10.4 Coulomb's Law

Ok, today we're gonna look at our sort of last topic for this first chapter 10 part of the electricity that we're going to be doing. Today, we're gonna look at Coulomb's law. Coulomb was a French physicist and did most of his work in the late 1700s, so we're kind of in the late 1780s/90s, that sort of range is when he came up with a whole bunch of theories. But his most famous one is Coulomb's law, that we now call it. Basically, what he did was he investigated the relationship between charges. So, if we have two different charges, what's the force between them? And he figured out that as the charge increases, so does the force. That makes sense, right? If we have a stronger charge, it should create more electrostatic force. He also figured out that if you move them apart, the force between them will decrease exponentially. So, he did a bunch of experiments and so on. But for all of his work, we've honored him by calling the unit of charge a Coulomb. So, if we're trying to explain how much charge is on an object, we could say it has three milliCoulombs of charge or whatever. Another thing he was sort of famous for is he was one of the key people involved in creating the metric system, which is sort of an interesting side note. How he did his experiment is he used a torsion balance. If you remember, we actually did the same thing in Physics 20 when Cavendish was trying to figure out how to get the big G number in Newton's gravitational law. So what Coulomb did was basically the same thing, and in reality, he actually did it first. So, Coulomb used the torsion balance to figure out the charge, and then Cavendish sort of looked at that and said, "Hey, maybe I can use this to figure out gravity." And we're gonna see that the two are actually very similar formulas and they're very closely related. Just a quick explanation, if you look at the picture here, what Coulomb did was he charged an object on each end of this sphere. So, what he did was suppose he put a positive two charge on that one, and he put the same charge on that one. He didn't know how much the charge was, he couldn't figure that out yet. But what he did was he charged them with a rod, kind of like what we do in class when we rub a nylon rod or something like that. And then he would touch the metal sphere, and then he'd do the same thing on the other side. So, he would get those equally charged, and then what he did is he swung them over to touch two fixed spheres that were neutral. And as soon as he did that, then he would have all of the charges balance out. So, we just kind of rewrite this here now. If we had a charge of two on each original one and we touched them, then basically what happened was he'd have a charge of positive one on all of them, or it could be negative, but it doesn't matter whether it was a positive or negative charge. And what happened as soon as they were equally charged, because they were close together, they would repel each other. So, these two on the left side would repel each other, and so would these two. And then what he did was he measured what that force was. So, he had a meter at the top of this central rod or string, and just by measuring how much torque was on there, how much force was twisting, he could then calculate what the force was between those two objects. So, he got them to separate, right? That was sort of what happened. By then separating, he created force. And because it's due to electrostatics, we call it Fe. He didn't know how much the charge was, but he could sort of measure the forces. And what he did was he would change the amount of charge by changing the size of the spheres. He figured out that it didn't matter how much he rubbed the rod, he would only get the same amount of charge. But if he made the sphere bigger, it would then take more electrons or whatever, and it would be a bigger charge. To change the size of the spheres, he changed the distance between them, changed all kinds of things, and he would measure the force that it created. So, by calculating the distances, the force, all that kind of stuff, he ended up coming up with what we call Coulomb's law. And the formula that's on your formula sheet is F_e = k(q_1)(q_2)/r^2, where F_e is our force in Newtons, Q is what we call our charges, so those are charges in coulombs, ours are just like we did with Newton's, r is our distance between the charges, and that one will be in meters, and our k is typically a constant, just like we had big G for Newton's gravitational formula. K is a constant, sometimes you'll hear it called Coulomb's constant, but basically we just say K, and the number it actually works out to the Coulomb calculated was 8.99 times 10^9, and the units to cancel everything up properly is Newtons-meter squared per coulomb squared. Okay, so that number is on your front of the sheet as well, so they'll have K listed with 8.99 times 10^9, and the formula will also be listed. Well, what we do notice on the front of the sheet though is it's typically written with absolute value signs around the F_e. The reason for that is we don't really care force is typically about the rates of a positive, negative, left, right, up, down, whatever, but we don't want to worry about it in this case because our direction isn't necessarily going to be indicated by the vectors of the force. It's going to be indicated by what the charges are. So by that I mean, if we were to suppose we have a positive charge and another positive charge, we just plug in our values of the charges and we plug in our K number, we plug in the distance between them, we calculate what the force is. So we would just get our force sensor or whatever and then by looking at the charges, we'll know what's going on. So if the question was saying, "What's the force acting on the charge on the left?" So if we were looking for this force, we would calculate using the formula and because they're both positives, we know that they're going to repel. So our force then would be to the left, right? It's gonna move away from the other charge. So we just deal with absolute values. We don't care whether they're positive, negatives, whatever. Just plug the numbers in to get your answer. And then we use the idea of positive charges attract or repel or negative one plus one minus that type of thing because normally with vectors, if we say plus or minus, that's indicating direction. But now we're dealing with charges. So, plus and minus is actually indicating what the charges are. So, a little bit different. We just have to be careful in that case. So, let's do a simple example. Example number one here in the bottom. It says calculate the force between hair and a comb. So, you comb your hair and you create static. So, we can actually figure out what the force of electricity is on those objects. So, we just plug in our K value, 8.99 times 10^9. Charge one, you can see it says negative 2 coulombs. We don't care if it's positive or negative, so we're just gonna put in 2. And it says microcoulombs. Remember, micro is 10^-6. So, it's negative 6. Charge two says the piece of hair has 3 coulombs of charge, so that'll be 3 times 10^-6. And our distance is 0.15 meters, 0.15 squared. Is about an answer. Okay, so plug everything in, like I said, it's important. Don't worry about the positive or negative. Like, that's gonna be dealt with later. So just solve it as absolute values or just whole. Numbers. We plug all this in, we get a force of 2.4 Newtons. Okay, now what we want to do is worry about the direction. So, remember our comb was negatively charged, our hair we said was positively charged. So now it says determine the force that the hair exerts on the comb. So the hair is going to attract the comb. So we're talking about in terms of what's the force on the comb. We'd say the comb has 2.4 Newtons to the right, or you could say towards the hair or something like that. Okay, so that'll be the force on the comb. If the question was what's the force acting on the hair, it would be the same calculation. But in this case, our hair being positive will be negative. The hair has been pulled to the left. This would be the exact same number, but our direction would be left. X, you get the idea. Just calculate the calculations. Use the idea whether they're positive or negative. Figure out which direction the force would go to. So a couple quick things you notice on the last example. We use microcoulombs times 10 to the negative 6, instead of just whole numbers. And that is our typical charge. So if you were to measure charge on an object, typically you're going to be somewhere in that range. You're going to get 10 to the negative 6 or somewhere close to that. For it to be something like 1 Coulomb, we're talking about something that has a massive amount of electricity or electric charge. Things like lightning bolts have things like 1 or 2 coulombs of charge. So huge amounts of power. So most of our questions we're going to be doing are going to be pretty small amounts of charge. And the reason for that is our particles that are making the charge, the electrons and the protons, are pretty small. So on your formula sheet, if you look at an electron, on your front of the sheet, the charge of an electron is they write it as -1.60 times 10 to the -19 coulombs. And a proton, obviously, they would be the exact same, but positive. And a neutron is neutral, so we say it as zero coulombs. Okay, so because the electrons are so small, at 1.60 times 10 to the -19, you know we're going to need a lot of electrons in order to come up with a decent amount of charge. That's why our numbers are going to be pretty small for working with. 1.60 times 10 to the -19 is an important number because it's the charge of an electron and a proton. We call it the elementary charge, right? That's the smallest we can get, just one electron. So we use the symbol e quite often. That's on your formula sheet. They have e = 1.60 times 10 to the -19. And let's think about what that means. So if we were trying to figure out how many electrons are in one coulomb of charge, we can actually figure it out by saying 1 coulomb divided by our elementary charge, 1.60 times 10 to the -19. If we do that math, we would actually see that there are 6.25 times 10 to the 18 electrons in one coulomb. I kind of messed that up. Hopefully you understand what I'm trying to get at here. So a lightning bolt strike, for example, has 6.25 times 10 to the 18 electrons being emitted in that bolt. So it's a huge, huge amount. Six point two five two times ten to the power of 18 electrons per one Coulomb of charge. So, the mass of Amun is... Okay, so if you haven't noticed that yet, we see that we actually get two formulas that look almost exactly the same. So, what we just did with Coulomb's law was K q1 q2 over R squared. If you remember, in physics, Newton's gravitational formula was G m1 m2 over R squared. So, if you look at the two formulas, they look almost exactly the same, and they are very closely related. In electricity, we're dealing with K, Coulomb's constant, and in Newton's, we have big G, the gravitational constant. We had q1 q2 with charges one and charge two. In gravity, we have mass one and mass two, and they both have R squared. So, when you look at these formulas, you might think, "Hey, these are both exactly the same," but there are some pretty important differences. The key difference is that G was pretty small. If you remember, right off your formula sheet, G was like 6.67 times 10 to the power of -11, I think, something like that. And our K number that we just used was 8.99 times 10 to the power of 9. So, K is huge, right? K is a very large number. G is really small. So, because of that, and pretty much only that, the rest of it obviously plays a part too, but because K and G are so vastly different, our electric force ends up being a lot stronger than gravity. So, so far, out of all the forces we've talked about, electric would be massively bigger than gravity. Okay, if you think about it, well, that doesn't make sense like we saw yesterday when we're doing the experiment, that there wasn't that much force between our static rod and a little chunk of hole-punch paper. But what we do see is, you gotta think about it, is for us to take that static rod and lift up that little piece of paper, right? We're able to lift it up. We only had a little bit of charge on there, right? We had a little bit of charge by rubbing it, but it was still strong enough to lift that piece of paper up, where we were overcoming gravity, right? We've got gravity that's pulling that piece of paper down, and our electric force actually lifted it up. So, for that to happen, obviously, electric force has to be larger than gravity, right? That electric force has to be, otherwise, a piece of paper wouldn't have been attracted to the rod. So, just in a simple small example like that, we overcome all of the gravitational force acting on it, and the electric force was still quite a bit stronger. So, that's probably one key thing to remember in terms of our forces so far. Electric force is hugely bigger than gravitational force. And another kind of key difference, electric force can measure whether things attract or repel. The formula will do both, right? Whether it's positive or negative doesn't really matter, whereas gravity only deals with attraction. What's the force of gravity, how Earth pulls somebody down towards it, right? It's only attracting the two objects together. Okay, a couple of examples to go through that you're expected to be able to do. So, the first one is we've got three charges in line, so you can see the amounts of charge on each object. And the question is asking, "What is the net force acting on the one to the right?" So, we're trying to figure out what is our electric force on Q3, the one on the far right. So, because there are other charges, we have to take into account all of them. So, what we have to do for step one is we need to figure out the force between two of them. So, let's do the force between charge one and charge three. You can see charge one, I've colored it red to show that it's positive. It's positive 1.5, whereas the other ones are negative. So, sometimes colors will help you separate them, or just be careful about the positive and negative idea. So, in this case, I'm going to plug in our numbers. We'd have 8.99 times 10 to the power of 9, our charge one is 1.5 times 10 to the power of -7, and charge three is 3.5 times 10 to the power of -4. Remember, don't worry about positive or negative. And then, our distance is ready. The total distance we have is 3.1 meters, and we need to square it. When we calculate that, we get 0.05 Newtons. Now, we need to determine the direction. Q1 is positive and Q3 is negative, which means Q3 will be attracted to Q1, going 1.05 to the left. Next, we need to do the same calculation for charges 2 and 3. The value for charge 2 is 8.99 * 10^9 and charge 3 is 3.5 * 10^-4. Now, we divide by 1.7^2. After the calculation, we get 0.25 Newtons. Going back to the question, both charges are negative, which means they will repel each other. Charge 2 will push Q3 away, resulting in movement to the right. So, our net force is 0.25 to the right. In summary, the net force is determined by combining the forces acting in opposite directions. Let's look at another example with the same setup, except this time the charges are not in a straight line. They are in two dimensions. We solve this question similar to how we would solve vector questions involving momentum. We need to make sure we solve it correctly and then work with the triangles. In this case, we have 2 Coulombs and 1 Coulomb at a distance of 3 meters. Plugging those values in, we get 8.99 * 10^9 * 2 * 1 / 3^2. This gives us a large value due to the 1 & 2 Coulombs. The result is 1.998 * 10^9. We repeat the same calculation for the other two charges, the red one and the green one. Using 8.99 * 10^9, 2 * 1, and 4 meters squared, we get 2.248 * 10^9. Now, we need to be careful with the directions. Both forces are positive, so Q2 will experience repulsion. Q1 will push it downwards, and Q3 will push it to the right. We can represent these forces as vectors and combine them as a triangle. The downward force is 1.98, and the green force is pushing 0.247 to the left. All we have to do is draw a new triangle of those two vectors and solve. So, if we use the Pythagorean theorem to figure out our net force, we end up getting 3.0 x 10^9, right? Just 1.998 squared plus 2.248 squared, square your answer. And then, we also need to calculate our angle, just like we normally do for questions. We want to go tan of 227 divided by the 1997 seconds. Tan that answer, and we get an answer of 48 degrees. So, in this case, we would say 48 degrees west of... So, that's our final answer. Okay, so you can see, really, we're doing nothing different than we did before. Just a new formula, a little bit different than Newton's formula, same sort of idea. And there are a few examples showing you how to work it. And that's all.