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.  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?  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 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.  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?  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?  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.  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?  Wow, you guys come from a dark place in college. That's, man, I have to talk to the dean about that question.  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.  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?  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.  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.


I'm not saying go and have fun, but I'm saying that be part of the change that we are looking for on the continent. Entrepreneurship is the key, whether you like it or not. So this is the best channel for you to subscribe because I'm gonna bring inspiring people every single day.  I mean I told you that help us reach 700k, I mean after 700 straight to a million, but hey I've also been telling you that Nigeria is not just Lagos and Abuja, I'm here in the cold city, I'm here in O42, I mean I've never been here, you definitely don't know what I'm talking about. This is Enugu and where I am is called, is it Enugu state? No I'm not from here, I'm not gonna talk too much about places that I don't know. I've got a special guest today, one of the best YouTubers here in Enugu state and she's gonna be the one to take me around today.  Hello guys, hello, hello, hello the Mayans, the Mayan army, it is me, your girl, Tibera here with, what a Maya, in Enugu, can you tell us what his t-shirt is? The cold city? The cold city state of mind, he's here in Enugu, live and direct and on today's episode, I'm here to take him to one of the youngest guys doing great things here in cold city and like he rightly said, Enangir is not about Legos or Abuja, we also have Enugu and we are doing wonderful things here, so today we are at Rosette paint, Rosette is a company making premium paint here in Nigeria, like if you're looking for anything paint, building, real estate, whatever, Rosette is a place to look for and the interesting thing about this company is that it is owned by a young guy. A young guy from where? A young guy from Enugu state. He's an African too? He's an African. That's impressive. He's an Igbo man. Igbo Kwenu, please, my title, one day, I can't even remember my own name, do you know I have a chieftain's title now, one name, the number, say Igbo Kwenu, yeah, but hey, I got a lot of people who don't know anything about Enugu, if you have to say something about Enugu, what are you going to say? I would say that Enugu first of all is a state in the south eastern part of Nigeria. And it's also a state known for its serenity, cleanliness, and everything, ambience is of this state.  Come and do something here. I think Enugu has to employ you as their ambassador because within one minute you sold the whole city. That's what I do. Fantastic. That's what I do as Tibera, that's what I do as Tibera. Check her out on her YouTube channel, the link will be in the description box. Make sure you subscribe, don't disappoint me. I mean, she has over 6,000 subscribers, you know we can make that magic happen by making her get 10,000 subscribers. I don't want to talk too much. Can you take me to meet this young gentleman? Before we go there, I have something from Rosette from UK, so I think you should wear this, right? It's like I'm a brand ambassador now. Yeah, he's a brand ambassador. Rosette, you have to pay me for this one. Hey! Okay, you know I support African businesses, so I'm just gonna wear this one just to support the business. Wow, Maya. All right. So, what comes up with this, sir? Maya, this is the man I've told you about, the MPC of Rosette, so this is what the Maya. Wow, Maya, good to meet you. Yeah. It's nice meeting you. My pleasure. Everyone is talking about you. Oh, thank you. It's been three hours in Enugu and everyone is telling me that if I don't meet you, I can't leave here. Oh, wow, wow. What is so special about you that everyone is telling me to meet you? Well, first of all, my name is Casey Ogie. Okay. Yeah, I'm an architect and I happen to be the CEO of this company, Paint Rosette, and so many other companies that we have under this particular building.  Yeah, we produce here, we manufacture here. You produce here? Yeah, yeah, we produce here. Are you the CEO? Yes, I'm the CEO. The CEO of the company, yes. We produce here. We first of all started with importation. We imported part of it, then 100% importation. Then from there, we started the local production, 50-50. Then now, we almost have zero importation. We are doing all the productions here. And we have two stages of production. We have the primary production going on here. We have the secondary production going on here. And we have outlets outside of Enugu. We have in Lagos. We have in Abuja. We have in Oweri. We have in Portacort. We have in Uyo. We have in Asaba. But all those places, you have the secondary production going on there. But here, we have the primary production and the secondary production going on here. What I mean by primary production is this is where we do the production of the paint, the main production. Then the tinting, the tinting process is the secondary production. We move from here in white colours, in white containers, in containers to all our outlets. Then they do the tinting, which is another process of production before it gets to the customer. Let me understand. Yeah. The people who are producing the paint, are they Africans? Yeah, they're all Africans. We're all Africans here. We don't have any expatriate here. Yeah, we're doing it. Starting from even our technical team here, even before we started production, the installation of the machines, we're all done by Africans. Wow. We've not had to bring anybody in here to do any solution for us. Before you take me to your factory, I would love to get to know who you are before I start. But the sun is really shining. It's okay. I can take you to the factory. No, before you take me to the factory, can I find a place and sit down with you and talk to you about who you are and then from there take me to your factory. Okay, so let's go to the factory. Brahma, I have heard a lot about you. Somebody introduced me to you. Yeah. But personally, I don't know you. And my audience are from different parts of the world. Yeah.  My name is Keseoji. I'm an architect. I'm from Ibea State. But I live, I'm based in Enugu here. And that's where I do my businesses. And there may be some other places where we have our branches. So the paint company, I'm sure that's the one you've heard about. Although we have other companies, we have the KKH contractors, a construction company because I'm an architect. So I still do my business. Then we have the door company. We import doors. Then we have other companies, the Rosettes Oil and Gas. We're into oil and gas and all that. We have a real estate company. But then we have the paint company that everybody is talking about now. That's the paint Rosettes. I happen to be the CEO of the company. We have our branches, a lot of places like in Lagos. We have in Abuja. We have in Asaba. We have in Potacot. We have in Uwere. We have in Uyo. But these are outlets that we have those places. So this is the head office.  Yes, I'm an architect, yes. So after school, you practice? Yes, yes. Where? Yeah, in Enugu here. In Enugu. Yeah, but as an architect you get jobs. You do jobs a lot of places. You can go to them. What really inspired you to start a painting business? Well, I think it should be what inspired me to become an entrepreneur. Okay. Because right from time, I've always known I had a lot of people linked up to my vision and my dream. And I was able to discover it when I was younger. Okay. And when I was even in secondary school. Because I had to settle where some business is doing. I had a barben salon running. When you were in there? When I was in secondary school, yes. Barben salon? Yes, I had a barben salon. Yeah, I had a shoe mending factory. Just a mini small, mini shoe mending factory. Then I went to, I had a radio mechanic shop as well. So I knew I wasn't going to be on a paid job, on anybody's payroll. So, because a lot of people need to be employed. So, and I also got it clearer when I got into school. And we asked why we wanted to be architects. Because one of the things I told them was I needed a job and some other businesses where we can employ people. Reduce the unemployment rate in the society. We have a lot of issues in Africa. So, that was one of the responses that I gave to them. I wanted to do that. I wanted to do that. And some other things that would be able to make me employ more people. Yeah, and also I was inspired more when I got into the business of architecture and found out that a lot of people, almost every human being in the world has an option of colors. You can attest to that when you ask a little child what color was your favorite color. They already have their colors. So, even as a little kid, they have their colors. And we decided to have a paint company, a paint product that should be able to give everybody the option of the color that he wants. So, what I mean by that, it's made us to research even deeper and deeper and found out that most of the paint companies around the world restrict customers and people to the colors that they can produce. So, we needed to do something more to be able to produce any kind of color so that any person will have a place in our company to pick the colors that you want. And that's why we did a lot of research on all the products available in the market. We found out that there are so many things that need to be done for Africans. And we also found out deeper on the technical part that there are some short-changing of the qualities of products coming into the country, even the ones that are being imported. So, we had to work on those ones, analyze all the products in the market and they created a formulation for ourselves that is an improvement on every product that is in the market. And that's why you heard about a product. So, which means that you're doing something in Africa for Africans? Yes, for Africans. Because most of these companies outside Africa that bring their products to Africa they don't really know the environment very well. So, they don't know the heat, they don't know the weather in Africa. So, we needed to study it very deeply and we'll be able to take care of that. And we have done that already. I want to understand, before you achieve this, did you live in Nigeria all your life? I've lived in Nigeria all my life. I've traveled abroad, I've had reasons to travel to a lot of countries in the world. We have seen all those products and all that. I bet, I don't think I've ever stayed outside Nigeria for more than a month. My business is here in Nigeria. So, you're trying to say that it's possible in Nigeria? Yeah, sure, why not? It's very possible. It's very possible. What you're doing right now, having a paint business, do you think it's worth it? Yeah, yeah, it's worth it. It's worth it. It's worth it. Yeah, it's good. Yeah, that area, yeah, it's very good. Like every business, like if you study a business very well and know how to go about it, you should be able to find out that every business has a level of profitability in it. But then, the profit is not the key. The key thing was to make sure that our people get the best quality, our people get employed, and our people get the options that they have in colours, and our people get the durability that they want, that they pay for. They pay a lot. They pay for all those important products, they pay a lot. So, and when you pay a lot, you have a product that after like a few years you have issues with the product. It doesn't have quality. It doesn't have enough, the right dosage of anti-fungi and the weather resistance that everybody needs. So, we had to work on all these things and be able to, it's not on the profit. The profit margin is low compared to what they're doing but we have the satisfaction that our people are getting what they want. How many people have you employed so far? So far, I think we should be about 122 or so. Yeah, all our branches, yeah, 122 now. That's a class counting because we hope to take a whole lot of people. Amazing. I would love to go check out the factory. Let me see what goes on in the factory and how are the next customers. Okay, okay, no problem. Maya, you're welcome to our factory. Thank you. Like I said, this is the primary production section. Alright. So, we produce, we mix all the additives and the chemicals and all the raw materials together here and yeah, I do the meeting here to get the product in white before it goes to all our outlets for secondary production before the customer takes it except the customer that wants to take it in white. Okay. So, we actually have a smart production going on here. What I mean by that is we don't need too much space to be able to produce, run our productions because we don't produce the colors and keep. We produce in white. So, once the product is going down, we should be able to find out and we'll run more productions and start and keep. So, we don't have any space to keep all the colors, different colors like other paint companies are doing. So, this is actually a very big innovation on our own parts and it's helping us. Can you show me where you do the smart production? Yeah, this is where we do it. This is where we mix all the products, all the raw materials. So, you're going to get a white paint from there? Yeah, you're going to get a white paint from there. You take it to the printing room. From the printing room, the customers can take it from there. So, everything is from start? From here, then to the printing room, then to the customer. Yes, yes, yes. That's beautiful. That's there, yes. Wow. I know you're a young entrepreneur. Yeah. And I have so many young entrepreneurs watching us right now. So, I just want to know, what are the kind of challenges that you face as a young entrepreneur starting this whole business here in Nigeria? Yeah. The truth is that Maya, it wasn't easy because we are in Africa, we know how difficult it is for an entrepreneur to start in Africa because if it were other parts of the world, you'd be able to have access to funds. But in Africa, it would be very difficult. It was actually that difficult for us to start. But we had to start. Okay. What was propelling us was bigger than the constraints that we had when we were starting. Yes. Actually, here in this factory, we are saying we mixed products with bare hand here. That was how we started. We started with bare hand. Yeah, we stared it with bare hand here. That was how we did it. We didn't have machines. There were no machines. But we had to start. So then, like I said, I have a construction company and I was patronizing myself. I was buying all the products myself. When it wasn't to the standard that we needed to take to the market, I was buying all of them myself. But we needed to start. And when it was what we wanted, we started taking it to the market. We started telling people about it. So we had challenges. We also had some challenges from big companies per se that were on the side of bullying us because they didn't want us to come out because they didn't want us to come out. They saw us as a very big challenge and it was tough as well. But there's one thing that I think I really want to know. Does African governments really support young entrepreneurs? Maybe there could be policies by some governments and all that. But the truth is, up to now, we've not been able to assess any of those facilities or anything. Any assistance from them. But it doesn't detail us. We are moving and we are pushing forward. That's really inspirational, bro. I think your life alone is inspirational to so many people. But I just want to know, yeah, we have so many young Africans who are trying their best to be an entrepreneur. If you haven't missed it for such young Africans, what would that message be? Yeah, they have to just keep on. You have to follow your dream. You should be able to know where your areas of strength lie. And you should be able to follow it. And the most important thing is you don't have to, in Africa, you don't have to wait for the government. You need to start. Wow. You need to start small. And most importantly, from what I've seen recently, I've seen the government's their help. That's when you must have started. Okay. So it's always very difficult for you to meet the government and explain what you want and they will give you assistance. Maybe, no, no, you need to start. Then you can tell them, maybe possibly, maybe you can be lucky and you assess the front. My brother, where do you find a reset page? Okay, this is the head office. Okay. Then we have outlets in Lagos. We have in Abuja. Okay. We have in Oweri. Then, Porta Cots, Aba Uyo, Asaba and Aba Kliki. Then, we are very active as well on social media. On Facebook, on Instagram and Twitter. Then our website is up and running as well. Are you selling only in Nigeria? We are selling. We have customers buying from outside the country, but we don't have outlets outside the country. So you're looking for? Yeah, yeah, yeah. If we can have people from outside. I mean, you just have to tell them. Yeah, if we have people, you know, from Ghana, like anybody that is interested in being a distributor in Ghana, other parts of Africa, then we can do that. We have the capacity to do the production.  Yeah. You achieved everything in Nigeria? Yeah, of course. Yes, of course. You think it's possible to make it in Nigeria? Yeah, of course. It's very possible. We are making it here in Nigeria and we're making it big. Yeah, but you need to start small. Exactly. Then we time, give it some time, be on it, be serious with what are doing, be consistent, then work on the quality, and with time, you see a lot of people buying your products and referring to other people. And that's it. If you had a chance to change one thing in Africa, what would it be? Oh, the way we see ourselves, you know, that they cannot do with spirit in Africa. Something has to be done about it. Something has to be done, especially in our youths, you know. We just have to believe that we can do it. And we have to start. Yeah, that's what I... Why do you think Africans don't believe that they can do it? Yeah, because, most times, when you see the value of maybe some foreign products, you see that something is wrong. When we can actually have the better products down here, and everybody is thinking that it has to just come, it has to cross the border for it to be a very quality good. So that mentality has to change. We have the best going on in Nigeria and in Africa. And we are good. If we talk about the installation, the machines installation, we had reasons to bring those our expatriates to come for the installation of the machines. But we decided to do it here. We did all the installations and we have not brought any of them here. Africans, we can do it. If we believe we can do it. We can do it. We are doing it and it's working. Brother, I want to say thank you so much for talking to me. I really appreciate your time. You know, I'm not a brand ambassador for this company, but I think each and every African out there needs to support an African business. That is why I'm here. I'm even wearing their t-shirt. Bro, you didn't give me any money to be here. You heard about me. You have big passion for Africa. Exactly. Yeah, you're working so hard. And I came in here. So, hey, if you're living in Nigeria, you have to support this brother's business by buying the paint. If you are living outside a country and you want to do business with him, I mean, it's time. You know, they've already launched Intercontinental Trade. It's time for you to trade with a different African country. Let's make it possible. Let's everybody know that Africans can actually do business together. No matter where you're coming from, we all come together. That's one Africa. I'm a huge boy, Mr. Ghana, baby. And don't forget to like the video. Don't forget to share and don't forget to subscribe and be part of this awesome family. I'm going to see you in the next one. Aya Maya.


Derek doesn't make a big deal about the things that he does so a lot of times it's after the fact that that I learn what things have come from him. There has been an allotment of Nike gear that consistently comes in that he has donated. My first year here we did t-shirts and we did like these really sweet Nike half practice pants and we did shorts. There was an allotment of $5,000 for the weight room that Derek sent us. The other piece that he I know for sure has donated was he reconditioned helmets through some of the money that he sent us and that's important because you got to get those helmet certified every year. That's a player safety thing. I know that's big for on everybody's mind and it's a big piece of the puzzle. He just has that kind of spirit. He is a kind and generous person who doesn't forget people who took care of him when he was young. If we don't have what Derek has been able to do for us I've got to take money out of improving our practice field, improving our game field to get our guys some essentials like that Nike gear. Like it's important for us to feel like a football team. This past year we went with the jacket I'm currently wearing, pants that match, yule up yule up a side with the embroidery. Every day of the week you could tell a football player because they are proudly what strutting around in those tracksuits. You should see the kids faces when they got them. It was incredible. Like this jacket, this this jacket pant combination was the best thing that could have happened. This team didn't have those kind of things when Derek was here. They were literally sharing shoes and and belts and and things like that. But so I'm sorry that makes me a little emotional to think about that for him. The experience he's giving these kids that he didn't have for Derek to care so much about us. That's showing what he's about in his character, his selflessness for this community, his love for this community and where he grew up. He doesn't have to do the things that he does. He's doing this out of his own generosity. It's selfless. He's never forgot about yule, Florida.


I'm Robert Walkman, staff photographer at the Los Angeles Times. And I'm Allison Sheridan, the chief podcaster at the NoCillicast podcast, hosted at podfeed.com. We decided to take a look at the iPhone 5s camera. We wanted to look and see, is it really worth going to the iPhone 5s if you already had an iPhone 5 just for the camera alone? We're not looking at any other features except the camera. But then we thought, well, you can't just look at those two cameras, right? What's the big daddy in the iPhone, in the phone category for cameras? I would say Samsung with their S4, probably one of the more popular Android phones. Right. But then you've got the Nokia 1020. I know it's Windows phone, but the Nokia 1020's got that 41 megapixel sensor. We've got to take a look at that, right? That's certainly a lot of megapixels in the phone and even in a regular camera. That's true. That's true. And then the HTC One, I hear a lot of people say, well, yeah, but don't forget the HTC One camera. It's been a lot of people like it better even than the Nokia 1020. I knew this was going to be a big project. It's a lot of cameras. It's a lot of pictures and to try to get it done, you know, in a short amount of time. Even with help. That's why I enlisted Allison to help out because I knew I couldn't do it myself. So we thought we should take a lot of different kinds of pictures. One of the things I have a pet peeve with my phone cameras has always been trying to take a picture of a bright red flower. They tend to oversaturate and you just get these washed areas of all the exact same color of red. So we went in the search of some boogin value. We found some, I thought they were red, Robert, you said they were fuchsia. That's correct. Yeah, I don't want to be wrong in the whole color thing here. And so we took some pictures of those and we took the cameras and we took two or three or four or five sometimes of each scene that we looked at. And then we took the best picture by each camera. So we showed the best of the best of the best in each case. Then we tried outdoors in bright sunlight, boat scenes, reflected sunlight. And actually, those were the ones that were most similar. The biggest difference we saw, I think, was in low light. Right. So low light is where you take, what, at least half of your photos. You take them indoors, right? And it's really the weakness of most of the smartphone cameras. Because if you ever notice, how do your flash pictures look when you snap that and look at it? I like the green tinge that it gives Grandma. I think it's really attractive. I don't know. Sometimes it's almost hard to look at how bright and how washed out it can be. And the iPhone 5S, this is one of the big improvements they say. So we decided to take a look at that and does it make a difference? Is it going to make me want to get a new phone? And you have, we're also comparing these to what you know you can do with a cheap point shoot, not even expensive DSLR. So you were kind of looking at it in that light, too. Would you ever keep any of those photos that you got from one of the low light photos, right? Well, the problem is you're always carrying your phone with you. Good point.  We decided we would rank them and see which did the best and which was our favorite camera. Exactly. So let's head into that section next.  Yeah. I really wanted to take a picture of some red flowers because red is a really hard color, I should say, for cell phone cameras to take pictures of. Actually, I think it's pretty hard for most camera state pictures. It's a difficult color for those sensors to reproduce. Okay. What I find is that I'll take a picture with my older iPhone and the reds just get oversaturated. It's sort of like the whole area, like a whole leaf will all be the exact same color of red, even though my eyes can see a lot of variation in color there. And it was bright sun, so this was a difficult situation for that camera to capture. Awesome. It was perfect. Okay. So we're taking pictures of some fuchsia flowers. They were some bougainvélia. And the iPhone 5s was more muted than the others in the way the color was reproduced, but it was actually more accurate to what our eyes saw. I think it was the most natural. The iPhone 5s made it look like you really saw it. Right. Now, the Nokia 1020 was way oversaturated. There were big sections of the flowers that were all the same colors. I didn't think that it did a very good job at all.  I'd have to agree. And then the other three cameras were kind of in the middle. They were more vibrant, somewhat saturated, but not too bad. Next we headed over to the marina and we found some brightly colored kayaks. I think it was good. It really gave us a good photograph to take. It really showed a lot of contrast, bright sun, vivid colors, and actually some deep shadow. How do you think the cameras did? In this case, the iPhone 5s did a really good job of getting the brightness of the colors. They were bright yellow and bright blue, but it also had the dynamic range to pick up the rocks that were under the water in the foreground of the bay there. It definitely did the best. I mean, to see that shadow detail in the water was pretty good. The other cameras had a tendency to overexpose on the yellow kayak. The yellow kayak got wrecked in, I know it was in the 1020 and I think the Samsung might have done, Samsung did an interesting thing. The kayaks looked beautiful and there was no detail at all in the water. You might have looked at that and went, wow, that's an awesome photo and taken it home. Maybe a matter of taste there. If you wanted the full dynamic range, you would have been happier with the iPhone 5s. Yeah, I think the 5s was definitely the standout from all the other cameras. Next we decided to head over to the jetty. This was a perfect spot because there were sailboats in the background, beautiful blue sky, and this is where the Nokia Lumia 1020 really stood out. The photographs from it were really beautiful and you could certainly zoom into that sailboat a lot better than any of the other phones. Wouldn't you agree? Absolutely, absolutely. I do want to mention two cameras had a lot of trouble with this scene. The HTC One, three of the four photos I took were completely out of focus and I don't know why. It had a terrible time focusing with that scene and the one that was in focus when we got it back and looked at it in high res, it was very over sharpened by the software in the camera. So that was a disappointment. The real disappointment, we were really liking, both of us were liking the Samsung Galaxy S4 in bright sunlight and blue skies, it was really rich and vibrant, looked really beautiful, but the camera kept failing. We have a photo of it failing, actually, that we'll show you here. And we would reboot the phone and it would fail again. We'd get one photo out of it and it would fail again. So unfortunately the Samsung Galaxy S4 lost any chance in the running and it was kind of a disappointment. I thought the 5s and the 5 both looked about the same in the scene. Yeah, they looked good. And they looked good. They didn't look bad. You know, they looked okay. The Nokia 1020, just enough better to definitely take the spot. Then we started zooming in and we had to do a lot of math here to figure out exactly how to compare them. This is a little over my head, so hold on here. So let's get out our calculus books here. But the Nokia 1020 is a 41 megapixel sensor, but that's as a circle. So when you actually take a rectangular photo, you don't get as many pixels. And we were doing these at 4 by 3, so we'd have comparable photos because the 5s and the 5 take a 4 by 3 photo. So you actually get fewer pixels and I forget exactly, it's 30 some odd you get. So that's four times as many as the 8 megapixel sensor in the iPhone 5s. So it should be, you would think you could get four times closer. It turns out that's in two dimensions, so you have to square it to get four times, which means it's only twice as big. So when we zoomed in to 200% on the iPhone 5s picture, the boat we were looking at was the same size as the Nokia 1020 at 100%. It looked way better than the iPhone 5s, but it was only twice as big. If that makes any sense at all, half as pixelated. It doesn't make sense to me because I just go out and snap the photos. And I think that's what most people do. Don't you think? Exactly, exactly. But some people do want to zoom into photos. So I think it is an important factor. But it's not eight times better. No. Or 10 times better. It was about twice. It's about twice as much. Next, we decided to take a look at low-light photography. This is probably one of the areas I thought cell phones did the weakest, wouldn't you say? Absolutely. And it's sad because that's where you probably take more than half of your photos as indoors at parties and things like that. Yeah, it's probably never a great thing. I mean, the camera can only do so much. Now, the iPhone 5s did promise that their low-light photography would be a lot better. And it appears it was, wouldn't you say? Yeah, it was definitely better than the 5. By the way, we're talking low-light without a flash right now. We'll get to the flash in a minute. The 5s was pretty good. It had the least amount of grain and noise in it, I think. What was interesting, though, was that the Nokia was just dark. It was underexposed. A little bit, yeah. So we broke our rule. Yeah, we used the exposure compensation dial. And when you got the correct exposure, that photograph was excellent. It was way better. Yeah, it was a really good photo. It was really good for low-light. So I have to say, if there's situations where you would actually need to use a lot of low-light photography, having that Nokia Lumia 1020 really did a nice job. Yeah, yeah. So if you're someone who is willing to do that kind of thing, if you just want to be able to go click, click, click, click, you're not going to get a very good picture with a 1020.  And with enough light, with opening that up, it actually came out with a really nice photo. Yeah, I said. So straight out of the box, the 5S. But if you were willing to put a little effort in, you could go way above that with the 1020. I agree. Some of the other phones also didn't look probably, what would you say, was the worst of the group? Definitely the Samsung. The Samsung was horrible. If you didn't like someone, take a picture of them with a Samsung Galaxy CX-4 in low light. I'd have to agree. There was a lot of grain in that photo. Next, we tried flash photography. And again, this is one of my least favorite areas with a cell phone. Just not good. No, no. Attractive. And again, I'd say in this situation, iPhones knew to flash system did the best. I mean, it's better. It's not as nice even as your typical point to, I wouldn't say, or compact camera. I was hoping for more out of the 1020. But we got a really strange color cast. That was a little yellow also. It seemed like we got a color cast off of the others. I think which one was red and which one was green. Yeah, the Samsung, I think, was pink. One of the HTC One was green. The 5 was just a little too bright. But again, we didn't think any of these photos were ones you would frame. No, that's for sure. So in this situation, I guess they're improving. That would be the best way to look at the way these came. Yeah, it's getting better. But that's it. After looking at all the smartphones, I would say the iPhone 5s was the winner. If you're looking for a camera that you're just shooting on automatic in all types of scenes, what do you think? Yeah, I think so. I didn't think it would do as well as it did in so many different categories. Because we went back and forth a lot trying to judge these different cameras. The two front runners are probably still the 1020 and the 5s. But looking at the 5s versus the 5, I'm not sure I saw enough to make me go, wow, you've got to get a 5s just for the camera alone. I would agree. I would not probably upgrade. And that's actually, I wanted to see the camera to make that decision. Now, we didn't get into the advanced features, like we explained before. So on the 5s, it's got a slow-mo feature that's really fun. It takes pictures of a video at 120 frames per second. You can slow pieces down. I know we've been having a lot of fun playing around with that. And then it's got a burst mode where I'm not exactly sure how it works, but it's supposed to take the best of a lot of shots. So there are some other functions just within the camera. And again, like in the 1020, we didn't explore all of the different functions of that camera either in its advanced features. Yeah, I think the 1020 is a great camera for someone who wants to have features like.  Yeah, more control. Manual focus, manual. Yeah, the exposure compensation. It's got white balance you can change. Yeah, not many. Exactly. If you've got family and friends who will sit still while you do that. That's a good point. And we found that you were able to zoom in the photos more. So it kind of gives you what they're calling a telephoto just by having more pixels and being able to have a larger photograph. Right. And it doesn't sacrifice much in quality with the smaller sensor size. I mean, that's smaller sensor size, the smaller pixels. But it makes up for that in the way it combines those pixels together to make a better photo. So the 1020, I'd put it awfully close to the 5S. Yeah, it just depends. I just, it had a few problems in the low light and a little bit of a color shift in some of the photos, which I think overall the iPhone 5S just did a better job. I'm afraid I have to say so too. Taking a look at all these cell phones was a lot of fun. But it was a lot of work. And I couldn't have done it without Allison. You know, with her expertise, there were issues with the computers, the cameras. We had to figure out how to get Windows phone photos onto a Mac and Android photos. It was awesome. I think we're probably ready for next year's crop, right? Exactly. If we start now, we'd be done by then. That's a good idea. For the Los Angeles Times, I'm Robert Lochman.