**Concept:** Magnetic Force Between Two Moving Charges

Hey guys. So, in this video we're going to talk about the magnetic force between two moving charges, let's it check out. So, you may remember parallel currents feel a mutual force. So, if you have a current this way I1 and then you have another wire this way with current I2 they're going to feel a force, a mutual force, given by dissipation mu naught, I1, I2, L divided by 2pi, r and you may even remember, that if they are going the same direction, which is in this case here, they're going to have an attractive force, you'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 they end up having a mutual force between the two wires then charges that are moving by themselves will also have, will also feel a mutual magnetic force to 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 an equation for the mutual force, you may never need to use it. Now, I'm including 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 are, 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 your professor covered this equation then you probably don't need to know it, you might want to ask him to just to clarify whether you need it or not. So, for those of you that do need it, let's keep going.

The force equations is going to look very similar to this but it's going to be a little different. So, mu naught q1, q2, v1, v2 divided by 4pi, r square where r is still the distance so. Notice how here instead of I1, I have q1 v1 and instead of I2, I have q2, v2, cool? That's equation, plug it in and you're done. Now, directions are a little bit more complicated here you have two possible directions, you can go right and left and then here you're 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's actually 16 different combinations, there is 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, they're both going to the right then they will be attracted, they will attract, okay? Which is actually very similar, this is very similar to this. Remember, currents by definition are positive. So, in there two positives that are going to the right. This is identical situation to that so it is attractive but it turns out that they're both 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, okay? One of the ways you can do this, you can kind of figure this out is by looking at q1, q2, v1, v2, okay? Let me show you this real quick, q1, q2, v1, v2. So, let's look at this example here, the q ones are positive and let's say that because we're going to the right that's positive as well. So, you have a positive times a positive times the times a positive, that's a positive, in this situation here, you have a q1 that is positive a q2 that is negative, a v1 that is positive to the right and then a v1, and a v2 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 repulsive, okay? it's a repulsive force, cool? So, let's do look at this example real quick.

An electron is moving right with 1 times 10 to the 8, when a proton passes moving left and then it says here, that they are 3 micro meters apart. So, let's put the electron up here. So, and the proton is up here, remember, the charge of an electron is minus e and the charge of a proton is plus e and then the electron is moving right with 1 times 10 to the 8 and the electron is in the, I'm sorry, the electron is going to the right and the proton is moving left with 2 times 10 to the negative 8, what is the magnetic force between them. So, a, magnetic force FB, it's going to be, it's just the equation we wrote up here, right? So, it's just mu naught, q1, q2, v1, v2 divided by 4pi, r squared, 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 4pi times 10 to the negative 7, that's my mu naught, 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 in both as 1.6 times 10 to the negative 19, and in fact, it's that twice, it's one times the other. So, I can just square this if I want to, times speed, which are 1 times 10 to the 8 times 2 times 10 to the 8 divided by 4pi times the distance, which is 3 times 10 to the negative 6, don't forget that this whole thing is squared, okay? The 4pi's cancel, which is cute but we 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, what's 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 opposite charges. So, this force will be an attractive force.

Part B is actually old news, Part B is asking, what is the electric force between them and I'm just adding it to you because you might take this question as well, that's cute, the electric force between two charges. Remember it is K, q1 over q2 over r squared, 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 9, q1 is 1.6 times 10 to the negative 19 there's actually 2 of them. So, that's square 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 the larger force, right? This is a larger force of the tube because the negative exponent is smaller but if you do like that, if you were curious of 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 4.5 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.

Two 1.0 μC charges are moving with a speed of 350 m/s parallel to each other and 1.00 cm apart. Find the magnitude of the magnetic force that they exert on each other.

A. 3.50 x 10^{-7} N

B. 1.23 x 10^{-10} N

C. 3.50 x 10^{-10} N

D. 0.0421 N

E. 4.21 x 10^{-6} N

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What is the force on the charge at the center of the square in the following figure if the charge is moving into the page at 2v? Consider the side length of the square to be 2r.

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A 5 C charge moves at 15 km/s in the +x direction, and a –7 C charge moves at 10 km/s in the –y direction. What is the force on the 5 C charge due to the –7 C charge when the 5 C charge is at (0,5 cm) and the -7 C charge is at (-1 cm,0)? Is the force on the 5 C charge larger, smaller or equal to the force on the –7 C charge? Assume gravity has no impact on this problem.

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