 Two electrically charged objects are brought near each other, and before they physically make contact, somehow each is aware of the other's presence. It's as if some unseen force reaches out through even empty space and makes contact between these two objects even before what we perceive as physical contact can occur. The physicist Michael Faraday hypothesized that just such an invisible field of force was present whenever net unbalanced electric charge was present, and we'll begin to explore that idea now. In this lecture I'm going to cover the basics of the electric field and give an example of electric fields from a very simple charge distribution, specifically the electric field of a single point charge and what it does to other point charges around it. The concept of a field of force is something to which you were introduced specifically in the context of gravity. Mass is the source of gravitation as a force. It can act on other masses at a distance with no physical contact, and the way that we describe how mass responds to the gravitational fields created by other masses is through the concept of a field of force. At every point in space there is a force vector that points in a predefined direction and that direction is given by the law of gravitation. Similarly, as we have now learned to describe the electric force using Coulomb's law, Coulomb's law is the central law that tells us how one point charge exerts a force on another point charge through the electric force. We can adapt that idea into the concept, which is a more general concept, of the electric field as the fundamental object and then charges responding to those fields. We're going to transition now much as one does in gravitation from the idea of the gravitational force to the gravitational field, from the idea of an electric force to an electric field. The field is more fundamental than the force. The force is the consequence of the presence of the field. The idea of using a field of force vectors to describe the action at a distance that we see in the electric phenomenon is really credited to a brilliant chemist and physicist named Michael Faraday. He lived from 1792 to 1867. He is one of the most important figures in the last 400 years, 500 years of physics as it developed as a fundamental science. He came from origins in English poverty. He was an Englishman by birth, but he was not wealthy. He did not come from a wealthy family. And as was common in the time, if one was not born into wealth, it was very difficult even with an education self-generated or otherwise to break into the scientific elite of England of the day. The elite largely were monied individuals and because of the class separation in England at the time, it was very difficult for somebody coming from poorer roots to imagine a successful career even though they may be brilliant, even though they may be motivated, it was very hard to break the ceiling of poverty and get into the scientific elite. But through his own persistence and his own genius and his own abilities, he was able to break into England's nobility controlled scientific elite and he did it first by basically being an assistant to a well-known scientist at the time. But through experiments that he conducted as that person's assistant, he began to distinguish himself as his own scientist and eventually not only rivaled the person that had mentored him while he was simply a laboratory assistant but surpassed his mentor in ability and reputation. And he introduced the concept that one can describe the spooky action at a distance of the electric force reaching out across space with no physical contact through the field concept. That is that you can picture tendrils of force stretching out in space and any charges that cross those tendrils will be subject to Coulomb's law, to the Coulombic force, the electric force. We can begin to explore the concept of the electric field by first appealing to what we have learned from Coulomb's law about the force between two charges. Let's begin by considering two elementary charges. So the elementary charge is a magnitude of charge in Coulomb's equal to 1.6 times 10 to the minus 19 Coulomb's. So let's imagine that we have two charges, one negative E and another one up here, positive E. And let's also imagine that the negative electric charge is fixed in space. It's not allowed to move. So we'll call that fixed in place. The positive electric charge, on the other hand, is free to move. And what we know from Coulomb's law, what we know from observing the electric phenomenon, is that charge exerts a force on other charge. And in this case, we have dissimilar charges, a negative charge and a positive charge. And so we expect to have an attractive force between them. Now, since the negative electric charge is not free to move, let's consider what happens to the positive electric charge. We can imagine the line of force that connects these two particles. We can draw it here, for instance, as an arrow, a vector. The vector points from the positive electric charge to the negative electric charge and indicates the direction that the Coulomb force will point between these two particles. Again, these are oppositely charged particles, so we expect them to attract one another. And in order to attract, the force on the positively charged particle must accelerate it toward the negatively charged particle, and so that force must point toward the negatively charged particle. We can then imagine what happens if we were to consider another moment in time. So if we had labeled the original position of the positively charged particle as time equals zero seconds, and then we consider maybe a millisecond later, so at t equals one times ten to the minus three seconds, what has happened to the positive charge? Well, in the intervening time, it has been acted upon by the Coulomb force from the negative electric charge, and so we would have expected it to accelerate, and therefore it should have moved. It should have changed its position in space. And so it could be, for instance, that in that moment later, where time is now equal to one times ten to the minus three seconds, that the positive electric charge is now over here, continuing its acceleration toward the negative charge. And again, in that moment in time, we can consider what is the force on the positive charge due to the negative charge. The arrow that I originally drew has two pieces of information. One is the direction in which the force points, and the length can be taken as representative of the magnitude, the strength of that Coulomb force. We know from Coulomb's law that as the distance between two particles decreases, the force between them increases, and in fact, if the distance between them goes in half, the force between them increases by a factor of four because of the relationship that the force goes as the one over the distance squared. So we would expect now that, having moved a little bit closer to the negatively charged particle, if we were to now draw the new force arrow, well, it would still point toward the negatively charged particle, but its length, its magnitude, its strength will be much greater now because it is much closer to the negatively charged particle. So we see here that if we were to imagine more times in our picture that at some point, the positively charged particle would reach the negatively charged particle. And what we see here is a line, a line that, for a particle that starts out anywhere along this line, it will be accelerated along that line toward the negatively charged particle if it has positive charge. And so we can imagine that anywhere we dropped a positively charged particle on this line, it would accelerate along the blue vector that I've just drawn. What if instead I had started a particle somewhere out here? What if instead I had put the particle off to the right of the negatively charged particle? And again, this is going to be an elementary charged particle with positive charge. Well, again, it would have felt some acceleration, some force due to the Coulomb force, and it would have moved. And then as it approaches the negatively charged particle, the force will become greater and greater and greater and so forth until finally it reaches the negatively charged particle over here. And we might imagine that any particle that comes along this line here would follow this route. And I can draw this extended all the way out to, let's say, infinity. If we were to put infinity over here, infinity way over here, infinity is anywhere off the page. We begin to see a pattern. The lines of force between a positively charged particle and the negatively charged particle that is acting on it point inward toward the negatively charged particle. And no matter where we drop a positively charged particle here, if we were to start it down on this line, it would come in along this direction. If we were to start it over here, it would come in along this direction. If we were to start it up here, it would come in along this direction. Not my best work, but you get the idea. This leads us naturally into the concept of the force field, which you've already seen in gravitation. Instead of having to draw all these situations where we start with a particle and then we move it and then we see how the force has changed and we move it again and we see how the force has changed again, we can instead represent the force vectors themselves around the particle that is generating the force. And so again, if I were to place my negative electrically charged particle here, then those lines of force would all point inward like this from all directions. I'm only drawing a few of them here. We notice a few things. If I were to decide to draw, let me keep this symmetric here, so I will draw eight total lines. If I draw eight total lines of force that are meant to simply be representative of all of the lines of force that this thing can exert in any direction, we notice a few things about the drawing. First of all, the lines increase in density as you get closer to the negatively charged particle. They point inward toward the negatively charged particle and they draw closer together. The spacing between the lines here and here and here and here is a lot smaller than it was out here. In a pictorial representation of a force field, this helps you to understand what the strength of the field is doing. If the lines of force are closer together, the field is stronger at that point in space and if the lines of force are further apart, then the force field, the force that a particle would feel is weaker at that point in space. So in this pictorial representation of a force field, you have a lot of useful information that helps you to remember how the electric force field is going to work. Now if you were to repeat this exercise for a positive elementary charge, again putting in the space around it another positive elementary charge, you could map out what the electric field, the force field is going to look like around a positive elementary charge. When we used a little positive elementary charge to kind of probe the force field around the negative elementary charge, we found this picture that all the lines of force point inward toward the negatively charged particle. Again, using a positively charged particle as our probe, this is what the picture looks like. And in fact, it is standard in all of electricity, all of physics involving the electric field and electric force, to use a very tiny, positive electric charge as your reference point. Thanks to Ben Franklin who assigned the definitions of positive and negative and all of the then conventions that followed from that choice, these days we're stuck with the fact that anytime we talk about electrical phenomena, we're doing it with reference to what a positive electric charge would be doing in the presence of an electric field. It's a convention and it is one to be memorized because it is going to repeat throughout the entire course. Whenever you need to know what an electric phenomenon is doing, you have to think about what a positive electric charge would do in that situation, even if the problem on its face involves the motion of negative electric charge. This will seem a little weird, but it's convention, it's to be memorized, and it's something that's gonna come up a lot and I'll repeat it as we go forward in the course. Now let's go over here to the positive elementary charge that I fixed in place here in this grid and think about what happens to our little positive test charge. I'll go ahead and finally label this. This is our test charge. It lets us test to see what the force field would do to a little charge that's positive in sign at that point. Well, like charges repel and so we expect the line of force that the elementary charge fixed in place exerts on the elementary charge we're using to probe the field. We expect that line of force, not the point toward the positive charge, but away from the positive charge. And in fact, anywhere we put our little test charge around here, we'll find that the lines of force radiate outward, not inward from the positive charge. And so we have here our first pictorial representation of fields of force for the electric force. One can test the pictorial representations using the notions of Coulomb's law and see that it gets all the features right as one goes further away from the charge. The force field weakens significantly. If one increases the magnitude of the charge that would be akin to adding more lines of force. So for instance, if I wanted to look at the force field, not of a plus E charge, but a plus two E charge. So let's say that that's what this thing is here now. So I'll make this dot more significant to indicate that I've increased the charge. Well, all I have to do to represent that is double the lines of force that I've already drawn. That's it. So if you want to pictorially represent relative charges, compared to one another. And they have simple relationships to each other. One of them is plus one E and one of them is plus two E or minus three E. You could do a pictorial sketch of this by just drawing a few representative lines of force for the smallest charge. And then for the bigger charges, you can multiply the number of lines you draw. Obviously this gets unwieldy at some point. This is why we have computers to do this for us. And it would be better to use a computer at some point to do this work. And in fact, we'll walk you through a demonstration of exactly this. So we have here pictorial representations of the fields of force. And now we can explore some basic concepts mathematically with the actual electric field. We've been talking about a visual representation of force fields. At every point in space, if we were to drop a tiny little test charge there with positive electric charge, we could figure out what the force is on that charge due to any other charge in the space around it. And based on that, we could map out what the force field looks like. We just have to draw a little vector representing the force and then accelerate the particle along that line of force for a moment. And then in the next moment, repeat the whole exercise again. And by doing this, we could map out trajectories in electric fields. And in fact, I'll show you how a computer does this in a moment. It does it very fast compared to a human being. But we need to introduce some basic mathematical concepts here. It is obviously inconvenient to have to keep talking about a infinitesimal little charge that we drop around a field and then worry about what the size of that charge is. That's kind of nonsense. And in fact, in physics, we have a simplifying concept that lets us forget about for the moment what that charge is that we've dropped in around the electric field and actually just have a concept independent of the test charge's size that lets us write down a mathematical representation of the electric field. We know from Coulomb's law that for any two point charges, the force between them looks like this. There's a constant k and then we have the two charges multiplied. We have the distance between them squared and the denominator. And we have this vector, which indicates the direction from the source of the force to the recipient of the force in Coulomb's law. And it's merely a direction that has no magnitude. Its length is one. Now we could apply this equation to the picture drawn over here at the left. I've drawn q1 and I've said the q1 is a positive charge and I've drawn q2 and I've said q2 is a test charge and that also makes that one positive. And so we could go ahead and work through Coulomb's law if we knew some of the distances involved. Can't figure out what exactly the force is here over where q2 is located. But we're much more interested in the electric field and its behavior. And the electric field is a concept that should be independent of what charge we are using to probe the force around the central particle. So in this case, let's consider the source of the electric field to be q1. And all we care about is its electric field. All we care about is the direction that its lines of force point and the strength, the length of those lines of force. And we could always multiply by charge later, add charge in somehow to get the exact force out. So we can define a new concept here based on the electric force known as the electric field. And it will be written as a capital E with a vector hat over it. And it will be determined merely by taking the electric force and dividing it by the charge in question. In this case, it's q2. So I have an equation for the force up here. And if I would like to know just what the structure of q1's electric field is without worrying about exactly what q2 is, all I have to do is get rid of q2. And that naturally leads into this new quantity here called electric field. You'll note it is a vector just like force. So that means it has magnitude and it has direction. And quite generically, if you can write down what the electric force equation is for any simple or complex distribution of charges, you can write down the electric field that corresponds anywhere in space to that distribution of charges. Now, we're gonna pick a simple one and we're going to just look at that mathematically, how that looks and we'll exercise in class together, problem solving using this concept. So let's look at the point charge, which I've conveniently already drawn here. So I've drawn a point charge q1 and I've drawn schematically what its electric field ought to look like based on our exercises from a moment ago. And I can now substitute in with Coulomb's law, which is the force between any two point charges. I can substitute in with Coulomb's law to get the exact form of the electric field of a point charge. Okay, so let's go ahead and do that. So I know that in general, the electric field is equal to the electric force divided by the test charge. And in this case, I know exactly what the form of the force is. It is kq1q2 over the distance between them squared. Let me label that R12. And then I have the little unit vector R12 hat. And then of course, I still have to multiply by one over q2, which started out in the denominator over here. This is nice. You see right away, this is a really convenient concept. This serves to simplify Coulomb's law to a slightly simpler equation without a second charge. And that's what's really beautiful about the electric field concept. You have kq1 over R12 squared, R12 hat, and that's it. So this right here, a slightly simpler equation than Coulomb's law, has all the information we need in it to describe mathematically the electric field of a point charge. So let me be very clear about this. This is only for a point charge. So if you are working a problem where you can either directly use a point charge or approximate something as a point charge, then this is a great formula. If you can't approximate something as a point charge, then you have to go back to the original equation and figure out exactly what F is, and then use F to solve for E. So not all problems will necessarily be simple, but there may be some simplifying assumptions that you can make, or you might be handed a really easy problem involving just a couple of point charges, in which case this is very straightforward, and you can apply this directly. So the electric field of a point charge can be written directly as K, Q1 over R12 squared, R12 hat. And you can say here that we are looking at the electric field of the point charge, Q1. So let's see that this has all the right features. So again, let me draw my point charge, Q1. Now, I'm gonna draw another point. I'm gonna call this point P. It's just a point in space. There's no charge there. It's just a point. We're gonna look at what this vector, E vector sub point looks like at that point P. Well, we can use the lessons, Glean from setting up Coulomb's law problems to understand all the pieces here. So K is just a constant. It's positive, so it's just a number. And whatever it does at the end of this, it's just gonna multiply everything else by some positive number. It doesn't change the sign of E, doesn't change the sign of any other numbers in there. So we can ignore it. It's just a number that scales up a vector, up or down. We have Q1. Well, in this case, I said that Q1 was a positive charge. So I said that Q1 is greater than zero. It's a positive number. And let's keep that kind of in our back pocket. That's gonna come in handy later. So Q1 is also a positive number. It's just gonna multiply whatever else is in the equation to scale it up or down. Then we have one over the distance squared. Okay, well, so here's one and here's P, which we can call point two. So the point P is also point two here. And we know that the convention is, is that the vector R12 always points from the source of the force to the place where the force is acting. So let's go ahead and draw that. So here is, by convention, R12 vector. And it points from Q1 to the point P. Well, our hat is just the unit vector that points in that same direction. So there's our hat, one, two, and it has a length of one. And then finally, the magnitude of R12 vector is just the distance between these two points here. Okay, whatever this distance is in meters, centimeters, millimeters, whatever it happens to be, that's what you need to get for the magnitude. Magnitudes are positive. So we know that R12 squared is also positive. And our hat is just a direction. And in this case, it points from Q1 out to two. So this lets us now write down the direction of the electric field due to Q1 at point P. We see that the only thing that carries direction information is R12 hat. And it points from Q1 to the point P. And so we can take that vector and we can use it to represent the direction that the electric field points. So up here at two, we can transport that vector. And we know that that is what E hat is going to look like. That is the direction that E points in is the direction that R12 points in. And then finally, we have to multiply by Q1K and one over R12 squared to get the length of this thing scaled correctly. And so we don't know quite what this is gonna look like. I haven't told you what Q1 is. We know what K is. I haven't told you what R12 is. So I'm just gonna make up an arrow here. I'll just make it longer than E hat for now. And we'll call that E vector. And so what we found is K is a positive number. Q1 is a positive number. R12 is a positive number. R12 hat is a direction that points from Q1 to the point P. And it only carries direction information. So we know already that none of the numbers here change the sign of R12 hat. And so E vector, E point in this case, will in this particular instance point in the same direction as R12 hat. Now let's consider a slight modification to this. Let's consider a negative charge. So let's say we have negative Q1. So Q1 was a positive charge and now I'm gonna flip it sign. And then again, I'm going to have a point P where I'm measuring the electric field. And so I can set all this up again using the formula. K is a positive number. I've now flipped the sign of Q1 and made it negative Q1. So let's keep that in mind. We'll hold that in our back pocket as well. R12 squared is still the distance squared between these points where Q1 is located and where P is located. So that's R12 magnitude. We know that the R12 by convention, R12 vector still points from Q1 negative Q1 to P that hasn't changed. And we know that R hat 12 also points, indicates that direction from negative Q1 to the point P. So the picture hasn't changed. The vector R12 still points in the same direction it did before for the positive charge. But what has changed is the sign of the charge. And so now for a negative charge, if I were to write down E point, what I would find is this is equal to K negative Q1 over R12 squared, R12 hat. And I can, if I'm gonna denote this as the electric field of a negative point charge, I'm gonna put a little minus sign up here in the superscript. And I'm gonna pull this minus sign that's multiplying Q1 out. I can move it anywhere I want. If writing this here is equivalent to writing negative one times Q1. So I can move that negative one anywhere I want. All of these numbers are multiplied together and I can multiply the negative one by anything I like. So I'm going to choose to move it entirely outside of this entire expression and then group all these terms together. And what we see is that whereas the positive electric point charge field pointed outward from the positive charge, the negative electric point charge field has the reverse direction. That minus sign flips the vector of the positive electric point charge field 180 degrees. So over here, we had Q1, which was a positive charge. It's electric field lines point outward like this. Down here, we have negative Q1, a negative point charge. And all the only difference between these two, assuming that the magnitudes of the charges are the same, is that the arrowheads flip. And we see that this beautifully mirrors if you were to rewind the video and look at the sketch we made of the force fields using a positive test charge. This beautifully mirrors what we saw in those pictorial representations of electric fields using positive test charges. That is, this electric field and this electric field point in the directions that would represent the lines of force on a positive test charge dropped somewhere in space around the charge we're probing. But without actually having to have that test charge appear anywhere in the picture, that's the beauty of the electric field concept. And what you see here are in fact the electric fields four point charges, positive point charge and a negative point charge. Having motivated the pictorial representation of the electric field by starting from the force concept and then having defined mathematically what the electric field is in terms of force, that is, it's force per unit charge, force divided by test charge, for instance. We can now identify the units of the electric field. If the units of force are newtons, which you learned in 1307, and the units of charge are coulombs, then if one takes force and divides it by charge, the units of electric field must therefore be newtons per coulomb. And we can see that very clearly in this table, which comes from the textbook. We can then consider what different electric field strengths that may be common to you or perhaps unfamiliar to you would be like and how they compare to one another. So for instance, if you were to sit right at the surface of the nucleus of a uranium atom, remember a nucleus of an atom is a collection of protons bound together with a collection of neutrons. And it's the number of protons that determines the chemical element. If you were sitting right on the surface of uranium's nucleus, you would experience an electric field of strength in newtons per coulomb three times 10 to the 21. And that's because you're essentially sitting right on top of a ball of fundamental particles and you're right next to them. So you're getting almost the full effect of their electric field. And think of the protons as point charges and then think of putting yourself next to a whole clump of point charges. If you remember the field picture that we had a moment ago, you're going to experience the strongest possible electric field that you could get around a bunch of protons by sitting right there. So this is a big field. Three times 10 to the 21, newtons per coulomb is huge. Now, if you were instead to travel out from the nucleus and go, you know, let's say within a hydrogen atom out to a radius that's approximately the typical orbital radius of an electron in a hydrogen atom, something known as the Bohr radius, which is about 5.29 times 10 to the minus 11 meters, if you were to sit at that distance from the single proton at the center of a hydrogen atom, then you would experience an electric field of strength five times 10 to the 11, newtons per coulomb, significantly weaker. You're now further from a single proton because there are fewer protons in hydrogen than there are in uranium, and because you're further away, you would expect a significant fall-off in the strength of the electric field, and that's certainly what we see here. What about something we're more familiar with? What about sparks? Static electric jumps, the charges in the air, lightning strikes, things like that. What happens when that phenomenon occurs? When you see a spark in air, which I demonstrated in class, what you're seeing is the movement of electric charge as the nitrogen molecules in the air are ripped apart and become perfect conductors. So normally air is an insulator. It doesn't really allow electric charge to move very freely, but if you put a strong enough electric field on air, eventually you will tear the electrons, let's say in nitrogen, away from the nuclei in nitrogen, and you will separate those charges, and you'll create what's known as a plasma, and a plasma is merely a gas of electrically charged particles that are weakly or not at all bound to one another. And as a result of that, you'll essentially generate a perfect conductor. This is something known as breakdown. This is when you take a conductor and you turn it, you take a, this is something known as breakdown. This is when you take an insulator and you turn it into a conductor by subjecting it to such a huge electric field that it changes the properties of the material and completely alters the chemistry of what's going on in the physics of what's going on. So what does it take to break down air? Well, it takes an electric field of about three million Newtons per Coulomb, or three times 10 to the six Newtons per Coulomb, to cause air to become a conductor. This is bad. You never want air to become a conductor, especially for instance, if you're in the midst of storm clouds, if you are anywhere near air in the presence of an electric field that strong, you are about to get struck by lightning because the air molecules are about to break down, making a perfect path of conduction from wherever the charge is built up in the clouds to wherever you are, and you will receive some or all of the full brunt of that blast of electrons, as it moves, say, from cloud to ground, or if the electrons are building up in the ground from ground to cloud. It's bad either way. What about near the charged drum of a photocopier? Well, photocopiers make images by using electric fields and electrically charged droplets of toner. So those toner particles that you get in the cartridge and a laser printer or in a photocopier drum, those are designed to be attracted to specific locations by electric fields, and in doing so, make little dots, and building up all those little dots will replicate the original document. The electric fields inside of a photocopier drum are typically something like 10 to the five Newton's per Coulomb. So not air breakdown because that would be bad if the photocopier drum was capable of making electric fields at a million Newton's per Coulomb, three million Newton's per Coulomb, you'd have sparking and arcing all the time inside of a photocopier, and that would be really terrible. It would destroy the photocopier. What about near a charged comb? Have you ever brushed a plastic comb through your hair and then gotten a spark off of it? Well, try it sometime if you have never done it. On a nice dry winter day, run a comb through your hair. You'll notice your hair stands on end. That's because you've either added electrons from the plastic to your hair or soaked electrons up from your hair and put them on the comb. But this will cause a net charge on your hair that will make it frizz out. And you'll have a corresponding, equal but opposite sign charge built up on the comb. And if you were to move that near a metal sink fixture, you could get a spark that would jump through the air. So the electric fields near a charged comb are typically about 1000 Newton's per Coulomb. In the lower atmosphere where we basically live, typical electric fields are sort of the order of 100 Newton's per Coulomb. That's good because otherwise lightning would be a very common phenomenon. And if you were to go inside of the copper wire of household circuits where electric fields are used to move charges from one place to another. Remember I said earlier that electrons in wires, electrons in conductors are like water in plumbing pipes. You can move them around, you can make them do work for you. Well, copper wire is the plumbing of household circuitry. And the electrons are the water that flows through them. And the way you get those electrons to move is you set up electric fields. And those electric fields don't need to be very strong. They're typically a hundredth of a Newton per Coulomb. Electrons aren't that heavy, they can be easily motivated to move from place to place. And so you don't need a strong electric field to move things around. Nonetheless, the electric fields inside of home wires, copper wiring can be dangerous and you should never touch them because if you touch them and the other half of your body is connected to ground, as we'll learn later, this will cause you to get electrocuted. And it's not a pleasant experience. So having pictorially introduced the electric field concept and then having looked at the electric fields from singular point charges, we can now make a more complicated situation and look at the dynamics of charges in the presence of electric fields, just again, point charges. But we're gonna use a computer program to do this because it's a lot easier to have a computer do repetitious calculations than it is to do all this sketching by hand. And this is the last thing we're gonna look at in this lecture. I've been doing a lot of hand sketching but hand sketching only gets you so far. It's much more interesting to see the dynamics of electric charges and the fields around them, what happens to the fields as electric charges interact by using a computer. The computer is capable of taking the point charge equation, the electric field equation in general and implementing it and doing thousands or millions of calculations a second and updating pictorially and accurately the representations of the electric field. So I have here a grid which represents a space, a two dimensional space into which I can put electric charges. So let me go ahead and do that. And what I'm going to do is I'm going to put a negative electric charge. I can choose its charge, its sign and its mass. I'm just gonna leave these properties alone for now and go ahead and add one of those into the picture. And you see here that this creates a little ball which represents a negative electric charge. And what's kind of cool about this is that I can put it here on one of the grid points in the space. And what you see in the space around it are those little vector field representations of the lines of force. If I were to drop a little positive test charge in here, that would then tell me where the acceleration on that positive charge would point. That is where's the force point on that positive test charge. So you see as in my pictorial representations, the lines of force are longer and denser as you get closer to the charge and they spread out and they get shorter and further apart as you go further from the charge. In addition, what you also notice is that this is a negative charge and indeed the lines of the electric field, the vector arrows of the electric field point inward to the negative charge. So the negative charges are said to be the sinks, the recipients of electric field lines. Electric field lines end on negative electric charges and they begin on positive electric charges which are said to be the source. Now I can go ahead and remove that and instead I can add a positive charge just by changing the sign of the charge. And you'll see that everything flips around. But if I can get this back onto the grid point where it was on a moment ago, good enough. You see here now that the electric field diminishes in the same way as for the negative electric charge but the lines of electric field point outward from the positive charge, not inward. So positive charges are said to be the sources of electric field by convention. Negative electric charges are said to be the recipients or sinks of electric field by convention. And that's all because of this convention of using a positive charge to define everything regarding the electric force and electric fields and so forth. What I'm gonna do now is I'm going to put a negative charge back on the board. But I'm going to make its mass very large and the reason I'm going to do this is because I want to simulate making this negative charge fixed in place in space. Now there are two ways I can do that. I can have the computer literally fix it in space but unfortunately this simulation doesn't let me do that. Instead I'm going to use a physics alternative for holding something in space. I'm gonna make it super massive and as you know as you increase the mass of something you increase its inertia. Inertia is the tendency to resist changes in motion. So if I have a huge mass, I have a huge capacity for resisting changes in motion. So if I'm at rest, it will be very hard to get me to not be at rest anymore. And so I'm gonna make the mass a thousand for this negative charge and I'm gonna add it into the board. What I'm gonna do is I'm gonna put it down here in the corner of the board. You see the field is still there but it's very weak out here. Now what I'm going to do is I'm gonna add a positive charge with a smaller mass. I'm gonna set that back to one. And I'm going to add that into the board. Now I have paused the simulation so that no time is passing. And this gives me a moment to kind of drop this in. Now what have you noticed? You've noticed that when I added in this positive charge, let me remove it again. Okay, so here's what the field looked like out here and then I added in the positive charge. See the field has changed. It's bigger now than it was. And that's because I've added in a positive charge with the same magnitude charge but different sign. And so the electric field from that should look identical to that of the negative charge except it should point outward from the positive charge and inward to the negative charge. And electric fields are vectors. They add like vectors. So anywhere I put this positive charge, its electric field combines at a point in space. You can see this here, watch. So out here, there's very little of the negative charge's electric field present out here. And there's very little of the positive charge is electric field present here. But as I move the positive charge in closer to this point, watch what happens, it grows in strength. And that's because that point is getting an additional piece added into it, in this case in a positive way that increases the length of the vector. If I now move past this point, what happens is that the electric field of the negative charge, which points down to the left corner, has only a tiny length. But the electric field of the positive charge, which points outward from the charge, and on this side it points back toward this corner, it has a big strength because we're very close to the positive charge. And so the sign of this vector reverses because the positive electric field dominates. And you can see that flip happen again here. Watch, I'll bring this in. Watch this point right here very closely. Remember the electric field lines on this side of the positive charge all point down to this corner. And on the upper right side, they all point up to the upper right corner. And as I move closer to that point I was indicating, you see it grows in length. I'm now adding in the positive charge's electric field. And then as I pass that point, I'm still adding in the positive charge's electric field, but the sign has reversed and it wins. And as I move it closer to the negative charge, we see that that arrow will diminish in strength again, back to roughly what it was before. Now this is all being done statically. I'm moving this charge around by hand. If I was to allow time to pass, we could watch the acceleration between these two charges happen. Remember I've made this one very massive. It has a huge inertia. It will tend not to move very much, even though it is being tugged on by the positive charge. The positive charge has a tiny mass and so it's much less resistant to the force of the negative charge and it will tend to accelerate a lot faster as a result. Watch, I'm going to add time to the simulation. You see a slow acceleration, but as we get closer to the negative charge it picks up and then boom, they collide with one another. So this gives you a sense of how you can use a computer for instance, to represent the mathematics of electric fields and Coulomb's law and get all kinds of interesting dynamics out of it. So normally in class, when we're talking about problems, we're typically dealing with something called statics. That is the charges are fixed in space and we're just trying to figure out at that moment in time what are the forces on each of the particles involved or how do those forces add up and what's the total force. You can do dynamics as well. Dynamics means you let time pass and then you have to calculate the acceleration and how it changes and how the position changes at each step in time. That's very cumbersome. It's better for a computer to do that because computers are very good at repetitious tasks and that's exactly what this simulation program can do with electric fields. So I hope you've enjoyed this little demonstration and what we're going to do now is we're going to use the electric field concept in class and exercise problem solving so that you can see how all these pieces fit together.