, positive, negative, and neutral. Four, there is one kind of electric charge. So, just shout it out. What do you think the answer was? One, two, three, or four? One. One, yeah. And one is the correct answer. So, it is conserved as far as we know and we measure this relentlessly. I worked on an experiment to measure charge conservation within the last 10 years. Whatever total charge you start with in a system as long as you don't change your definition of the system it will remain the same forever. Charge is neither created nor destroyed. You can start with a neutral system and then divide it into positive and negative charge and then recombine it back to neutral later, but the total charge will always remain the same. This is useful because it lets us understand nature at a deep level. We can always predict what's going to happen when charge is involved. And there are two kinds, positive and negative. So, that's the other key thing about charge. Let me see if I can get this back up. There's always little glitches at the beginning of the semester. That's what I get for using free software. You get what you pay for. Okay. Alright, so, let's just do this here. Okay, so, which of these is a true statement about the electric force as described by Coulomb's law? Alright, so, one, two, three, or four? What do people think it was? Three is the correct answer. So, force, just like in semester one, force is a vector. It has direction and magnitude. Okay, so, electric force has a scalar component. That's its magnitude. But it also has a directional component. That's the described by a unit vector that points in the direction of the full vector. Yeah. Actually, go back to the first one. Yeah. I put the right answer, but do we have a rod and we use one of those silk claws? Right. So, the rod is a system. The paper and the rod are the system together. Yeah, so that's why you have to be very careful about what you mean by your system. Whatever you're using together, that's the system, right? So, you include it, right, because you can also soak up or give up a little bit of charge. So, you kind of have to, like, isolate all three of those things from the environment. No water vapor to soak up charge. So, a physicist always has to start with the simplest assumption and then think, how could that fail? That's the trick with being a physicist, right? It's diagnostic in nature. So, you start with a simple assumption, just like with diagnostic medicine. And when somebody comes in with vomiting and a fever, you don't necessarily assume you bowl it right away. Okay, but you keep an eye on it. You think, okay, well, how could my assumption that this is just norovirus coupled with influenza be wrong? And then you think of three things that you might do if it turns out that person doesn't respond to normal treatments for that sort of thing. So, physics and medicine have something in common and it's diagnosis. Is there a way to know the reading? It's that we figured it out or whatever, but is there a way to tell if the cloth is getting worse? Off the top of your head? No. What you'd have to do is kind of go back to using when he assigned positive and negative and use those materials and then you'll know the answer and then kind of relate everything to that choice. It's tough, right? So, when we do problems, it'll be specified this is a positive charge, this is a negative charge because I don't want you to have to sit there and figure it out, right? That's really difficult actually to do. Okay, so force is a vector and I always like to remember it from the villain vector from Despicable Me. What did his crimes have? Direction and magnitude, right? Okay, so that's why he's called vector. He's also a terrible geek like me. So, which of these is true about this course? Physics 1308. According to the homework policy, I'm not allowed to work with other people to solve the homework problems. Is that true? No, you are. As long as you acknowledge your collaboration and say who you worked with and that write-up is your own independent work, your understanding of how the solution works, everything's fine. According to the grand challenge problem policy, my grade will only be made from the group grade on the solution write-up. Is that true? I see shaking heads. No, that's right. It's not true. Your grade will be 80% the group grade because it's a big write-up. Okay, it's like 10 pages. But on the final exam, you'll have some questions customized to the members of your team to make sure everybody on the team knew what was going on. Okay, so get your story straight before you go to the final. According to the syllabus, attendance during the class period is not assessed and doesn't count toward my final grade. Yes or no? True or false? Shaking heads. Yeah, I assess using these quizzes, so that should have been self-obvious by the fact that you had to put your name on the quiz. And according to the homework policy, all final answers to the homework problems must be boxed, have appropriate units, and adhere to the rules of significant figures and by the process of elimination, that must be true. And in fact, it is true, okay? Okay, so let's talk about solving problems involving coulomb's law. Now, I don't expect you to work in here, walk in here today having read chapter 21 and know exactly what the hell I'm about to do. You're going to learn by watching and then you're going to learn by doing, okay? The way that I've heard other physicists term this, this is what's known as the flipped classroom, okay? That's the formal name for this, where you do learning outside the classroom, which you're expected to do anyway, but now it's formalized, and then you come into class prepared to absorb a process based on some of the things that you remember from the reading, okay? So when you're doing your reading, when you're watching the videos, you should be taking notes, and especially if you don't understand something, you should come in with that question. And as I'm working through a problem, if we hit that question, then you should stop me called Dr. Sikul, I have a question because my back may be turned, and then let's talk about it, okay? So the other informal way I've heard physicists describe this is learn by watching the master, which I think is bullshit, okay? I hate that. I'm not a master, I'm a teacher. I make mistakes too, so you're going to learn a lot by watching me, including how to fail, which by the way is a very important life skill. If you don't already know how to fail, you may learn in this class, although I hope not in terms of your grade, just in terms of the process as we go, okay? I'm trying to prevent that as much as I can by having lots of opportunities in the class. But ultimately, you know, it's important for you to learn as we go through the course. There's going to be tough material. There's going to be stuff that just doesn't click. I don't expect everything to click the first time. I expect it to start clicking maybe the second time you see it, and then by the time you get to the end of the class, you feel comfortable and confident, almost like you could predict the outcome of the cosmos. That's where I want to get you, okay? I want the universe to seem rationally intelligible because crazy fact it is, which is already weird in and of itself, all right? So this is Coulomb's law, and that thing looks terrifying. What the... It's all of that, okay? So let me step a little bit through Coulomb's law. I'm an experimental scientist, okay? I have tremendous respect for my colleagues that do purely theoretical math-only calculations. But the truth is that we only make progress in the universe and understanding the universe together. You have to have people that are willing to go out and make risky observations with possibly seeing nothing, and then you report that, or possibly seeing something, and you report that. And then it's the job to sit down and figure out, well, what the heck just happened? Can I make a hypothesis that explains my observation? A hypothesis is a testable explanation that can be falsified by doing another experiment. That's the key feature of a hypothesis. You can prove it wrong. If you can't prove it wrong, it's not in the domain of scientific investigation, right? So some of you have heard of string theory. Anyone raise your hand if you've heard of string theory? It's very popular. We have no idea if it's true, and actually nobody knows how to falsify it. So is it even science is a fair question? Well, I don't know, right? But these are things that we worry about as scientists, okay? This is a mathematical description of a great deal of observation that happened in the 1700s and 1800s, mostly by chemists, but chemists and physicists and mathematicians and biologists, they were all kind of the same thing for a while there, until you needed such specialized knowledge in each of those disciplines. There's no way you could learn all of those things at once, okay? We're getting to a point now with like bioengineering and biophysics and biochemistry that disciplines are reemerging to solve even more difficult problems. But there was a time when you could be a person who knew everything in the field and do something with that. So people made observations. They made observations of putting charge on things and then how much force that exerts on another charged object. And a French scientist with the last name, Coulomb, Augustin de Coulomb, okay? He is the one who finally figured out he wrote down the mathematical description of what was happening in the laboratory. And then he tested it. Lots of people tested it. We've been testing Coulomb's law for a few centuries now. And as far as we can tell, it still works. It works great. And if you really want to boil it down, all of chemistry, adding in a little quantum mechanics, can be explained by that equation. And I know the chemists down the hall are like, oh, sure. It's much harder than that. It is much harder than that, okay? But at its essence, that's what's going on. So that equation and the gravitational equation are very similar. So how are they related? Did one come before the other? That's a good question. So in fact, the gravitational equation did precede this equation. It was, in one form or another, it was essentially Newton, Isaac Newton, working in the 16th century who wrote down that equation. But this, as far as we know, it was very exciting to see this equation in many ways because you look at it and think, oh, yeah, this is just like, you know, G, the gravitational constant, times mass 1 times mass 2 divided by the distance between the masses squared. And then this thing has all the direction information and about where the force points. But as far as we know, mass and charge are completely unrelated phenomena and all efforts to get the theory of gravity to play nice with the laws of electricity and magnetism and other forces we've discovered since then have failed. This is a great puzzle. How does gravity fit into the picture that's been built up from what we understand about electricity and magnetism? We have actually united a great deal of the cosmos into one equation starting from the study of electricity and magnetism. But gravity has always been its own thing and all attempts to merge them have not led to any testable consequence that we can actually go into the lab and assess. String theory is an attempt to do this, to unite all of these things. But again, as I said, no one's figured out what the test of string theory is yet. So it's interesting. Maybe there is a clue deep in here someplace that we just haven't been smart enough to figure out. But as far as we know, there's no direct relationship between the two. Right now it's coincidence that they have the same force. All right, so there's this number, K. Let's talk about that. K equals 8.99 times 10 to the 9 Newton's unit of force times meter squared unit of distance squared per Coulomb squared unit of charge squared. What's the job of K? It is known as the constant of proportionality. It was very clear from experimental measurements, observations of electrical phenomena, that more charge equals more force, that more distance equals less force. So those pieces were actually fairly easy to imagine shoving into an equation. But forces measured in one system of units and charge and distance are different units from that. And so you have to have some number that relates Q1 and Q2 and that distance in the denominator R12 squared to force measured in Newton's. And that's why it has these units. Coulomb squared in the denominator to cancel out the Coulomb squared you get from Q1 times Q2, meter squared in the numerator to cancel out the meter squared you get from R12 squared. So this is just known as a constant of proportionality. It must be measured. It is not determined by thinking. You have to go to the lab and blood and sweat and money and get it done. And so that's what was done. It was measured. And that's the value that was determined, at least in our current system of units. That's its value. So it happens to be related. We learned later to a more fundamental constant known as epsilon naught. And I'm going to write this down because it's a very important number. But let's just kind of hold it in the back of our minds right now and have some good gospel minutes. It is 8.85 times 10 to the minus 12 Coulomb squared for Newton meter squared. So you notice it has the flip of the units I wrote out there. It was one over those units. And they're related to each other. K is one over four pi epsilon naught. Epsilon naught turns out to be the more fundamental of the two numbers. It will come up again later. I'm going to read this in a few weeks for some other things. But I'm going to tend to use K just because I don't like having to shove one over four pi epsilon naught into that equation all the time. I think that's always a calculator at a time. So it's a good idea to keep this number in the back pocket. But this number turns out to be more fundamental. And in fact, it has deep consequences for the cosmos. That's a teaser. I like to drop those in there. No one cares. It's Thursday. Everyone wants to get out of here. No Friday classes. There's a problem now. And I'm going to use this. And as I go through the problem, I will motivate what's going on with each of those pieces. And I want you to see the overarching theme of solving these problems. Divide and conquer. Divide and conquer. There are five things in that equation on the right-hand side that if you want to determine the thing on the left, you have to figure out. You got to figure out K. No problem. That's given. Constant never changes. Q1 and Q2. Well, that kind of depends on the charges you're given in the problem. Well, hopefully you're given those. We'll see. This thing. R12 squared. So this is the distance between charge 1 and charge 2, that quantity squared. That's what that means. So it's a distance. It is a scalar. It's a number. And hopefully you'll be able to figure it out from the problem, or you'll be given it directly and maybe have to figure something else out. And then there's this terrifying object over here. And I remember the first time I saw one of these. And if we had WTF as an expression at the time, because I'm that old, we didn't have that. I would have said it, because I don't know what that is. What is that thing? Does anybody know what that thing is symbolically? Commit. Is it a unit vector? It's a unit vector. Exactly. What is a unit vector? It describes its position in like a 3D. Yeah, 3D or 2D. Yeah, some system of coordinates. Yeah, all it does is it tells you where that vector f points. So it contains the pointing information. So in a sense, I like to think of vectors as directions to your friend's house. If your friend, you say, oh, I'd like to come over to your house, but I've never been there before. How do I get there? And they say, oh, no problem. You go north and you go east and you're there. What did they do? They gave you a unit vector. Direction in direction and you're there. That's useless. How far? How far do I have to go north? How far do I have to go east? So if somebody gives you just, hey, you go north and then you go east and then you're right there, it's my front door, no problem. They've just given you a unit vector, which is great because you needed that information, but you also need to know when to stop. So the rest of this stuff, all these numbers out in front of the unit vector, they tell you distances. How far north? 10 blocks. How far east? 6 blocks. So a proper full vector involving directions to your friend's house are 10 blocks north, 6 blocks east, and you're there. Great. Just 10 blocks and 6 blocks doesn't help. And just north and just east doesn't help, but that information together will let you pinpoint a place in a two-dimensional space. So the unit vector, north and east, the magnitude, some combination of 10 blocks and 6 blocks, and then you can put that all together into a single vector. So we'll come back to that. But yeah, that's a unit vector. It has only direction. Its magnitude is length 1. It only has a length of 1. That's the great thing about unit vectors. Now, I'm going to spice up the problem a little bit. I'm going to throw a little biology out here. But fair warning. I'm a physicist. And that's what biology looks like to a physicist. Bird, bird, bird. That's bird over there. There's some bird. So yeah, you probably know more about biology than I do. But what I like about biology is deep inside of it, there's embeddits in physics. So in particular, one fascinating structure that I like as a physicist, probably for different reasons than biologists care about it, is the cell membrane. The cell membrane is this really beautifully complex structure. And when I look at it, I see this sea of interesting physics that's going on. Ions can be pumped across the cell membrane to change the charge concentration inside and outside the membrane. And those pumps are really awesome. I mean, there are little electric engines that are capable of accelerating charges into or out of the cell. And so here, for instance, you have a pump that's capable of pumping potassium ions, K plus, from the inside of the membrane to the out. There are also chlorine pumps and sodium pumps. They change these gradients, these charge concentration gradients across the cell membrane. I'll come back to the cell membrane later. It's actually an interesting example of a physics device known as a capacitor. And in fact, its thickness was first measured using physics because no one could actually see with a microscope the thickness of a cell membrane a long time ago when people cared about how thick these things were. So instead, one had to appeal to physics to figure out the answer. But that's a whole story we'll come to later. I'm more interested in those ions. Ions are merely neutral atoms that have been robbed of electrons to make them positively charged or had electrons added to them to make them negatively charged. It kind of depends on the chemistry involved as to whether or not there's any place to shove an electron or take one away and how hard that would be. Okay, but we have potassium ions, K plus, we have sodium ions, Na plus, chlorine ions, Cl minus, and then these anions, which are essentially proteins that don't permeate the cell membrane through those pumps. And in here, at least in this little depiction that you can pull off Wikipedia, the ones that they've chosen to represent here have a four minus charge. That is, they carry four negative elementary charges. Okay, so let's think about ions near a cell membrane. So here's a physics problem involving these that really gets down to it at its heart. It has Coulomb's law. Okay, so what's the force that the potassium ion, Na plus, okay, exerts on the sodium ion in this picture, Na plus up there. All right, so you've been given a few pieces of information. We're told that the distance from the anion, which is sort of in this lower corner of the picture, is L and then another distance, L. So it's 2L from here to here and it's 2L from here to here. All right, so we essentially have, and you can see that this is a nice 90-degree angle here, okay? So we have a nice two sides of a right triangle, all right? So as a physicist and somebody who likes math and has to use it all the time, I look at this and without even starting to think about exactly what I'm going to solve, I'm like, oh, cool, okay, great. So there is definitely a right triangle in here. Maybe I can use some geometry later to figure some stuff out. I've been given distances, okay? I've been told that this L, okay, and the full distance here is 2L, but this L here is 2.8 times 10 to the minus 9 meters, okay? So that's a useful piece of information. I've also been told that the charge of the sodium ion, potassium ion and the sodium ion are both the same and they're both plus one elementary charge. So again, E is 1.6 times 10 to the minus 19 coulombs. And since the potassium ion is positive and the sodium ion is positive, they both have the same electric charge. So without doing a single calculation, I can already look at this and go, like charges repel, dissimilar charges attract. These are life charges. I expect the sodium ion to move away, to be pushed away by the force from the potassium ion. So I've done no math. I've used physics. I've used the physics of like charges. And I've already come up with a hypothesis or at least something I can use to assess my math later. And that is that I expect the sodium ion to be forced away, pushed away, repelled by the potassium ion. The potassium ions doing the pushing, the sodium ion is the recipient of the push, okay? So the potassium ion is the source of the force. That's what we've been told to consider. That's the source of the force. That's the thing doing the pushing in Coulomb's law. The sodium ion is the recipient of the push, and that's where we want to figure out what's the force. Okay? So the thing to do now is, I think what I will do is I'll start over here, is to jot down Coulomb's law. So I'm interested in finding force. Force is a vector. If I'm asked for the magnitude of the force, then I only have to find a number. If I'm only asked for the direction of the force, then I only have to find a unit vector. But if I'm asked for the force, I've got to find both. I've got to find the whole thing. So the force I'm interested in is the one that the potassium ion exerts on the sodium ion. So this is a bit of a clunky notation, but hopefully it's crystal clear. The force that potassium exerts on sodium is going to be equal to, alright? And now we need Coulomb's law. These are given to us as charges. There's no information that tells us we need to do anything other than assume that these are point-like. So if you think you have point-like charges, you bust out Coulomb's law and get started. Alright, so I'm going to do that. Busting out Coulomb's law. K is a number. We've got it. One-fifth of a problem done already. QK, so this is a different K. This is potassium K. QNA all over the distance between sodium and potassium squared and then a unit vector that points from the sodium to the potassium and has some actual form to it based on my coordinate system. Okay, well, let's look at some of these other things. We've got five things we have to figure out. Charge of the potassium ion. Give it. Done. So we have K. We have QK. Charge of the sodium ion. Give it. Done. Okay. The process. What's the process? First, look at the problem. Maybe make a new picture for yourself. Okay? I'll do that in a moment. Maybe make a new picture for yourself. Redraw it. Get to know the pieces of the drawing a little bit. This distance here is 2L. That distance there is 2L. This looks like a right triangle. File that away. It might be useful later. What are the knowns? The knowns are K. The knowns are QK and QNA. We've got all that. The mass was given here. Now, so those of you who are doing this for the first time, you might think every piece of information is always useful. Every piece of information is not always useful. Okay? It turns out because, and this is a good question that Jody asked earlier, all right, is there a relationship between the gravitational force and the Coulomb's law that describes the electric force? There's no mass in here anywhere. We don't need mass. Now, given the force, what could we use mass and force to find? Acceleration. Acceleration. Right? Newton's laws still apply in this course. F equals MA. You will use it. All right? So file that away as a pro tip. Okay? We're going to be using stuff from 1307 in this class. Newton's laws are still in effect. All right? And that doesn't change. But we have a new force that we can shove into F equals MA. And given M, we can find A. All right? So yeah, so mass might be helpful for acceleration if we had to find it. But am I going to do that? All right? I'm just going to focus on this. Now, we need to know the magnitude of the distance between the potassium ion and the sodium ion squared. And we need to know this distance unit vector. We do not have these things given in the problem. So we're going to have to calculate. And let me do a little asop. Vectors. I like the following notation for vectors. Hopefully you've all seen this. Okay? If I have some vector, V, I can write it as an algebraic sum of its components using unit vector notation. So for instance, let's consider a two-dimensional space where I have a y-axis and an x-axis. I can denote directions along x using the i-hat vector. This is a unit vector that points along the x-direction. And like all unit vectors, its length is one unit. One. That's it. So if I wanted to write down the length of a vector, the really pedantic way of doing it is to say, okay, I have a vector i-hat. I want to find its length, which is denoted by putting these vertical lines around the vector. And that means the magnitude of. Okay, so, you know, the English version that goes along with that symbol is magnitude of the thing inside of it. Okay? So I like to think in terms of language, human language sentences when I'm looking at a math equation. What is this equation telling me in my own language? Because math is a language and you do have to translate from it sometimes. Okay? So that's a unit vector, and that means the magnitude of a unit vector. This is written most pedantically as the square root of the dot product between the unit vectors. i-hat dotted into i-hat square rooted. The dot product is where you just take the components of the vector, multiply the x component by itself, the y component by itself, and the z component by itself, and sum them up. That's what the dot product is. So in two dimensions, it's like the Pythagorean theorem. To get the length of the vector, you take its components, square them, and sum them. It's just the Pythagorean theorem all over again. Okay? Because this thing only points in one direction, it only has one component, and that is the x direction. And since the length of a unit vector is always one, this thing must be one, which means that it's equal to the square root of one, which is also equal to one. Okay? So this is the cool thing about unit vectors. Dot it into themselves, they always give you the number one. I tell you this, not because it's a trivial fact about unit vectors, but because it lets you assess whether or not, later on, when you've calculated a unit vector, if it truly is a unit vector. Okay? So this is the really pedantic way of writing all this out. It can save you. So you don't always have to do this. If you know, look, it's a unit vector. Its length is one. I'm just going to write that down. Then write it down. That's good. Okay? But if you ever need to test that assumption, now you know how to do it. Do the dot product formally. Any vector can be written in terms of unit vectors. So v vector can be written as the x component, vx, times the unit vector that points along the x direction plus vy times the unit vector that points along the y direction, which is written as, this is my crummy j. This is j hat. If you have a z component, then there's also a vz times k hat. K hat is the unit vector that points along the z direction. Okay? So what's nice about this is if I ever need to calculate the length of v, I just do that really pedantic thing I just did. I take the dot product of v with itself and then I take the square root of that dot product. So the dot product of this is going to be vx i hat dotted into vx i hat plus vx i hat dotted into vy j hat plus vy j hat dotted into vx i hat plus vy i hat dotted into vy j hat. All I'm doing is distributing the multiplication. Okay? So that's what v looks like. It's got two components and if I dot it with itself I get four terms in the result. I'm just distributing the multiplication. Okay? Now here's another cool thing about unit vectors. If the unit vectors point along directions that are at 90 degrees to one another like i hat and j hat do an x axis and a y axis their dot product is automatically zero because the dot product tells you how much of the second vector lies along the first. So I give you a number. If none of it lies along the first then the answer is zero. And that's nice because if you see any term that has i hat dot j hat that's zero. j hat dot i hat that's zero left with this. The square root of vx squared i hat dot i hat plus vy squared j hat dot j hat. What's i hat dot i hat? One. What's j hat dot j hat? One. Okay, this is great. So surprise, surprise you've recovered the Pythagorean theorem. Very exciting for a Thursday morning, right? I do this not because I expect you to do this every time with vectors. I do this because I want you to understand the power of this particular notation. It lets you do algebra with unit vectors and all you have to do is remember a few simple rules. A unit vector dotted into itself gives you a length of one. A unit vector dotted into another unit vector that's at a right angle to it is zero. That's the great thing about using unit vectors to denote coordinate directions like x and y and z. Okay? Questions on that? Because that's an essential ingredient in what we do next. I know, thrilling, right? Trust me, you'll be thankful when you start sitting down and solving problems that I'm doing this. This is what I don't normally get to do during the semester because I'm lecturing about Coulomb's law and not solving problems. What are the pieces we need? We need r, k, n, a squared. We need r hat, k, n, a. We do not know what those things are yet, but we can figure them out from the information given. Let me draw the picture. Since we're only interested in the sodium ion and the potassium ion, I'm just going to draw them. When you draw a picture, it's very convenient for you to decide what you'd like your coordinate system to look like because everything you do with vectors after this will be done based on the coordinate system you choose to solve the problem. This is a problem involving a bunch of straight lines and right angles, so Cartesian coordinates are an excellent choice for this. An x-axis, a y-axis, and they're at 90 degrees to one another. The only other choice you have to make is where to put 0, 0 in the coordinate system. I like to put things so that, like in this case, because that lies at a 90-degree angle to that, I'm going to put the origin of the coordinate system right here below the sodium ion and at the same level as the potassium ion. If I'm going to draw my coordinate axes, I've drawn them up there in a different place, but I can move them anywhere I want, as long as once I write down my coordinate system, I stick to it, like glue. You commit to your coordinate system and you will not go wrong. You start changing your coordinate system mid-calculation, you're going to get all kinds of crazy results. So once you pick it, stick with it. And I'm going to commit to this. That's a choice. You can make a different one. There are other choices that may make this problem easier. But I'm going to pick this one. Okay? Okay, well, I know a few things. I know that this distance along the y-axis is 2L from the picture. And I know that this distance along the x-axis is also 2L. So I have a triangle with two sides that are the same length. And I have to figure out the distance between the sodium and the potassium ion. And that forms a hypotenuse on this right triangle. So Pythagorean theorem is going to come in real handy for this. Now, a convention in Coulomb's Law Problems. The vector, R1, in this case, R, K, and A. The full glorious vector with direction and magnitude by convention always points from the source of the force to the thing being pushed or pulled. Okay? That's a convention to be memorized. That convention will remain the same throughout the entire course. And when we do magnetism later, we'll have a similar equation that describes magnetic force. Always the same. The vector that tells you the distance and direction from the source to the recipient always starts on the source and ends on the recipient. The tail of the arrow is on the source, the head of the arrow is on the recipient. So by that convention, this vector R must point like this. That must be it pictorially. This is why pictures are great. You get all this down and you can start breaking the picture into numbers and algebraic symbols and so forth. So that pictorially is this full glorious R vector whose magnitude is R and whose unit vector is R hat. Okay? Okay, great. So now I need to know the magnitude of that thing because that's one of the unknowns. The magnitude squared. So if I can just figure out the magnitude, I can get the magnitude squared. I can figure out the magnitude of this. What's that? Okay. What theorem should I use to do this? Okay, a squared plus b squared. Pythagorean theorem, yeah, exactly. a squared plus b squared equals c squared. So a squared plus b squared equals c squared. c squared is the hypotenuse length. Okay, so I know that R, k, and a squared will be equal to 2L squared plus 2L squared 8L squared. 4L squared plus 4L squared is 8L squared. Awesome. And then I can go one step further. I can take the square root of that and just get that length. So it's this annoying thing squared of 8 times L. All right, so let me box all this. All right, so with a little geometry, little Pythagorean theorem, we're done. We've got four of the five pieces of Coulomb's law now. Our only thing to do is to get the unit vector. And this is the last thing I want to demonstrate to you here. And then we'll just press ahead. So unit vectors, how do you get R hat? Well, a unit vector is a vector with length 1. So if I take any vector and divide it by its length, that by definition will give me a unit vector. So if I square that thing and take the square root, it will always yield 1. All right, so to be formal about this, R vector divided by its own magnitude will automatically yield something of length 1. I can demonstrate that very quickly. R hat dot R hat equals R vector dot R vector all over R R. So this thing up here is just R squared and this thing down here is just R squared and surprise, surprise they cancel 1. So any time you need a unit vector, if you have the vector and you have the length, you can get the unit vector. If you have the unit vector in the length, you can get the vector. We're going to go from the full glorious vector to the unit vector. So the full glorious vector we need to write down and I'm going to do this in unit vector notation. We need that thing, that beast, the full beast. Well, R vector points from here to here. It has an X component, it has a Y component. To go along X from the potassium ion direction to the origin. And then to get to the sodium ion I have to walk up in the direction of the positive Y to the sodium ion. The distance I have to walk here is 2L. The distance I have to walk here is 2L. These are like directions to your friend's house. You go west, you go north. You go west 2L, you go north 2L and you're there. So we can write that down. You go west 2L. Now I hat. That's what west means on this picture. Negative numbers. And then you go north 2L. J hat. Done. That's the vector. I can simplify it a little bit. I can make it look prettier. I can write this as just 2L negative I hat plus J hat. Okay? That's a little bit prettier. That's about as far as I'm going to go. Because the only thing I have to do now to get the unit vector is divide that by its length. 2L negative I hat plus J hat all divided by square root of 8L. Now what's nice about this is L cancels with L here. Okay? And we have a dimensionless no meters left in this thing. A dimensionless thing. This is a direction only quantity. It should not have meters lurking in it. Or coulombs or any of that stuff. If you see any units left in this there has been a terrible, terrible mistake. That's another way you can check your math. Okay? So we have all the pieces. We got K. We got QNA. QK. We've got the magnitude of the well, we've got this thing, right? We've got the magnitude of the distance between K and NA squared. We've got the unit vector. And now I can write this all down in coulombs law. K, QK, QNA all over R squared. So that's going to be 8L squared. And then 2 over root 8 negative I hat plus J hat. Now I can simplify that further. I can cancel that 2 out but really all that's left at this stage is to actually plug in numbers. So this is where you bust out your calculator. You notice I never took a calculator out this whole time. I strongly caution you to never enter numbers into your equations as you write them too early. Why? Because now you have to carry numbers like 6.5 times 10 to the minus 26 KG line after line after line after line or the charge 1.6 times 10 to the minus 19 coulombs line after line. You will make mistakes. You'll make transcription errors. Algebra was invented for a reason so that we can symbolically move things around without having to carry all the big numbers around as we go. So I stuck with algebra and vectors in this notation right through until I can't go any further really with algebra. Again I could simplify this. I could change that into a 1 I can plug that in my calculator too. So I know I have numbers for Q's I have numbers for K so I can get this number out in front by plugging into a calculator. So before I set you on a task for the rest of the lecture questions you've only seen it once now I do not expect you to have simply mastered the osmosis you're going to master by practice. So the last part with the 1 over part of the force that's not part of the unit That's right. This is all kind of bound up in one formula now but this thing out here is just a number and it multiplies the I hat and J hat thing over there. So the last step usually in a problem like this is to figure out what that number is up in front that's usually what you have to enter into Wiley or you might have to enter the X component and the Y component so whatever number is in front of the I hat at the end whatever number is in front of the J hat you enter those into Wiley. Let's just check one last piece of physics here because our intuition from knowing the physics of like charges is that this vector ought to point in such a way that the sodium ion is trying to get away from the potassium ion and sure enough the force vector points in the same direction that the R vector points so no sign flips occurred if any of these charges were negative if either of these were negative then I would have one more minus sign floating around in here that would flip the directions of these vectors but I don't have that all the signs that you need are written there already and so the force points in the negative X direction that's its X component and the positive Y direction that's its Y component which means it points this way away from the sodium ion and the charge physics intuition should have suggested so that's the last check at the end did I get all the signs right did I shove any negative charges in when I wasn't supposed to because if you did that it could have screwed the whole thing up so those are the little things you plug the number into Wiley first and wrong start asking okay wait was my unit vector a unit vector so I could try taking the dot product of this with itself do I get one my force vector pointing in the direction I would have expected for these charges et cetera et cetera so here's the problem I want you guys to work on so pair up and ask questions as you start on this same picture same L what's the force that the potassium ion and the sodium ion exert together on the anion the anion has charged negative 4E so my tip calculate the force vector due to the sodium ion first calculate potassium ion calculate the force vector due to the sodium ion second and then what do you do to get the total force if you have multiple force vectors involved add them like vectors so again divide and conquer find one force find the other force and then add them together as vectors divide and conquer so pair up with somebody next to you most of the cases there are even numbers of people there should be an odd number of people in the class but I'm not sure all 35 are here so pair up start talking to each other and work through this I want to hear noise and if I don't hear noise I'm going to start asking you questions about how the calculation is going so come on better than me talking at you for the next 15 minutes so