Hey, guys, how's it going? So I wanted to work this problem out together. We're supposed to figure out where we need to put a one Coulomb charge so that the force on it is equal to zero. So what does that mean? We're just going to assume that the net force we're asking for needs to be equal to zero, because we have two Coulombs, or, sorry, we have two charges on the outsides. So what this problem is saying is that we need to put a one Coulomb charge somewhere around here in order for the forces on it to cancel out. Before I actually start plugging anything in, I just want to figure out qualitatively, meaning no numbers, where this thing needs to be in order for those forces to cancel out.
Let's take a look at the left side. So imagine I dropped a one Coulomb charge here, and I want to figure out if those forces would cancel. So all I have to do is just figure out the directions of the forces on it from the two Coulomb charge. Remember, all of these things are positive, so they're all going to repel. They're going to have repulsive forces on each other. This thing has to move to the left, right? So this is the force from the two Coulomb charge that's going to point to the left. But the force from the three Coulomb charge also has to point to the left. It might be stronger or weaker, we don't know, but it's going to point to the left. So in other words, there's never going to be a situation where these forces are going to cancel out. They're always going to add together and point to the left so it can't be here, right? And so I'm writing here that these things never cancel. There's never going to be a situation in which one will point in one way and one will point in the other. So, it can't be in this area in this region right here. So let's look at the same situation on the right. Who got this one Coulomb charge here, and you do the same thing from the three Coulomb charge. You have a force pointing in this direction. The two Coulomb points points in this direction. So for the same exact logic, this thing will never cancel out. These forces will never cancel out. So that means it has to be somewhere in the middle.
The reason for that is if you work out the forces here, the force from the two Coulomb charges is going to point in this direction, and the force from the three Coulomb charge needs to point in this direction. So that means that there is some magical distance in which these things will cancel out. Now, we have to be careful because it won't exactly be in the center. If this thing were in the center, then this stronger charge here would produce a stronger force. So this thing actually needs to be slightly closer to the left so that these things will balance out. So, that means that the condition that we're trying to solve for the rest of the problem is that we need the magnitude of the two Coulomb force to be equal to the magnitude of that three Coulomb force or the forces from those two charges. In other words, this is the condition that we need to solve, right? They're going to be pointing in opposite directions. But if their magnitudes are the same, they're going to cancel out.
So let's go ahead and write out the expressions for those two. F 2 for the two Coulomb charge, where I need a distance between these two charges. By the way, this is a one Coulomb test charge that we're going to drop right here. So I'm going to call this distance here, \( X \). And that means that if this distance between the two and the three Coulomb charge is, \( R \) so this is \( R = 10 \) centimeters, then this is just \( R - X \), that's the distance between \( R \) minus that little chunk of \( X \), \( R - X \). So I've got the \( K \) constant times the two charges that are involved. In other words, this is going to be the two-column charge and the one-column charge divided by the distance between them. That's \( X^2 \). Now I set up the equation for \( F3 \). So that's this guy right here between the one Coulomb charge. So \( K \) and then I've got the three and one. So that's three Coulombs one Coulomb divided by the distance between them squared \( R - X \) right and then squared, so I don't know what those distances are. Now again, what we're supposed to solve is that these two things are supposed to be equal to each other. So in other words, these expressions right here that I've just figured out have to be equal to each other.
So let's just go ahead and actually set those things equal. K 2 1 / X 2 = K 3 1 / ( R - X ) 2 So this thing is an equal sign, so we can actually cancel out anything that appears on both sides of the equation. We're going to get the \( K \)s will cancel, and also the ones will cancel. I mean, one doesn't really do anything, so I mean just cancel it anyway. And so we come up with an easier expression. Now, what I'm trying to solve is where I need to put this one Coulomb charge. So that's a distance. So in other words, I'm going to go ahead and solve for \( X \). \( X \) is my target variable here. So just because of this, the easier one to work out like I could actually go ahead and solve for what \( R - X \) is and refine. But it's just that \( X \) is an easier variable to solve. So let's go ahead and do that. If I wanted to solve for \( X \), I had to get all the things involving \( X \) to the one side. So, I'm going to move this \( R - X \) over to the other side, cross multiply, and then this two needs