 Okay, let me go ahead and get started with some exposition at the beginning here. Today we're going to set up and solve problems using electric potential as the key idea with the focus on the electric potential of point chargers, since that's what got really expounded upon in the lecture video that you had to suffer through. So some concepts again, just building up the stuff we've learned so far in the class. You know, sciences are layers of information and all the great discoveries are built on the things you already know, you just have to do something new with them, okay? So what do we already know in the course? We know from experiment that electric charges exert forces on one another or even that there is such a thing as electric charge. They do so via the electric field and that field, which is a concept that was, so I finally got around to watching a lot more of the Cosmos series over the weekend. Well, is that my nephew's insane seventh birthday party? It was a good way to hide in the basement and not be noticed by them if they were looking for me. And so one of the episodes later in the series is about Michael Faraday, who you've met briefly before. He is the person credited with the concept of fields. Newton posited the existence of the gravitational force. He was at a complete loss to explain how it is that the Earth and the Moon communicate with one another when they're not in physical contact. And to most people, any proposition of something like a force that reaches out and grabs from a distance was a little too spooky to be considered science. It was maybe more theology or religion. But actually Faraday is the one that recognized that in the electrical phenomenon there was also a spooky force field reaching out to other charges, but it was a mathematician that put it on its firm basis, all that vector stuff you have to do now. You can thank the person that read Michael Faraday's work and put it on its first firm mathematical foundations. And that's when the scientific community started taking the idea seriously. So we're talking about the late 1800s, the last third of the 1800s before this idea was even taken seriously because you could calculate with it. So this field exists independent of whether or not anything is being acted upon. It's always present. The electric force, much like gravity, appears to be a conservative force field. That means it has an associated potential energy. You can store energy in the field and the field can do work and release energy stored as potential. So that's the basic idea there. The energy available to do work per unit charge is called the electric potential. So work divided by charge or change in potential energy divided by charge. That's actually how we define delta V, the change in electric potential. It exists whether or not there is a charge present in the electric field. Okay, so the potential is always there. And if you want, based on, you can look in the book. They have some nice pictures of this. You can look on Wikipedia. There are also some nice pictures of this. If you plot, if you make a graph of the strength of the potential of a point charge versus the radial location, let's say an X and Y around the point charge that you're located in a plane. A positive charge has a giant mountain-like potential. It increases with strength as you get closer to the positive charge. Negative charges are more like holes in the XY plane. Their potential dips down to big negative numbers. And so you can, independent of knowing what charge is moving around, you can already think, well, if I drop a positive charge on that plane and try to move it toward the other positive charge, it has to climb a mountain to get closer and closer and closer to the positive charge that was already there. It takes energy to move positive charges together. So that conceptually kind of makes sense. If I want to move the positive charge closer to the negative charge that's located in that plane, it's falling down into a well. It wants to be closer to the negative charge. They attract each other. So the potential picture gives us all this powerful energy-based terminology for understanding in a sense why it is that negative and positive charges attract and positive charges repel each other and negative charges repel each other. It takes work to bring like charges together. And it doesn't take work to bring unlike charges together. They want to be together, okay? So it's like falling down a hill. It's easy if you just let gravity do its thing. If you let the electric field do its thing, negative and positive charges will find each other, okay? So those are the basic ideas so far. Let me do some announcements here. So remember your next assignment, I had to toy around with this a little bit. The video was recorded last semester. So I had to think about what exactly I wanted you to watch from it. So you have to read chapters 25.1 to 25.3. You're going to get introduced to the first electric workhorse device, the capacitor. This has the ability to convert charge and electric fields into work. You can store energy in a capacitor. You can release energy from a capacitor. They're very useful devices. How many people have a touchscreen device like an Android phone or an iPhone or an iPad? Yeah, a lot of you, okay? And if you don't, then you probably know somebody who does. When you touch that screen, you are becoming part of a large capacitor. So the touchscreen technology that's most common these days is known as capacitive touch. That's not an accident. It's based on the capacitor idea of storing and releasing energy, okay? So capacitors are fundamental to everything that we do. They're buried inside of all microelectronics. They are one of the main workhorses of modern electronics, because they can store energy and they can release energy. So I want you to watch the associated video on capacitors. There's some demonstrations in there with a simulation tool of what it means to increase the capacity of a capacitor or decrease it, how you can do that. And I want you to just watch the first 43 minutes or so. Basically, when you hit the discussion of shoving a material into the capacitor called a dielectric, stop. Okay, that's for next time. You'll read about that next time. And there'll be another little video on how you can take a whole bunch of capacitors and add them up to get one big capacitor. So that's the only bit that we're missing from this video is the sum them up issue, which you'll read in the next part of the book, all right? So homework three is do Thursday by 9.30. Homework four will be assigned on Thursday, but it's due two weeks later, okay? So maybe you can breathe a little sigh of relief, but not quite because we have exam one next week, all right? So that's why I'm giving you guys two weeks to work on the next homework is to give you some breathing room to study for the exam, okay? So grand challenge problem before I get to the exam. So team lead editors, a lot of you have already done this. Thank you very much. Thank you teams for appointing an idiot to lead you. They are foolish for having volunteered for the job and they will now pay the price for their mistakes in believing that they wish to be leaders in the community, okay? So I want you to, the team leaders need to mail me the minutes if they haven't already done it from your first meeting and if you haven't already have it, just shoot me an email by 5 p.m. today and let me know what's going on with your team, okay? And if there is no lead editor for your team, random team members, email me and gossip, tell me what's going on. This sounds exciting. I wanna know why you aren't functioning as a team yet, okay? Come on, you can tweet it if you want. If you have a Twitter account, I'll follow you. It's cool, all right? Some of you made the mistake of following me so you found out about all of my stupid adventures over the weekend, all right? I went shopping, yeah? So, first team meetings with me. So your team meeting with me, one on one, well, one on five, okay? So that we can talk about how things are going. That's gonna be the week after the exam, all right? So February 23rd to the 27th. So team lead editors, what I want you to do is start working with your team to find out what time slots you were all available that week and on Mondays and Wednesdays I can meet as late as 7 p.m. All right, so 7 p.m. to 7 30, I can schedule a meeting. I'd like to do it in half hour increments with each team, okay? So I need to know from your team, don't have to email me this yet. Well, I'll send out a poll and you guys will just sign up. So the lead editors will just sign their teams up knowing what times are available for the team. Okay, so here's your first leadership task. Find out what time slots are available in half hour increments for your team, write it down and be ready for a doodle poll, okay? So again, February 23rd to the 27th, all right? And then watch for my emails with signups for the meetings. I'll just email the lead editors and then I'll mention the link to the poll next class if I haven't heard from everybody yet, okay? Exam one, next week, February 19th, Thursday, in class right here, okay? Same bat time, same bat channel. We will use Tuesday to review. All right, so come to class on Tuesday with doubts, fears, uncertainty, all the creeping suspicions you've had about your ability so far in the class. Now is the time to air them. Come meet with me privately if you're not ready for exam one or if you don't think you're ready for exam one. It will be on homeworks one, two and three. All right, so whatever materials on homeworks one, two and three, that's what will be reviewed on exam one, okay? I can't, obviously, since I'm giving you homework for this Thursday and eight due until after the exam, that violates time paradoxes or something like that. So I'm not gonna quiz you, I'm not gonna test you on material, you haven't actually had homework on yet, all right? So we're gonna use Tuesday to review, solve problems in class, and prep for the exam, bring questions. There'll be some problem solving. I'm not gonna do any demonstrations, but I will take questions on homeworks and things like that if people are scared or confused. And we'll just talk about the problems and I'll work what I can at the board, but I also wanna give you all problems, exam style problems from me to see what that exam's gonna be like on Thursday, okay? All right, so here's what I know about the team so far. So I don't know who the lead editor is for Team Taki on yet. I have no information about Team Echo. So again, by 5 p.m. today, I want you to rat on all your friends, all right here. And then Team Alpha, you've organized, you've got a name, just pick a lead editor, do it by email or Facebook or whatever, okay? Snapchat, I don't care, whatever it takes for you. Vine, if you need to use a little animated GIFs to figure it out, go ahead and do it, it's cool. So everyone else has given me a lead editor name and I have, I basically have minutes from everybody. There was a technical snafu with one of the minutes sending, but that'll get fixed easy. So that's basically where we are. Questions? Yeah, so. Do we have a quiz on Tuesday? Will we have a quiz on Tuesday? Yeah, I think I will still do that because I'm still gonna have you watching and reading things as we go, okay? There'll just be a little delay in terms of class setting up and solving problems, but we'll get there, okay? My plan is basically that you will learn all about capacitors before the exam and then we will start using capacitors in in-class problems and you'll have them on the homework and then you'll be basically ready to hand in homework for by that next Thursday afterward. So it'll be a little bit of catch-up for me, a little bit of catch-up for you, but we'll do it together, so yeah. And I sent you a team echo on Thursday. You did, okay, great, then I just missed it. Sorry about that, so I will fix that. All right, great, so team echoes got their act together and I'm an idiot. We've all now, okay, well, actually, hang on, hang on. I didn't bring the bucket with me today, but all there goes in the error jar, okay. Boy, you guys are so close to donuts right now. You just, you feel it, right? All right, any more questions? And Elizabeth, are you the lead editor? Okay. Okay, well, then, it's happy quiz time. Hooray, happy quiz time. Okay, well, let's take a look at those. Let's take a look at those, shall we? So, okay, so how does one determine the total energy stored in an assembly of charge? This is something we're gonna utilize today in class for solving problems. Is it one, measure the distance between the charges? Is that alone sufficient? No, okay. Two, compute the work required to place each charge and sum them all together. Raise your hand if you think it was that one. Okay, three, compute the total electric charge in the system. Put up the charges. No? And four, compute the work required to place each charge and multiply them all together. Anyone for number four? All right, so we had two or four, sound most plausible, but of course, if you multiply a whole bunch of works together, you're gonna get joules times joules times joules times joules, which gives you joules to the nth power. So that doesn't give you energy. They give you energy to some huge power. So the correct answer is two, to compute the work required to place each one. So how much work does it take to place the first one? How much work does it take to place the second one? And keep going until you've placed them all and then add it up. And that gives you the total energy stored in the assembly of charge. That's gonna be related to either the work in the field or the negative of the change in potential energy. Okay, you just have to state what you're calculating. All right, question two. Which of these is the electric potential of a point A? Some point A. A distance R from an electric point charge, Q. Is it one VA equals KQ over R squared? Anyone for one? Two, VA equals KQ over R cubed. Anyone for two? Elizabeth? Big fan of number two? No, no, no. Sam's just being a jerk. Is that what's going on over there? Am I gonna have to separate you three? Is that how this is gonna work? All right, VA equals KQ over R, number three. Okay, and VA equals KQ times R, number four. Okay, so yeah, you can even just use unit analysis on this. If you remember that volts are joules per coulomb, you could calculate real fast that K is Newton meters squared per coulomb squared, and then Q is coulombs, and R is meters. And so you just have to get Newton meters per coulomb or joules per coulomb, and so the answer is three for that one, right? So I put this in here because not only, I mean maybe you memorized the equation, that's fine, maybe it was obvious to you from the beginning, fine, but a strategy for tackling questions like this where you need to eliminate answers quickly. It is useful to memorize the units of things and then use what's called unit analysis to whittle out the unreasonable answers. All right, so it couldn't possibly have been any of those other three, even just from the point of view that joules per coulomb is the target unit you need to get. And if you know all the units of the things on the right, you can multiply, divide, whatever you need to do to check the answer. So unit analysis will save your bacon, all right? Not only would that have prevented, for instance, at least one probe from crashing on Mars, but it constantly is used by engineers and physicists to check each other's answers. If your answer to a problem doesn't even have the correct units on both sides of the equation, it's just wrong to begin with, okay? The book I used to teach out of in this had this student solution manual, and the student solution manual is always written by some underpaid grad student who's looking to not hang out with their advisor terribly much for about a year or two, and so it's sloppy work. And one of the solutions that students kept handing in, it had the units on one side, not equal to the units on the other. I mean, the solution was just patently wrong. If the grad student or whoever wrote this had just checked it, they would have known their answer was wrong. Or if there was an editor for the student solution manual, that would have been helpful. So students kept copying out of the student solution manual handing in. It was incredibly obvious that they had done that, because the only way you could get that answer in that order was if you followed the student solution manual, okay? All right, so finally, circular coordinates are written in terms of what variables? One, X and Y. Two, X and theta. Three, R and theta, okay? And four, X and R, and yeah, the correct answer is R and theta, okay? So I just tossed that in to make sure people had been paying attention either in math class or in the video, that's it. Okay, any questions? So this is some of the more difficult material at the beginning of the class, because we have this weird concept of electric potential and differences in electric potential. You know, ironically, it's one of the more familiar things outside of the direct study of physics that most people have encountered, because most people have at some point had to purchase a battery, plug something into a wall socket to get electric power. And so we're all very used to thinking in terms of volts. How many volts do I need to run this gadget? How many volts do I have to have for this toaster to operate properly? But nobody really knows what volts are and they are just merely the potential to do work for some amount of charge and that's why everything's quoted in volts. We don't know how many charges this socket's gonna have to move to do work, but we do know it's potential to do work per unit charge. So that lets an engineer design a new device that can take advantage of that kind of voltage or a different voltage, for instance. Okay, so the basic ideas here are that the change in electric potential between two points, I and F, initial and final, it exists whether or not there's a charge there and it's related to changes in potential energy between the two points by this fairly straightforward algebraic equation. The change in the electric potential, V final minus V initial, is just the change in the potential energy, U final minus U initial, divided by the charge that's moving through the potential energy difference. Okay? Now one has to compute the electric potential for any new electric field type configuration. So I spent a lot of time on the electric potential of the point charge. We're gonna exercise that today. But really the only ones I care about you guys exercising right now are what do you do in a uniform field and what do you do in a point charge field? All right, so we're gonna do point charges today because uniform fields are fairly straightforward. So we'll do something that's a little bit less straightforward. All right, so you have to consider to get a new potential, you have to calculate the work required to move a charge from some place A to some other place B. That's what you'd have to do from first principles. You'd have to do the work done by the field is equal to the integral of the force which could change as a function of position. So this might be a function of R and that force vectors dotted into dr vector and then you have to figure out, okay, in my path from A to B, word of the little dr's point, I only consider the dr's that have components along the field, okay? So some non perpendicular component along the field and then I have to know the function f at every point and there can be some pretty messy, you know, fields that you're gonna have to deal with and then the nice thing is here no matter what, you just have to know what charge is moving and then write down the electric field equation in terms of position like R, x, y, z, whatever, okay? So these can get quite messy in general but we're gonna focus on two archetypal things, the uniform fields which can be constructed, for instance, inside a capacitor as you'll see in the next video and point charge fields, you know, from which essentially everything is made anyway. So it's convenient to treat these two things. Now one other thing, just like gravitational potential energy, it's very convenient to define in a given problem as zero point. That is your reference point for doing relative measurements of electric potential. Electric potential is based on electric potential energy and just like gravitational potential energy, there is no absolute energy in the cosmos. So one has to say, look, I'm just gonna choose in a gravitational field this table surface to be zero gravitational potential energy so I can say that this marker when it's left to sit there, okay, and it's just sitting, so the table is pushing back with an equal but opposite force to what gravity is pulling it down to the center of the earth with, that's sitting at zero potential energy. If I apply a force and then bring it to rest somewhere else in the field above the table, at that point I have done positive work to raise its potential energy. The field has had negative work done on it, so work done on it by me and now there's energy stored in the field. If instead I did work and I moved the marker down to a lower point below where I started, now it has negative potential energy and there's absolutely nothing wrong with that. So there's also consequently nothing wrong with having in an electric field a negative voltage. Okay, so you could have an electric potential difference of negative nine volts and that tells you something about where the zero point is considered to be. Now for a point charge, we choose the zero point for electric potential and electric potential energy to be an infinite distance away from the point charge whose field we're thinking about. That's convenient because the electric field goes as one over r squared and so at r equals infinity, it is by definition zero electric field. It exerts no force on a charge of infinity or very, very far away. And so that's a convenient place to say that it has zero potential energy. You know, to escape Earth's gravity, right? If you want to launch a rocket into space, you have to calculate the energy required to do that and the energy you calculate would be the energy to get it from the ground off to a point in infinity where it completely escapes the Earth's gravitational potential. Now in principle, you don't have to go to infinity. Okay, you just have to go far enough away that the Earth's gravitational field doesn't play a big role anymore in the behavior of the rocket or satellite or whatever, okay? But that's what you do. You calculate that energy required. If you want to know how much energy is required to ionize a hydrogen atom, a hydrogen atom has a ground state energy of negative 13.6 electron volts. This is something that probably got passed along in chemistry, okay? What does that mean? It means I have to dump in 13.6 electron volts to take that thing to zero potential. In other words, to take the electron orbiting the proton in a hydrogen atom and move it to infinity requires just a measly 13.6 electron volts. So when you say an atom is ionized, it means you've taken an electron or all of its electrons and moved them off to infinity by putting in the amount of energy required to take those electrons to zero potential, which is an infinite distance away. All right, so this pops up everywhere. You don't know that you're doing this, but you've been defining zero electric potential for a long time, starting in chemistry and then moving on from there. You've also known about zero electric potential whether you realized it or not. If you've ever heard somebody talk about grounding something, electrical. So I've mentioned it in the class, right? When I had that scary-ass Vandegraaf generator out here that gave me bigger sparks by touching the knob than by touching the top of the thing, that's allegedly been fixed now. I did what was called grounding myself. I'm a conductor, not a great one, but good enough that electricity can cause problems for me. What I wanna do is I wanna give the electrons that might build up on my body a place to escape. And so that place is referred to as ground and ground is literally the ground. It is a sink for free charge. The earth can soak up lots of electrons and so that's what makes it great. The earth is literally at a lower electric potential than I am, especially if I, well, if I do this, if I take these rubber-soled shoes off and then scrape my feet across the carpet, I'm now soaking up excess charge from the earth. And sorry, let me see if I can do this without actually touching anybody. Hey, we'll use this gas jet over here, that's safer. Nope, nothing, it's too humid in here. Water in the atmosphere will soak up that net charge that's maybe building up on me. It was dry in Wisconsin, I got statically shocked all the time. So this is a pleasure by comparison. But all I have to do to get that charge off means touch something that's at a lower potential so the charge will want to move to that place, okay? It's all about energy, it's all about going from a high energy to a low energy point, all right? So grounding, if you're talking about circuits or electrical systems, grounding means you define a location in the system as the zero electric potential point. You can go below it, you can go above it, but typically that's where things would like to go, okay? So you'll see more of this when we go into circuits. But for today we're gonna think about the zero point for point charges as being an infinite distance away from a point charge. Okay, so let's do an instructor problem here so I can demonstrate what I'm talking about. Let's go back to our archetypal friend, the humble point charge. And particularly let's look at a single electron, okay? So we're gonna be looking at just one electron and we're gonna be probing its electric field just using the electric potential concept. So there is no other charge present in the problem, but nonetheless we can think about, for instance, the work per unit charge that would be required to move from a point like A to a point like B in this point charges electric field, okay? So we have an electron. Let's start writing down what we know. We have an electron and that has a charge of negative one elementary charge which is negative 1.6 times 10 to the minus 19 coulombs. Now, a fun fact about my exams, you'll be given an equation sheet for the exam that will have more than you need to set up and solve problems. That's because I still firmly believe as a teacher that what matters is that you know how to use the things that you've learned and not simply memorize them and then try to bust them out randomly to solve problems. I expect if I give you a toolbox, you know to drive a nail into the wall not using a socket wrench and that you know to tighten the bolt, not using a hammer, okay? That matters more to me than the fact that you have access to a socket wrench and a hammer, okay? So I don't care about giving you equations. I care that you demonstrate to me that you know what the hell is going on with them. So yeah, Sam, this is the stunning revelation. Are you ever going to give us false equations? No. Jesus. Wow, what am I, a bloke bee? Wow, okay, that's it. No more Marvel movies for you, A, all right? No, I'm not going to give you false equations. Now, okay, if I do, a dollar goes in the jar because I didn't mean to. That doesn't mean I won't make mistakes. I'm a human being, right? But I would never intentionally give any of you a fake equation, okay? Wow, you guys come from a dark place in college. That's, man, I have to talk to the dean about that question. Would you ever bring a knife and threaten us with it during an exam? I've never waved a gun around during an exam. No, I've never done this. Who do you have teaching you? Okay, so you've all been educated by the Unabomber? That's good to know. Okay, wow. All right, what do we know? We know that. That's harmless, all right? Let's see, what else are we told? We're told that we're moving from a point A in this electric field. Wow, Sam. Damn. Okay, and that point A is the distance rA from the electron, and that distance is given. So the shortest distance between A and the center of the electron, okay, is given as one times 10 to the minus eight meters, okay? What else do we know? Well, we're also told the distance that B is located from the electron. So we have rB, and that's four times further away. Not necessarily in the same direction, though, and I did this on purpose. This is to reinforce the fact that when you're talking about moving through an electric field, just like moving through a gravitational field, and you're thinking about, okay, well, how much work would it take to move a charge from A to B? The first thing you confront is, well, does it matter what path I take? I mean, maybe there's no one answer to this question. Maybe I could just go straight through the electron from A to B, or maybe I could go around the electron and go to B, or maybe I could go way over here and kind of loop around and then come back to B, and does that make a difference? And what I tried to emphasize in the video that I gave you guys to watch is it doesn't matter. That's the beautiful thing about a conservative force field. It doesn't matter what path I take, so I should take the simplest path if I'm gonna go through all the effort of calculating work equals the integral of force times displacement, okay? But what I wanna emphasize today is that we can just use the fact that we have a conservative force field with an associated potential energy and this concept of the electric potential. We don't have to calculate the work. We only have to know what is the electric potential of point A, what is the electric potential of point B, and then all I have to do is calculate the difference between B's electric potential and A's. And this is equivalent to going through this whole mess of doing work equals the integral of force times displacement. I gotta pick my path, I'm gonna pick a straight line, I'm gonna bust it into tiny little pieces, I'm gonna calculate the vectors in each piece, I'm gonna calculate the force in each piece. You can spare yourself that by simply taking advantage of the fact that this is a conservative force field and all you have to know is the potential of point A, the potential of point B, and to take the difference, and you're done, okay? So these problems get remarkably simple, you don't even have to use vectors anymore. That's the great thing about energy, energy is just a number, it has no direction, okay? It just has a magnitude at some point and all that nature really cares about are changes in energy. Now changes in energy mean forces, but we can just set the force stuff aside for a second and think totally about the energy of these problems. So with that in mind, I could go ahead and I could do the work thing, but I'm not gonna do the work thing. I'm just gonna use the fact that for a point charge, here's the other thing we know. For a point charge, and I went through this laboriously in the lecture video, okay, we're just gonna use the result, the electric potential difference in general, like for going from someplace to A, we can pick the someplace and we can pick a convenient someplace with respect to which we measure all electric potential differences. And I'm gonna come back now to that concept of the zero point. You choose a convenient zero point and you say, well, I'm always gonna measure, so if I'm talking about like delta VA, just the electric potential difference between some reference place in space and A, I'm gonna pick a convenient reference point and I'm gonna pick the one known as the zero point, okay, so V zero. Now, V zero for a point charge is, as I've said before and I'll emphasize it again, the zero point of electric potential is also the zero point of electric potential energy for a point charge, for a point charge, for a point charge, it doesn't apply to every situation, but for a point charge, okay? And that is infinity, an infinite distance away from this electron. So the zero point here occurs when R is equal to infinity. And the nice thing about that is you don't even have to bother writing down that second term. All we have to know is what the heck VA is with respect to infinity or the zero point and that was calculated in the video. All right, so VA is going to be K, Q, whatever the Q is of that thing there, which is this, so here's Q, okay? Is that's the thing that's causing the electric potential to be formed in space, is that electron? Divided by Ra, the distance that we are from the thing that's forming the potential in space. That's it. And then there's corresponding VB, K, Q over RV. So let me go through the effort of writing down what you really wanna calculate now using all this and then show you how the zero point neatly fits into the calculation so that, this is really equal to delta VA with respect to infinity. This is really equal to delta of VB with respect to infinity, okay? Let me show you how those appear in the difference just between VB and VA itself. And this is a trick that all physicists know and mathematicians know it too, that's where we get it from. And it's called either inserting a clever zero or if you have to insert a one, inserting a clever one. So let me show you how to insert a clever zero. So to insert a clever zero is pretty much about as simple as what I'm about to do here. Okay? If you have some number A and you want to insert a clever zero, you would simply say A equals A plus zero. And please go ahead and laugh. It really is that simple. But let me show you where the clever part comes in. What if you knew A plus B, okay? And you knew B, but you didn't know A. No problem. If you put in a clever zero, like B minus B, now you have A plus B in your equation and B. That's what a clever zero is. A clever zero is a zero that you define that makes a problem easier. So let's say you don't know it, you don't know A, but you have A plus B and you have B. No problem. You would just, in your head, you would just subtract B off of this and be done. But what you're really doing and you're skipping a step, you're, in your mind, you've learned a shortcut doing this. Okay? So we're gonna do that for what we really want. What we really want is delta B, B, A. The electric potential difference between point B and point A. That clock makes no sense. It's got the minute right from the hour, totally wrong. Okay? Anyway, good. It's a minute if anyone looks at that clock back there. So this is just gonna be the potential at point B minus the potential at point A. Now let me show you the clever zero here. I could rewrite this as VBVA, okay? Plus zero. And here's the clever zero. VB minus VA plus V at infinity minus V at infinity. Okay, which also, those two things are zero. So I just did zero equals zero minus zero. Nothing revelatory here, all right? Nothing earth-shattering. But watch what happens. I'm gonna pull the negative V infinity over here next to the VB. So I have now VB minus V infinity. And then I'm gonna pull this positive V infinity next to the VA and I'm gonna pull a minus sign out of them. So I'm gonna have minus VA minus V infinity. I've just rewritten that equation, just moving some terms around and pulling the minus sign out of two of them. That's it. I didn't do anything magical here. But suddenly what I have is I have the difference between VB and infinity and the difference between VA and infinity. I have exactly that and that in my equation. Now mentally what you need to learn to do here is simply to say, okay look, in a point charge problem, I'm always measuring potentials at a given point, A or B, with respect to infinity. It's implicit. I don't have to write this down every time. But I want you to see where this comes from. So I could have just taken KQ over RB minus KQ over RA and been done. I went through this rigmarole so that you could see that sometimes in order to see how pieces you know match up to shortcuts, you have to put the long cut down first. So the shortcut is just to say, look, this thing here is KQ over RB. This thing here is KQ over RA. Okay, I've got K, it's a constant. I got Q, it's negative 1.6, 10 to the minus 19 coulombs. I was given numbers for RA and RB calculated, okay? But I want you to see the long cut on this, not the shortcut, because I want you to appreciate how much time you save just by doing this. These equations already implicitly are measured with respect to infinity. And if you go back through the video and watch how I derive the potential difference between any two arbitrary points, you always wind up having to solve this integral and then at the end of the integral you get 1 over R1 minus 1 over R2. And you have to pick, well, where's 1 over R1 and where's 1 over R2? And it's convenient to pick one of those Rs to be at infinity. Again, that's just setting the reference point for zero potential to be an infinite distance away from the point charge that's creating the potential, okay? So at this point, all I have to do is put a minus sign between these and calculate, all right? So that's what I'm gonna do. And when I crank this out, leaving all the units in, so K, right, is our friend 8.99 times 10 to the 9, Newton meters squared per Coulomb squared. So I have Newton meters squared per Coulomb squared times Coulombs, gets rid of one of the Coulombs and the Coulombs squared in the denominator, times 1 over meters, which gets rid of one of the meters in the numerator. So I'm left with an answer that's in Newton meters per Coulomb, okay? And the number is 0.11. A Newton meter, if you work it out, is a joule, okay? It's a kilogram meter squared per second squared. So joules is up here. Coulombs is down here. And this thing is just joules per Coulombs or volts V, okay? So that's your friend, the volts. One joule per Coulomb, one Newton meter per Coulomb, one kilogram meter squared per second squared per Coulomb. Okay, they're all the same thing. They're all synonyms. And this is actually, not a big number, not a small number. That's a sort of reasonable one-sized number. This is also why it's convenient, for instance, to work in things like electron volts, as you'll see in your own problem. If you get tiny numbers in joules on the scale of things that are subatomic or atomic-like electrons, then in joules you might get a really, really tiny number, but in electron volts you might get a reasonably-sized number because electron volts are teeny, tiny little units of energy and the sort of typical energy scales of atoms. So you'd expect that to be the typical energies of things that you'd observe in like biological systems involving atoms moving around or molecules moving around, okay? All right, so questions on this? I'm gonna amp it up with you guys now. All right, well, here, put this to work. Here's your problem. We're gonna revisit something that we visited a couple of weeks ago. When I gave you guys a problem on calculating, I think it was the total force on different ions near a cell membrane, all right? So you have anions, which are basically proteins that can carry various large charges. So for instance, consider an ion with a four minus charge. That means it has a negative four times the elementary charge in its total charge. You've got a sodium ion and a plus, which carries plus one elementary charge and a potassium ion K plus, which carries plus one elementary charge. And they're arranged as shown here. So they're each on a side, two L away from each other, two L and L is given here, 2.8 times 10 to the minus nine meters. So what I'd like you to calculate is how much energy is stored in this arrangement of charges? But after all, energy and changes in energy are the things that drive biological and chemical systems. Chemical reactions are driven by changes in energy. Biological systems constantly seek to move and rebalance energy, okay? Or convert one form of energy into another. Those are just chemical systems at their heart, all right? So the question is, how much energy is stored in this arrangement of ions? A sodium ion, a potassium ion and some protein. So an anion with a negative four times the elementary charge. And I want the answer ultimately in electron volts. So I want you to practice a little bit converting to electron volts, okay? So go forth and calculate. Work together in pairs of triplets. You're supposed to calculate the total energy stored in that arrangement of point charges. You can treat each of those as a point charge, okay? And I give you a hint. Assume the ions were brought together from an infinite distance away and treat them as point charges. So I already gave you a part of the hint, all right? But imagine that these were assembled by first bringing them from very far away from each other in together, all right? And see if you can build the problem up in pieces, tackle each piece, and then put the pieces together. That's always the strategy in a physics problem. Break it into pieces with some assumptions, tackle each piece, bring the pieces together at the end using some physics motivation, okay? So start talking, get some ideas from each other and see if you can make some progress on this. You guys have got 35 minutes basically.