 Okay, let's go ahead and get rolling here, just some introductory stuff. So a few concepts to get us kind of moving here today, we'll exercise some new stuff today. Let's remember that electric charges, that's the heart of this whole course, they exert forces on one another via the electric field and that's always present whether or not a charge is there to be acted upon. So as long as there's charge somewhere in the universe there are going to be electric fields somewhere in the universe and it's possible for those electric fields to exert forces on other charges even if none are present. Okay? You have to put a charge there to see the force but you can calculate the potential for a force or the potential for energy even if there's no actual charge around. That electric force is a conservative force field that means it has an associated potential energy which we call the electric potential energy. Energy can be stored in and released from an electric field and we're going to look at that in a little bit of detail today. The energy available to do work per unit charge is what's called the electric potential. It's often very easy to confuse electric potential or electric potential difference with electric potential energy. This is an unfortunate nomenclature but we're stuck with it, it's the convention that the field uses and by the field I mean physics. So we'll have to kind of live with that and just exercise our brains a little bit. So just practice it. There's potential energy and there's potential and potential is the energy per unit charge that's available in the field at some point in space, okay? And that thing just like the field exists whether or not there's a charge present in the electric field. Okay? Alright, so boilerplate stuff. So some announcements, you've got some reading for next class. This begins our discussion of capacitors. This will be the first device that we look at that takes advantage of charge and electric field and has the potential to do work on the system. We'll kind of construct a protocapacitor today and then we'll look more at big capacitors going forward. So there's a video associated with this but you only need to watch the first three quarters of it basically. So you can just watch the first 43 minutes. So if you want to keep watching, that's fine but the rest of this will be signed on Thursday. So you want to get a little ahead, that's your business but you can quit at 43 minutes. Basically when you hit the word dielectrics, stop. Homework three is due Thursday by 9.30 in the morning and homework four will be assigned on Thursday but it's not due for two weeks because you have an exam next Thursday, okay? And the exam will cover homeworks one, two, and three. Alright, so more about that in a second. One challenge problem. So team lead editors, if you know who you are, please mail me the minutes from your first meeting by 5 p.m. on Thursday. You should be having a second meeting sometime this week. So again, once you take some notes at those meetings, email them to me when you're done and I'll enter them into a little folder where I keep records of how things are going. Yeah. By minutes, do you just mean we tell you what we did and what we talked about? Yeah. And it doesn't have to be detailed. It doesn't have to be so-and-so said this and so-and-so argued this and you can just keep it high level, like what things were discussed and roughly what order and were there any to-do items at the end, anything, any questions nobody could answer and no one knows how to find an answer to, things like that. So this is all fodder for our first team meetings. So segwaying into that, our first team meetings, those are meetings between your team and me and this is 30 dedicated minutes you get of my time to talk about anything related to the grand challenge problem. We'll start having those meetings. I'm going to try to schedule these for the 25th all the way through the end of day on Wednesday, September 30th. So lead editors, please watch for my emails with sign up slots for meetings. Okay. So I'll send you my schedule availability and if nothing works from that, we'll find a way to make it work. Okay. All right. Exam one. So we have our very first of four total exams in the class. It's next week in class, this room, Thursday, September 24th. We're going to use all of Tuesday to review. So the thing I hate about exams is that it adds a dimension that you don't really have to the homework unless you're a pig procrastinator and that is time pressure. So doing physics when you've got seven days to do it is very different from doing physics when you have 80 minutes to do it. All right. So the style of the exam and you'll see a sneak peek of this on Tuesday. The questions will be very similar to the kinds of questions you have in Wiley Plus or the kinds of questions I'm assigning you guys to work in class during these periods. Okay. So there may be some biology, chemistry mixed in, but basically what I'm looking for is for you to get at the physics of the problem. Try to use physics ideas to solve the problem. It will cover the material that was tested on homeworks one, two, and three with all the associated class notes and problems that go with that. If you want to study for this, I recommend looking over all the material. I just suggested and look at any practice problems in the chapter as well. They'll often have steps worked out in a very detailed way. And so if you get lost and you need help, you can get help that way. They're also, you know, Wiley gives you access to study problems and things like that too. You know, take advantage of this stuff. But again, we're going to do Tuesday, we'll do an in-class review and it will be partially timed and partially relaxed because I want you to feel that time stress before Thursday shows up. I want you to get used to that so that you're ready for it on Thursday when it's all you and we'll go from there. What's up? Yeah. Question? What's your stance on formula sheets? Oh, that's a good question. I will be providing you with a formula sheet that has all the formulas you're likely to need to solve the problems, including things like converting micrometers to meters and all that stuff. Yeah, look at that. Jacqueline just huge relief, right? Great. I don't know if I remember what a centimeter is anymore. Yay. That'll be on there. But there'll be more information than you require. So you're going to have to understand. What I'm testing you on is not your ability to regurgitate a formula and plug in a bunch of numbers. This is an engineering school. Okay? No. Okay. All right. Nothing. All right. So this is physics and I'm trying to test your ability to think your way through a problem that you may not have seen before using things you have seen before. So my goal in testing you is to see if you can put ideas and concepts and math together to solve a problem. That's what I want to test. Okay? Yeah. And I should say that the exams consist of two parts. The first part is multiple choice. The second part are problems you have to solve. The multiple choice will be drawn in part from quiz questions that are given during class and there'll be some new stuff on there too that wasn't already phrased in the form of a quiz question. Okay? So again, what you want to do is to study for the multiple choice, you want to review the core concepts in the class. So you know, look at these little slides I give you here on concepts reviewed from the lecture. So Sophie. Will we have all of our previous quizzes and homework to your back? That's my goal. Yes. We'll find a way to get it back if we see if we miss a class period for it, but that's my goal. So okay. All right. Any, yeah. Lucy. How much of like homework is zero? Because it was like calculus review. Yeah. Well, that underpins the stuff you've been doing on the homework problems. Right. So that's foundation. Yeah. But I mean, you know, in principle, we've exercised at least a bit of everything in homework zero so far, right? We've done at least one integral, we've, well, I guess we haven't really done any derivatives per se yet, but they've been implied in many places. Vectors we've used, things like that. So top products have been in there. Yeah. So in principle, we've exercised a lot of homework zero already. Homework zero is to spot check it. So if there's still something you feel weird about, like integration, talk to me or practice it or do both. Okay. Any questions? All right. So here are the team names so far. So thanks. If I missed somebody, say so right now. But this is what I found in my inbox after going back through my archived mails. So we've got Team Alpha, now known as the Kugul Ohms, and they have a lead editor, Darcy, that's you, right? So if you have some notes from the first meeting, which I know you guys had already, just shoot them to me and I'll put a min next to the min means I got minutes from the team. Okay. Team Bravo, the thick neutrons, I got a lead editor for that in minutes and Team Charlie, the home lighting physicist, I don't know, lead editor in minutes. And then I just need Delta Echo and Foxtrot. So yeah. Okay. All right. So, okay, so I forgot to write that, isn't that? All right. So apologies. Okay. Yeah. I'll just send you an email. Okay. Oh yeah. All right. So there's two more that I have. Evan, did you have a question as well? Sorry. I thought I saw a hand come up from over there. Okay. I'm just getting seen off. Okay. Anything else? All right. Let's go to quiz times. We've time traveled briefly into the future, but now we're back again. So time travel sucks. Such a tease. All right. Make sure you put all your notebook in there. Thanks. Okay. All right. All right. Let's look over, we'll travel back to the future now and we will look over these, something, something. There we go. Answers to the quiz questions. Okay. Question one. Here we go. Question one. You have a group of charges. Okay. How does one determine the total energy stored in their electric fields? Is it one, calculate the work that was required to place each charge in the group, for instance, one at a time, and then sum the individual works to get the total energy? Anyone for one? Okay. A few. Two, calculate the work that was required to place each charge in the group and then multiply the individual works to get the total energy. Okay. Three, compute the average distance between the charges. And four, compute the total electric charge in the system. Okay. So we have some takers for one and one indeed is the correct answer. So in general, if you have an assembly of charges, and then any of them could be five, could be Avogadro's number, in which case you're going to want a computer or some other trick. We calculate the total energy is stored in their configuration. All right. So however they're assembled, laid out on a piece of paper on grid lines or not. Okay. What you want to do is kind of imagine what it would have taken to assemble that group of charges in the first place by putting all of them out to infinity to begin with and pulling them in one at a time and placing them. Okay. Actually, what's cool about energy, because energy is just a number. Energy's not a vector like force. And energy really is the fundamental thing in nature. Because energy is just a number. It doesn't matter what order you assemble the charges, as long as they all wind up where they're supposed to be at the end. So you could bring in charge, what you label charge one first, then two, then three, then four. Or you could do 50, and then 47, and one, and then three, as long as you put them all where they were at the end of the game, no matter what order you put them in, you'll get the same energy out at the end. Okay. So that's the way we're useful as later in today's class. Okay. Question two. Which of these is the electric potential of a point A, okay, instead of P for a change? Some distance R from an electric point charge Q. So we have a point charge Q, we've got some point A that's some distance R away from the point charge. What is the formula that describes the electric potential at that point A? Is it one, VA equals KQ over R squared? Is it two? VA equals KQ over R cubed? Is it three, VA equals KQ over R? Okay, a few on that. And is it four, VA equals KQR? Okay, and the answer is three. Yeah. So this looks, again, very similar to Coulomb's law, which is not an accident. You can figure this out, right? This was discussed in the lecture video, but if you have a point charge, you, or if you can look in the text, if you have a point charge and you write down its electric field and you want to figure out what's the electric potential of any point, assuming, let's say, start from infinity and move out to that point, which may be very close to the charge, this winds up being the formula you get. Okay, we'll exercise this a little bit in a moment. All right, finally, circular coordinates are written in terms of what variables? One, X and theta. Two, R and theta. Okay, three, X and R. And four, X and Y. Okay, the answer is R and theta. We've already utilized this a little bit, but I just wanted to make sure that people are thinking about this and don't forget this stuff. It's very helpful going forward, especially in circular problems. So, for instance, if you have a single point charge that has an electric field either radiating into it or out of it that has a circular symmetry, it doesn't matter what direction you look in. It's the same way the field falls off with distance. That's very handy for point charges then to think about in terms of radius and angle. All right, so let's get into the meat of today, which is solving problems that involve electric potentials. Electric potential in general is a foundational idea in physics. So it took a lot of work to get to the point where you define this thing, but once you define a potential, let's say associated with a charge, it turns out that just about everything we've ever encountered in nature so far that we understand by doing experiments on it can at its heart be described using some kind of potential language. So the electric force has an electric potential, but there are other forces in nature, the strong nuclear force, the weak nuclear force, and so far as we can tell, they all can be described and understood using the language of potential. So this is not just the convenient thing that we do with electric charge. It's actually nature seems to really like being described using this language. And so far as we can tell, for everything we can make in a laboratory, like all the stuff we can produce so far at the Large Hadron Collider, all of that stuff can be described at its heart by talking about potential and then from that deriving things like motion. So that's actually kind of astounding that you can start from something which has no direction of potential. It's just energy per unit charge. That's just a number fundamentally. And from that, you can derive the full range of motion of every particle we've ever produced. So we can tell what an electron emitted from the surface of the sun is going to do in the magnetic and electric fields of the sun, let's say, on its way to Earth on some kind of journey. So this is a very foundational concept and we're going to exercise it today, but in the context of the safe electric force discussions we've had so far. So the basic ideas. If there's an electric field present, there is also an electric potential. And in fact, in the more fundamental language of physics, all fields derive from potentials. So if you understand the potential of something, you understand the field of something. I study something called the Higgs boson. And we've discovered the boson, the particle. So we've discovered the little manifestation of what we think is a much more large and pervasive Higgs field, much like the electric field. The Higgs boson has a field associated with it. That field has a potential that causes it. And that potential, so far as we can tell, essentially tells us about the energy of empty space. So that's, in a sense, why things like you and I fundamentally have mass. We interact with the Higgs potential, which fills all of empty space, and it appears to slow our matter down, giving it the appearance of mass, substance. So mass is just another form of energy, something you learn in third semester physics. And in that language of just mass as another form of energy, you can think of mass as the energy of in motion, no motion at all. If you're just at rest and you have any energy, it's mass. So this is a really fundamental concept to understanding basically everything back to a millionth of a billionth of a second after the universe came into being. And that's about as far as we can probe with instrumentation at this point. All right, so let's get back to boilerplate stuff for this class. Now, changes in electric potential, much like changes in potential energy are the only things that matter, because energy doesn't have a exact zero point that we can talk about. We can assign zero anywhere we like for potential energy. And similarly, the change in electric potential, which is related to the change in potential energy, can have its zero point conveniently defined, and then you can just talk about movement relative to that zero point. So let's imagine we have two points, I and F, initial and final, so we start off at initial, we end at final. The change in electric potential between those two points exists whether or not there's actually a charge that we're moving between those points. We can just move between the points as a neutral object, but still talk about changes in potential, even if we're not feeling a force. And of course, as I just said, the change in electric potential is related to that change in potential energy between the two points. It's basically delta v is delta u divided by q. Whatever q you're moving through that, if you know it's changed in energy, you can figure out the change in electric potential. So one has to compute the electric potential for a new electric field type. So it could be a point charge, that's not bad. It could be just a uniform electric field, that's not a bad one to handle either. You do that by considering the work that's required to move a charge from some point A to some point B in the field. And it's very convenient to measure electric potential, of course, with respect to some conveniently defined v equals 0 point. This is known as ground in an electric circuit. So when we get to circuits, we'll talk about, OK, what's the electric potential difference between this point in the circuit and ground? And ground means v equals 0. And usually in a circuit, ground is defined. That 0 point location is defined with a symbol that looks like this. Looks like a little staircase going down into the ground. So you'll see a little line, which is like a copper conductor connected to this symbol here. And that means ground. Or in the language of potentials, that's where we define v to be 0. Now, of course, we could have a point that's actually at a lower electric potential than ground, then it would have negative potential. And one that's higher would have positive potential. That's OK. We know that negative potential energy is all right, because we just chose 0 at some location. If we go below that location, we get a negative number. Not a big deal. As long as we carry the signs through, we'll be all right. OK, so those are the basic ideas. And we're going to exercise them with a little instructor problem first. And the instructor problem is going to be to talk about changes in electric potential in the presence of a negative point charge. So this cartoon is what we're going to think about. So imagine we have a negative charge, this blue thing, located somewhere in space. It doesn't really matter where. Its electric field lines all point into it, because negative charges are where electric field lines end. They start somewhere else in space from positive charges, and they end on these negative charges. So we've zoomed in on one of these guys. And for the purposes of this problem, we're just going to assume that this is a lonely negative electric charge. There are no other charges anywhere nearby that we have to worry about. And in fact, let's say all the way up to infinity away from this charge, just to make it so it's a very lonely universe this charge is in. Our goal is to calculate the difference in the electric potential at A with the electric potential at B in the presence of this field. Now this could seem like a really nasty problem, but the language of electric potentials gives us a very convenient energy-only way of solving this problem relatively quickly. So we have here a simple case, a single point charge. And so there is a single electric potential present in this problem, and it's due to that negative charge. So we have the negative charge with its associated electric potential, which is a function of your radial distance from the charge, r. We already looked at that in the quiz question. That's going to be given by k q over r. Let's parse that for a second. What is this telling us? Well, if q, just hypothetically, if q were a positive number, then this would be some function where q's positive, k's positive, r is a positive number or 0. It only ever goes from 0 distance right on top of the charge all the way up to infinity. So there are no negative numbers for radial distance. It's only ever positive. So we have a positive number, a positive number, and a number that can either be 0 or positive. So this whole thing here can really only ever be a positive number. And so if one looks at the electric potential, if you were to plot this function around r equals 0, so here's v of r, it would look something like this. So and positive r's, well, so it doesn't really matter, because we're going around r equals 0 here. But essentially, you have this shape that looks like a big barrier. So you can imagine the potential of a single point charge like this looking like a barrier for a positive charge sticking up, like running into a mountain. And if you use that analogy, you can see why it is that, for instance, it's hard to bring another positive charge close to a positive charge. You've got to run that charge up this mountain. It doesn't want to go up the mountain. Positive charges don't want to go up the mountain. They don't want to be near the other positive charges. If you have q less than 0, then the whole problem just flips around. So here, again, is r equals 0. The y-axis is v of r. Now, r is still 0, we're a positive number. k is a positive number. q is a number that's less than 0, because it's a negative charge like that one. So in this case, we have a well, not a mountain, but a pit. And again, if you think about a positive charge brought near this negative charge, a positive charge wants to be closer to the negative charge. It's like dropping it down into a well. It wants to go down to the bottom. The whole picture flips, of course, if you talk about a negative charge coming close to this thing. A negative charge doesn't want to be here, and it does want to be here. It wants to go to 0 for the positive charge. So this is why charge and mass and gravity and the electric force, you have to be a little careful when you're thinking about potential and potential energies. Remember, there's negative charge in the electric force, and so that will flip things around for negative charges compared to positive charges. For gravity, there's only positive mass, and so it's very easy to think about. Balls don't like to roll up mountains. Balls like to roll down into pits in the ground. That's really a nice, easy analogy. You have to be careful stretching out to the electric force, because if you had a negative mass ball, it would like to roll up a mountain, and it would not like to roll down into a pit. So Newton's gravitational equations tell you that, but we've never seen negative mass, so we don't worry about it. We don't worry about objects that would roll up mountains and avoid rolling down into pits. That'd be cool if they existed, though. I'd love that. Maybe some really useful machines if we had something like that. All right, so focusing on our negative charge, we have this pit in terms of energy in the center of that picture. So again, if you're some positive charge, and somebody drops you here, you're going to start to accelerate down to the center of where this charge is located, like rolling down hill into a pit. Our goal, independent of what other charges are present, is to figure out what's the difference in electric potential between A and B. And that is the electric potential at any location, R, around a point charge. So our goal is to figure out this, the difference in electric potential between point B and point A. So that's it, just VB, whatever that is, minus VA, whatever that is. Now the nice thing is that we're given RA, the distance that A is from the negative charge is 1 times 10 to the minus 8 meters. And the distance that point B is away from the negative charge is 4 times that, 4 times 10 to the minus 8 meters, in some other direction on the other side of the charge. Now this formula for a point charge was determined by measuring the electric potential difference between a point R and infinity, yeah, Sophie. Okay, you write the delta V and you put which points it's going to, you put the final one first and then the initial one second. Yes, that's a good question. So these deltas, any time you see delta like this, it means final minus initial. So delta velocity is V final minus V initial or something like that. Like right at the bottom, it says delta V and then V to A, does that mean that B is the final and then A is the initial and then you're out? Yeah, so notationally, I mean that's the way I'm doing it here, but yeah, you usually put the final and then the initial point in the subscript. All right, so part of being a physicist is, and this is also part of being a mathematician as well. There are many tricks that all of us know in our respective languages. We all know ways of padding, I mean think about politicians for a second. We all know ways of saying paragraphs of words with no actual content. It's an art, right? There are many places in rhetoric where that's an important skill to basically be able to talk and convince your audience you're saying something, but actually the content of your speech is zero. That is an art form to be able to do that effectively. I mean it takes a lot of practice to do that and do it well. Listen to deans, I'm tenured I can say this. Listen to deans, listen to provosts, right? How much substance do they actually transfer to a mass audience, right? They're very, they keep, like all people in positions of authority, your goal is usually to keep your cards a little close to your vest and only play the ones that you don't care so much about, right? This is also true in mathematics, okay? You can insert things into math equations that have zero content, but actually can help to clarify in a weird way the picture, all right? So in the same way that zero content speech can actually tell you a lot about the person who's saying those words, more than it does about the substance of the words themselves, you can do the same thing with mathematics and actually learn a little bit about what's really going on in an equation. And the trick is, I mean the way I refer to it is insert a quote unquote clever zero or a clever one, okay? So let me give you an example. Let's say we have a fraction, like A over B, okay? And what we've been told is we've been given A over C and we've been given C over B, okay? So one way to rewrite this equation, without really changing anything mathematically, but to get to any information we know is to insert a clever one, okay? A clever one would be to multiply this by C over C. That's a clever one because now it lets us do the following, it lets us regroup the numerators and denominators, A over C, C over B. Okay, well we know A over C and we know C over B. I didn't change a damn thing about this equation, but I inserted information I know in a clever way using the number one. That's a clever one, okay? It's just a trick. It's a trick that lets you simplify an equation into something you do know to solve a problem you didn't originally know the answer to. We're going to use a clever zero in the language of potential differences. A clever zero would be to rewrite this difference, delta VBA as VB minus V infinity minus VA minus V infinity. Now in a point charge, we define zero electric potential as being an infinite distance away from the point charge. So if I go all the way out to infinity, if I set R equal to infinity, okay, put that infinity here, I get one over infinity, which is zero. All right, one over infinity, nothing. Okay, so if I go to infinity, my potential is zero. But for point charges, V is like VA are always measured with respect to zero, okay? So by putting in VB minus V infinity and VA minus V infinity, all I've done is I've taken zero off this and zero off this, I've changed nothing about this equation. But I've illuminated what the next step is. The next step is merely to insert this formula, which was defined to be with respect to infinity away from the point charge. Okay, so quite literally all I have to do to solve this problem now is say delta VB A is equal to KQ, so there's our Q, over RB minus KQ over RA, slide this over. And I can pull the common stuff out in front. I've got K and Q common to both terms, okay? Well, K is a constant, I know it. Let's see, this is a single electron. So Q is negative E, which is negative 1.602 times 10 to the minus 19 coulombs, okay? So I know Q. And I know my Rs. I covered them up here, but they're written on the screen. I know RA, I know RB. I know everything I need to go ahead and calculate that potential difference, okay? And if you plug in the numbers, what you find is that delta VB A equals 0.11 volts, exercising this new unit, volts, okay? Joules per coulomb, volts. Okay, so it's not so bad in the end for one. Let me, I didn't do this on here, but let me ask you, any ideas what would happen if I put a positive charge right here, okay? So let's drop a proton in to this picture. So it's got the same magnitude charge as the electron, but it has opposite sign. So I'm going to put a proton here. I've already got this electron here. I'm going to pin them so they can't move, right? So maybe it's like a hydrogen atom, right? A hydrogen atom. For all intents and purposes for this class, you can kind of think of a hydrogen atom as an electron and a proton with their relative distances fixed, okay? They're fixed by fundamental physics called quantum mechanics, but they're pinned there, and they can't get any closer to each other, all right? So yeah, it's sort of like making a hydrogen. Electron and proton, done, got a hydrogen atom. But they're separated a little bit. So it's a dipole, okay? Darcy, yeah? Why is the delta V not negative? Is it negative, for instance? Ah, good question. Let me back up for a second there, okay? So Rb is 4 times 10 to the minus 8, okay? So 1 over Rb looks like 1 over 4 times 10 to the minus 8 meters, okay? And Ra is 1 times 10 to the minus 8. So 1 over Ra, 1 over 1 times 10 to the minus 8 meters, okay? So if I'm gonna pirate this thing for a second here, I'm gonna steal these symbols out, I'm gonna put in some numbers. So I'm gonna put a 4 here, I'm gonna put a 1 here, and then because I've got times 10 to the minus 8 in the denominator of both of these terms, I'm gonna put that out in front here, okay? So both of those terms were multiplied by 1 over 10 to the minus 8 meters. So what do I have? I have 1 over 4 minus 1 over 1. So I've got a small number minus a larger number. So that's 0.25 minus 1. So I get a what kind of number? Positive or negative number rather than negative. And then I have a negative charge on the front, okay? So in principle, if this were a positive charge, yeah, I would get a negative potential change. So actually that's kind of cool. If I were to suddenly swap that electron for a proton, just replace it with a proton, that potential difference would also change sign right away. No work required. All you have to do is flip the sign of the charge on the front, flips the whole thing, okay? All right, so does that answer the question? Okay, yeah, that's a detail I shouldn't have skimmed over. So it's perfectly reasonable for this thing to give you a negative number. And then what determines the overall sign is what's going on here with the Q, okay? All right, so imagine I have a proton and an electron here. And again, I wanna calculate the difference between point A and point B or B and A. VB, delta V, BA. How do I get the potential in which I want to calculate these differences? I have another charge thrown in, what do I do? Let's think about VB for a second, okay? Is there just one charge anymore that's generating a potential? No, Thurman's got it, right? No, there's two. How do I get the total potential from two charges? Just add them, just add them, yeah. This is the cool thing about potentials because this is the cool thing about energy. Energy's just numbers. You just add them together to get totals. That's what I love about energy is why it's way better than force, okay? So if you have two charges, you just have to add their potentials together. That means you're gonna have the R from point A to the negative charge and you're gonna have an R from point A to the positive charge. So there's gonna be two R's involved but also two potentials and you just add them together for A, add them together for B and take the difference. That's it, okay? This is why potentials so useful. Complicated systems can be reduced to adding a bunch of numbers together. All you have to do is figure out what's the distance from where I'm measuring potential to the charges in the problem. You can do this by hand. You can do this using your computer, okay? So let's exercise that. There's a reason I went through all that nonsense, okay? Here's your problem. Building on the point potential and what I just talked about, what happens if you have two, what happens if you have three, okay? So here's three. So how much, so this goes back, this is sort of a cartoon ripped straight from the headlines of an earlier lecture where I showed the cell membrane and a bunch of ions around it and then I threw the cell membrane away and said, physics, let's just look at the charges, let's look at the force that different charges exert on each other. But now, instead of looking at the force, we can look at the electric potential and the energy stored in this configuration, okay? So this is a configuration of charges and let's freeze time and imagine, at that moment in time, this is where they're located, all right? This one's two L away from, so the anion's two L away from the potassium. The anion is two L away from the sodium. You'll have to figure out the distance between sodium and potassium, okay? That's the distance between those two. It's the hypotenuse of a right triangle. That shouldn't be too bad. So you've got this arrangement of potassium, sodium, and anion, so some protein or something like that, okay? Now, what you want to do is figure out how much energy is stored in this arrangement, okay? Now, to get energy, you have to think about what it means to put these charges on the game board in these locations, okay? So what I'd suggest is a hint, okay? Assume that these ions were once infinite distances apart from one another. So imagine a big game board where infinity is on the outer edge of the game board. You throw the charges out to infinity and now you're gonna bring one of them in and place it. How much energy does that require, okay? You're gonna bring a second one in and place it. How much energy does that require? You're gonna bring a third one in and place it. How much energy does that require? And then the total energy for making this assembly is the sum of those energies. Again, you're just summing numbers once you figure them out, okay? So, go ahead and start working on this. See if you can figure out how much energy is required to assemble this, and I want the answer in electron volts. That's a convenient unit for things that are roughly molecular or atomic sized, okay? So electron volts is your target unit for the energy stored in this configuration, okay? So, go for it. Work together, talk. Morgan, yeah? So what you're saying is you want us to find the energy of each particular point and just act together. Yeah, and it can be specific, it's the energy required to place each of those charges in some sequence.