In the demonstration we just watched, we used magnets as a model for charged particles at different distances. Let's plot what we just observed. Let's graph those electrostatic interactions as a function of the distance separating the two particles with liked-charges. So this time we're going to be plotting what happens when two charges that are the same, they're either both negative or both positive, approach each other. First I'm going to start off by putting the energy on the Y-axis. So I've got low energy down at the bottom of the Y axis, and high energy up at the top. So, again, what we're doing here is, we're looking at what happens when like-charged particles are brought close to one another. So, in this case, we're going to say, what if they're both negatively charged? Well, we have to start at some distance apart from each other, so the distance here we're going to call r. So, small distances down here, this is low distance. Right? This is close. And far apart is over here, on the x axis, the right side of the x axis. So zero energy happens to be right where I have that x axis drawn. So I have positive energy up here and I have negative energy down here. Remember we can have negative delta E. Right? That's what happens when we have a spontaneous process. Okay at large distances, so here r1 the magnets are very far apart. The like charges are very far apart form each other. I can plot that on the graph and you can see that the energy is around zero. You see how it's just about at the, the zero point, right there. Now if I move those like charges closer together, r2 is a distance that's closer together than r1. Do you see how r2 is smaller than r1? RIght? r2 is less than r1. If that's true, then the energy of the system will go up. In this case, the like-charged particles, the fact that they're both negative means that they repel each other. They don't want to be close together. As I move them closer and closer together, here's a third point that's even closer together. The energy of the system goes up, and if you've ever had large magnets and you've turned them so that the opposite, so that the like poles are lined up. When you try to push them together, you can actually feel them resist. And you might not even be able to push them together. The energy of the system goes up, as like charged particles are brought close to one another. So the spontaneous thing to have happen for the like charged particles is for them to move far apart from each other. Right? Nature wants to be at low energy, right? Nature wants to be down here. Nature does not want to be up here at r3. Okay, now let's change the system. When I flip that blue magnet over, what I was doing was changing it so that the opposite poles of the magnet were then facing each other. So let's do that on the next graph. What happens when oppositely charged particles are brought close to one another? Let's graph that this time. So last time I had the same charges. They were both negative. But this time I want you to notice that one of them is positive, and one of them is negative. I've still got energy on my Y axis, and I still have distance on my X axis, right? So this is far apart, this is close together over here, right? This is negative energy, negative delta E, this is positive energy of interaction. Okay. So now, let's move those particles closer together. Large distance, right? The energy is essentially zero. Because they're too far apart to notice each other. But as they get closer and closer together, r2's going to be a slightly smaller distance. This time the opposite charges are attracted to each other. They want to be close together. So their energy goes down as they move closer together. Here's r3. A third point that is even closer together. So the minimum energy for the oppositely charged particles is for them to be close together. Okay, so nature wants the oppositely charged particles to be closed together, it wants to be at low energy, the higher energy state is not as favored. These electrostatic interactions between charged particles have been well known for a couple hundred years. The mathematical equation that describes the interaction between electrically charged particles is known as Coulomb's Law, shown here. This law has been tested many, many times and all observations are consistent with this equation. In fact, the SI unit of electric charge is called the coulomb. It is named after Charles Coulomb and his portrait is shown here. It's interesting to look at the different variables in the equation and think about what they mean. I hope that's something you start doing every time you see a mathematical equation, if you don't already do it that way. F here stands for the force. That can be the force of attraction, as we saw with the oppositely charged particles, or it could be the force of repulsion, as we saw if the two charged particles have the same charge. So F is the force. K is a constant, it's a positive number that's always the same number. We don't actually need to worry very much about what that number is, unless we're doing a calculation. Otherwise, all we need to know is that it's always the same number, and it's always a positive number. Q1 and q2 are the charges. So the charges here are multiplied together. Now the charge has both a magnitude, and a sign. The sign can be positive or negative. So for example, the charge could be minus one or minus two, or it could be positive one or positive two. The charges are multiplied together in this equation. Say that's the product of the charges. In the denominator are the dielectric constant or the shielding factor and that has to do with what is the medium that is between the two charges. Are the two charges in water, which has a high dielectric constant. Do they have carbon dioxide between them, which would have a relatively low dielectric constant? So that value, that number that's in the denominator has to do with what type of fluid is in between the two charges. Some fluids shield the charges from each other and make it so they can't feel each other as much. And other fluids don't shield the charges, and they have a stronger interaction, if you have a low dielectric medium. R, we had before, in our graph. R is the distance between the two charges. And this particular form of Coulomb's law, in this particular expression for the force, the distance is squared, okay. So, how can the force of the interaction becomes stronger, how can it become either more attractive, or more repulsive. Look at the equation and think about the different things that you can change. It can make the attraction either greater, or make it more repulsive. Some of the things that you might have said included the fact, well, it can become more attractive or more repulsive, if the magnitude of the charges increase. So if the charge changes from minus one to minus two, then that would increase the force of either attraction or repulsion because the charges were in the numerator, so as the things in the numerator get larger. The force gets larger, everybody agree with that? What about the variables that are in the denominator? Well the dielectric constant, we can change by changing the medium, but let's just assume that the dielectric constant is the same for everything here. What would happen if we changed the distance? Well, since the distance r is in the denominator, as the distance gets smaller the force gets larger. Because you would be dividing by a smaller number. So, if we want to make the force either more attractive or more repulsive, two of the things we could do would be to increase the magnitude of the charges on our charged particles. Or to decrease the distance between them. That's what we saw when we did that graph. There we just always had minus 1 and minus 1, or minus 1 and plus 1. So we waren't doing much with the magnitude of the charges, we were actually changing the distances there. Let's look at those graphs again. Here are the two graphs that we drew. I don't have the dots here, I just have the curve. These graphs, again, are energy as a function of the distance, okay, and we're looking at the interaction between charged particles. Which of these two graphs is showing the interaction between oppositely charged particles as a function of distance? Well some of you are thinking, the formula that she gave me before wasn't for energy. It was for force. There's another form of Coulomb's law that is for energy. In this case, I'm no longer am using the constant k, and instead I'm just saying that the energy is proportional to the product of the charges divided by the dielectric constant times the distance. And notice now, that the distance is not squared, do you see that? It's just r. So in the, the force depends on the distance squared but the energy only depends on the distance. Did you answer that situation B, is the case where there are opposite charge particles. Because those opposites attract and as they get closer together the energy of the system goes down, right, this is close together. And this is far, apart. Are you ready to do some more practice examples? Let's look at some of those. Okay, so this is sort of a multiple choice question. In this case assume that the dielectric constant is the same for all of these systems. And I have four different systems here. Right? A, b, c, and d. So four different systems. Let's ask some questions about these systems. So k is constant, right? [NOISE] Right? And now I'm saying that epsilon, the dielectric constant is constant. They're all on the same fluid, for example. And the first question I want to ask is, in which of these situations is the force the most attractive. Alright, so we need the equation for Coulomb's Law. I'm just going to write it up here in case you've forgotten. It's Coulomb's constant, times q1 times q2, divided by epsilon r squared. Okay, so k is a constant. And epsilon is a constant, so in which of these cases is the force the most attractive? Alright, first thing is, attractive means opposite charges, right? Because, like charges repel. So, b and d are opposite charges, right? And then, r is the same for both b and d, see that? They're, in fact, for all the systems, r is the same distance. So the only that's different is the magnitude of the charges. In b the charges are larger so the force of attraction is higher. So b is the situation where the force is the most attractive. Let's look at this equation for Coulomb's law and ask ourselves, for attraction, is the force of interaction positive or negative? And now I've lost the equation so let's rewrite it again, remember what it is? By the time I'm done, you'll have it memorized. Alright, for attraction, is the force of interaction a positive or a negative number? Well for attraction, q1, we'll say is negative and q2 is positive, right, for b and d. So force is a negative number, right? In which of these two situations is the force repulsive? Well, in order for the force to be repulsive, that means we would have to have like charges. Both a and c are situations where the charges have the same sign. So both of those situations are repulsive. And finally, I'm going to let you answer this one on your own. In which of these situations a, b, c, or d, is the force most repulsive? Probably not the best way to end the lecture, but I want you to go ahead and answer that. In which of these situations is the force the most repulsive.