Find Zero Magnetic Field

by Patrick Ford
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Hey, guys. So in this example, we would have finds where Between two wires. The magnetic field is zero. Let's check it out. So we got to horizontal wires that are 6 m away. They're both horizontal, which means they're parallel. I'm gonna make the first one red in the second one blue. And it says here the currents are four amps at the bottom. I'm gonna call that I to four amps and five amps at the top on. They're both going to the right. I one equals five amps. So it looks like this there distance 6 m away from each other and we don't know where, where or how far from the bottom wire is the net magnetic field going to be equal to zero. So what I'm gonna do here first is look at the direction of these magnetic fields. Um, that will be produced by these currents. So if you have a wire and it's pointing to the right, you're gonna grab your right hand, right? Your right hand right here, and you're gonna grab it, and it's going to be pointing. Your thumb is gonna be putting direction of current, which means it has to look like this. Now, if you go back and you go in again, notice that my fingers under the wire are going into the plane away from me, right into the plane, away from me. And then they come back around here towards me. Okay, So what that means is that I won will generate and into the plain field below it. Rights. Over here, this is B one due to higher one and then on top of the wire, it's going to be towards me. So I see a god's coming towards my face, so that's gonna be be one. Be one. And by the way, this extends out anywhere below. The wire is going toe have an X B fields. So I keep going here. This is also gonna be X B one x b one. Just like how here keeps going. This is a dots. Be one thoughts. Be one. Okay, so it's sort of a separation above and below the wire. Now, if you do the same thing for wire to wire to is also going to the right so you don't even have to grab the wire again. It's gonna do the same thing below the wire, you're gonna have X so x becoming due to be two X. I'm sorry, X direction of B to do to this current. And on top of the wire, you're gonna have a dots on top of the wires coming towards you. So this is gonna be this is gonna be be to magnetic field due to current to, and it's the same thing over here. Okay, so what, this ends up creating is sort of three zones. There's the top zone, which is everything above the top wire, the bottom zone, everything below the bottom wire. And then there's this middle area here, okay. And what you notice when you have two wires going the same direction is that the net magnetic field at the top has to be out of the page towards you because they're both dots, so they're gonna add up to be out of the page. And over here, the magnetic field at the bottom zone has to be into the page because both magnetic fields produced by those two hours air going into the page, which means that these guys can never be zero. Okay, the magnetic field will never be zero Here. You can Onley get zero in the middle because that's where you have different directions. You have opposite directions, so being that here could be zero is zero somewhere. And where is it? Zero. Well, that's what we're trying to find out, right? So the idea is that there is a line here somewhere that is just the right distance between the top wire and the bottom wire so that the magnetic fields at that line can so perfectly okay. And what we want to know is how far from the bottom wire that line is. So if you want, you can call this. You can call this distance a and I wanna know what is a and you can call this distance be or we can call it. We can also just call it, are one or are too right. And this is our one. And what we're looking for is our to. And by the way, keep in mind that are one plus r two equals m. Okay, equals 6 m. Cool. So what do we do now? Well, if we want the magnetic field to be zero, this means that the magnitude of B one equals the magnitude of B 22 things with same magnitude. Same number, but opposite directions will cancel themselves out perfectly. Okay, So what are the equations for? B? If you have a wire, it's more mu not I divided by two pi r. So we're gonna do this twice now. Obviously the first case here. We're looking at B one. So this is current one and distance one current two in distance to OK. These guys are just constant, by the way. So they get canceled out, which is nice. So you end up with I one over R one equals I to over our two. And we are looking for these numbers we're looking for are two okay, now, if you notice I can quickly replace. I can quickly replace the eyes. Eyes are five and four. Our choose what I'm looking for. What about our one? So the problem with our one is our two is my variable. That's what I'm looking for. But I don't have our one. So what I have to do is I have to write an expression for our one. And if you look at our one here, I can rewrite our one as six minus are too. So six minus are too. And the good news here is that here you had two unknowns, two variables and that's bad news with just one equation. Okay, here you have one unknown, which is good news. Now, now you can actually solve this. So now this is just an algebra problem. We have to cross, multiply and get our our choose, um, solved for So if I cross multiply to get five r two equals four times six minus are too. And I can expand this five r two equals minus four are too I'm looking for are too. So I'm gonna move it over to the other side. Five plus four r two is nine are too. It goes 24. So are two is 24 divided by nine, which is 2.7 m. Okay, And that is the final answer. What that means is that this distance here is 2. m. If the whole thing is six, by the way, it means that this is 3.3 m and it should make sense that the wire is a little closer to I two than two I won because the magnetic field comes from this equation. Notice that these two guys air Constance so they don't really matter right now. So the stronger my eye, the stronger might be, Um, and the stronger my are, the weaker might be. So the bottom one has a weaker I has a smaller I. So to compensate for that, you also have to have a smaller are so that it's a little stronger, so the smaller I makes it weaker. But then the smaller our means that it's closer, which makes it stronger. Okay, so anyway, weaker. I means he wanted to be closer, which means you have a smaller are. Okay. And then that way, the balance, That's it for this one. Let's keep going.