Hey guys. So in this video, we're going to talk about the magnetic force between 2 moving charges. Let's check it out. So you may remember parallel currents feel a mutual force. So if you have a current this way, I_{1}, and then you have another wire this way with current I_{2}, they're going to feel a force, a mutual force given by this equation,
μ_{0}I_{1}I_{2}L
2πr
. And you may even remember that if they are going the same direction, which is the case here, they're going to have an attractive force.

Well, remember, realize that currents are really just charges that are trapped in a wire. So if charges trapped in a wire end up having a mutual force between the two wires, then charges that are moving by themselves will also have, will also view a mutual magnetic force. They'll apply a force on each other. Okay? So you need to know this. This is a big deal, and you should know this.

But before I go any further, I want to have a big disclaimer for this video, which is a lot of professors and textbooks actually skip this part out, and you may never need to know this equation. I'm going to give you an equation for the mutual force. You may never need to use it. Now, I included this video, I'm including this video for all textbooks, even textbooks that don't particularly have this topic explicitly, because I want you to know that these things happen. But if you don't need to know this, you should probably stop here and not learn more equations. You don't need any more equations in your life. So if you haven't seen this, if you haven't seen your professor cover this equation, then you probably don't need to know it. You might want to ask him to just clarify whether you need it or not.

So for those of you that do need it, let's keep going. And the force equation is going to look very similar to this, but it's going to be a little bit different. So,
μ_{0}q_{1}q_{2}v_{1}v_{2}
4πr^{2}
, where r is still the distance. So notice how here instead of I_{1}, I have q_{1}v_{1}. And instead of I_{2}, I have q_{2}v_{2}. Cool.

That's the equation, plug it in, and you're done. Directions are a little bit more complicated. Here, you had 2 possible directions. You can be going right and left, and then here you can be going right and left. Here you still have right and left, but the charges could be positive or negative, which throws things off. But I figured this out for you. There are actually 16 different combinations of positive, negative, right, left for all these different things. But I worked out everything for you, and all you need to know is that if the charges have the same direction and the same charge. So, for example, here, you have let's say this is a positive and a positive, and they are both going to the right, then they will be attractive.

Which is actually very similar, this is very similar to this. Remember, by definition, they are positive, so in these two positives that are going to the right, this is an identical situation to that, so it is attractive. But it turns out that if they're both in opposite directions and opposite charges, it will also attract. Okay?

So this is this situation and this is this situation here. Where you have a positive in one direction and a negative in a different direction, and they will also attract. All other combinations, you should know, will repel.

One of the ways you can do this, you can kind of figure this out, is by looking at q_{1}q_{2}v_{1}v_{2}. Okay? So let me show you this real quick. Q_{1}q_{2}v_{1}v_{2}. So let's look at this example here. The q ones are positive, and let's say that because you're going to the right, that's positive as well. So you have a positive times a positive times a positive times a positive. That's a positive. In this situation here, you have, a q_{1} that is positive, a q_{2} that is negative, a v_{1} that is to the right, and then a v_{2} that is negative to the left. If you multiply all these guys you end up with a positive. That's why in these two situations, the positive here means that they will attract. That's another way that you can do it, but honestly, I think it's just best if you just remember these three things. And by the way, all the other combinations that you have, if you were to multiply the q's and v's, you end up with a negative, which means that it is a repulsive force. Okay?

It's a repulsive force. Cool. So let's look at this example real quick. An electron is moving right with 1 times 10 to the 8th when a proton passes moving left. And then it says here that they are 3 micrometers apart. So let's put the electron, up here. So and the proton is up here. Remember the charge of an electron is -e and the charge of a proton is +e. And then the electron is moving right and the proton is moving left with 2 times 10 to the negative 8th. What is the magnetic force between them? So a magnetic force F_{B} is going to be it's just the equation we wrote up here. Right? So it's just q_{1}q_{2}v_{1}v_{2} divided by 4πr^{2}. The distance between them is 3 micrometers. So 3 times 10 to the negative 6. So this is going to be a gigantic number here. So let's start plugging in. 4 π times 10 to the negative 7. That's my μ_{0}. The charges of these guys are both 1.6 times 10 to the negative 19 coulombs. Remember, in almost all or maybe even all magnetism questions, you always plug everything as a positive because your direction is always given by things like the right-hand rule. Okay? So even though these two guys have different signs of charges, we're just going to plug them both as 1.6 times 10 to the negative 19th. And in fact, it's not twice. It's 1 times the other, so I can just square this if I want to. Times the speeds which are 1 times 10 to the 8th times 2 times 10 to the 8th divided by 4π times the distance, which is 3 times 10 to the negative 6. Don't forget that this whole thing is squared. Okay. The 4 π's cancel which is cute but you still got to do a lot of work here. And if you plug all of this into your calculator, you get 5.7 times 10 to the, times 10 to the negative 18 newtons. Okay? It's a tiny tiny force. Cool. That's that.

By the way, what is the direction of this force? Well, they have, it's going to be an attractive force. Hopefully, you thought this would be an attractive force because they have opposite directions. Right? And they have opposing charges. So this force will be an attractive force.

Part b is actually old news. Part b it's asking what is the electric force between them? And I'm just adding this here because you might take this question as well, it's cute. The electric force between two charges, remember, it's just kq_{1}q_{2} over r^{2}. This is one of the first things you learn in electricity. And I can plug in the numbers. K is a constant 9 times 10 to the 9th. Q_{1} is 1.6 times 10 to the negative 19. There are actually 2 of them, so that's squared divided by the distance, which is 3 times 10 to the negative 6, also squared. And if you do all of this in the calculator, you get 2.56 times 10 to the negative 17. And that's part b. By the way, if you divide this one is a larger force, right? This is a larger force of the 2 because the negative exponent is smaller. But if you do like that, if you were curious about how much stronger one is versus the other, and you were to do a ratio of the electrical to the magnetic force, sometimes you see a question like this also, you would see that it's about 4.5. So even though the electric force is stronger than the magnetic force, it's only 4 and a half times stronger. It's not like a million times stronger. Cool? So they're pretty close. They're both really, really weak in this situation. That's it for this one. Let's keep going.