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Physics

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29. Sources of Magnetic Field

Magnetic Field Produced by Straight Currents

1
concept

Magnetic Field Produced by Straight Currents

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Hey, guys. So this video, we're gonna talk about how current in a wire will produce a magnetic field. Let's check it out. All right. So you may remember that if you have a moving charge, that moving charge will produce a new fields away from itself. So let's draw a little charge Q. Here and it's moving. So it's got a V, and that charge will produce a magnetic fields up here or over here, right in a number of different places away from itself. So and the magnitude of that magnetic fields, if you remember, is mu, not Q V sign of data divided by four pi r squared. But this is the old news. That's why I said, remember, right. And what we want to talk about now is how, just like moving charges will produce a new fields. Well, currents will also produce a new field because currents are just charges moving in a wire, right? So if you have a wire and it's got a current this way, I you can think of it as well. There's lots of little cues here that have V's right, lots of little cues with these. That's what current is. So if a Q movie with V produces a B field, Um, then on I will also produce a B fields. Okay, so currents also produce new magnetic fields away from themselves. Okay, so if you have a current just like up here, you have a current I you're also going toe have a magnetic field somewhere over here, and any number of distances, the magnitude of that equation, it looks a little cleaner than this one. A little nicer. It is that b equals mu, not I divided by two pi r and this equation is super important. You absolutely have to know this one. Okay? I mean, you should know all the equations, but this one is is really, really important. You're gonna see this all the time. I should know that this only works for a very, very long wire on did most of the problems. You're just going to assume that the wire is very long, even if it's not said, Um, even if it's not, say it. If you have a short wire, you gonna get a different equation from this one. That's a little more complicated, but most of you are gonna have to deal with that. Okay, remember, you not is a constant in this are over here is a distance. Little art is always a distance and never a radius. In fact, if you have a wire, it would be this right Here are some people even use the letter d or sometimes a, um, to make a distinction. Cool. So that's how you find the magnitude. And then what about the direction? The direction comes from the right hand rule. And what we're gonna do is we're gonna grab the wire with our thumb in the direction of currents, and this is pretty consistent with everything we've done so far. Which is you always want the thumb to be in the direction of the charges, the charges in the direction in which the charges air moving Well, the charges move in the direction of current. So you're always gonna wanna use your thumb to point in that direction. Now, what's gonna happen is we have a wire and I'm going to grab the wire in the direction, right. So if the currents going to the left, I'm gonna grab the wire like this. If the currents going to the right I'm gonna grab the wire like this. And by the way, you're always going to use the right hand rule when you have wires. It's always the right hand rule, never the left hand rule, because current is always by convention positive quote. And then if you have two fields in the same location and the fields were going in the same direction, we're going to add the magnetic fields. And if they're going opposite directions, we're going to subtract, and we'll see this in the second example. So we'll get back to that point for the first example. We're just looking for the direction. So we're gonna get to use this rule here, and I'll show you how it works. So I wanna know what is the direction of magnetic fields? Um, produced by a current on a very long wire. If the current is oriented up, meaning, if you have a wire and the currents up like this, what is the direction of the magnetic field that is produced? And what if you have a left current or what? If you have an end to the beach current into the page, remember, means that it's going away from you. So if you look at your sheet, it's going away from you, right? Do it yourself. So you follow. And that means that you're looking at the back of the arrow, so you see an X, and this is the symbol for into the page. Okay, So imagine that this cable is going away from you like this, right? Like that. Cool. So what happens with direction? Magnetic field? Well, we're gonna grab the wire with my thumb, pointing the direction of currents. So if I grab the wire with my thumb up, it's going to look like this. Okay, Now, this is super important, and I'm actually gonna move it over here so you can follow a little bit better. So this is super important. Here. Is we're gonna do this is going up, which looks like this is the direction of the cards. So I'm gonna grab with my right hand, always. I'm gonna grab it, and then my hand should look like this. Please do this right. Please do this so that you can see. So what you're gonna do here is notice that my fingers air curling into the page here into the page here right there, going like, in that direction, away from my face into the page. And when they come back around, they come out out of the page on the left side. Okay, So, please, I'm gonna do this really carefully. Hopefully, you can follow. So into the page here and out of the page. Whatever your thumb is up, you're always gonna get the effect. So what does that mean when you are drawing it? Well, what it means is that this is magnetic field anywhere to the right of this cable is gonna be into the page, and anywhere here is going to be out of the page. Okay? So you could draw a bunch of little excess and dots. Cool. Now let's do left, and I actually want you to do left. And just as a hint, the top of the wire will be either an X or a dots in the bottom of the wire will be either an extra dot figure out real quick positivity. If you have to figure out real quick which one you think is which you're gonna do the same thing I just did. I'm going to assume you pause the video and I'm gonna do it over here. So my my thing is gonna be my current going to the right. I'm gonna grab it with my right hand, which looks like this. Right? Notice that my fingers were going into the page out here and out of the page back here. Okay. So what this looks like is what this looks like is into the page here and out of the page. Here, Quote. You gotta be good at this stuff. What about here? This is into the page. So again, I think you have a good chance of getting it right. So just pause the video, give it a shot. I'm gonna do it over here. And this is going into the page, which means it's going away from you right into the page. Please do this grab like a pen or something and pointed into the page, which means my hand is gonna grab it like this. I'm actually gonna get rid of the pen in my hand, is gonna grab it like this. So you should have your hand where your thumb is going into the page. And if you do this, look what's happening with my curved fingers is that they're going in a clockwise direction. Try to get the angle here. Right. So I'm gonna go in a clockwise direction. Please do this yourself. It looks silly. Gotta do it, uh, to make sure that you're getting this right. So what? That means that actually, the direction of the current is going. I'm sorry. Direction of magnetic field is this way. Okay. What if you had What if you had the current coming out this way? Well, obviously, if the current flip directions in the magnetic field with flip directions as well. Okay, so in this case, you would have your thumb pointing at you and you can't see from my hand. But if you look at your thumb at your hand with your thumb pointing at you like this, right looking really silly right now, you're gonna see that be goes counter clockwise. Hopefully got that. Let's look at example to which is a computational example. Um, it says two wires are shown below 4 m away from each other. So this distance here is 4 m, and I wanna know what is the magnitude and direction of the magnetic field that it is produced at the center between the two on the center between the two is somewhere over here. Let's call it P. And it is a distance of 2 m away from both of them. And what I wanna know is essentially what is the magnetic field that Point P? I want the magnitude and direction of the magnetic fields. This is really the Net magnetic field. It's gonna be a combination of two because this guy produces a magnetic field and this guy produces a magnetic fields. They both produce magnetic fields there. So really, this is you can think of this is the net magnetic fields. So it's gonna be the magnetic fields due to the first current, plus the magnetic field due to the second currents I one and I to except that these guys are going in different directions, so they're gonna have different signs. Okay, so we'll get to that in a little bit. But essentially, you're adding, it's just that you might be adding a positive to a negative. Cool. So if you look at current one over here, it's got a doc, which means it's coming towards you. And that means that the balance coming towards you. Current is my thumb towards me. Which means that the direction of the magnetic field is gonna be counterclockwise. Okay, so it's going to create a magnetic fields, right? At that point, that's gonna be counterclockwise. So this is gonna be be one is counter clockwise and then be, too is an ex. It's going into the page. If it's going into the page, it's going to be looking my thumbs up, my, my fingers over here. If it's going into the page, it's gonna be clockwise. So please do this to yourself clockwise, which means it's gonna go in this in this sort of direction. B two is going to be clockwise. They're going in opposite directions. And what we set up here is that if they're going in opposite directions, we're gonna subtract them. And that's because you're one is you can think of. One is being positive. The other one is being negative. And if you're adding them, you have to subtract. So let's let's calculate B one and B two and then we'll figure out how to combine them. B one and B two. So the equation is mu, not I, and by the way one divided by two pi R One cute way to remember this is it spells Moy, right? So it's totally silly, but b equals Moy. Or if you want to get fancy in French, it's ma right? Which is me. I think I don't know S o ma on, baby. By making weird sounds, I help you remember these equations that's worth it. So B equals ma to part two. Pi r mu is four pi times 10 to the negative seven the I If it's b one, it's I won and our one So it's three amps divided by two pi. The distance is not four distance from this guy to the middle is true. So look what's gonna happen here. The four cancels here the pipe cancels here. So you're left with three times 10 to the negative seven Tesla because it's a magnetic fields. What about this guy? Be too. Is ma Um hopefully not saying that aloud with me. People would think you were four pi times 10 to the negative seven. The current is five divided by two pi and the distances to just play it in the numbers. Same thing happens. The four cancels, the pipe cancels, and now you're left with five times 10 to the negative seven Testa. Now we want to combine the two. And this is super important up here. I wrote that You just Adam, but it really meant is that you combine them. Okay? Which means you might have to actually subtract so you can think. Well, B two b two is this clockwise B one is this counterclockwise? Be too has the bigger magnitude. So it wins, right? It wins. And then now you can just subtract and say the net is gonna be winner minus loser. So five minus three. Big minus small, which is to true times 10 to the negative seven Tesla's. And if this is the winning direction, that's the final direction. You get clockwise. Okay? That's one of the ways that you could do this. And that's the way that I prefer to look at the big If they're different. Look at the big number. Subtract the two. Now. Another thing you should know is that you can assign assign to these things, and you may remember from rotation that counterclockwise is actually the positive direction. And clockwise is the negative direction. So what you could also have done, or what you could also have done is you could have said, Well, be one is counter clockwise, which is positive. So it's positive three times 10 to the negative seven be, too, because it's counter clockwise. B two is clockwise, which makes it negative five times 10 to the negative seven and then you can actually hear just add the two and say that the next one is gonna be the addition of the two. So you have a minus seven with a plus three, which gives your minus two times 10 to the negative. Seven and minus means it is clockwise. Okay, so you could have just looked at the big one and said, big minus small. Or you could have actually assigned signs to them. And then based on right, the sign came from the direction. And then the final sign also gives the direction I want to show you both, because different professors different textbooks do a different way. Um, in different people, just prefer different things, and they might understand differently. Cool. That's a lot of difference in one word in one sentence. Alright, guys, let's keep going
2
example

Find Field due to Two Perpendicular Currents

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Hey, guys. So in this example, we have two perpendicular wires, and we wanna find the magnetic field that they produce at a point. So let's check it out. So it says two very long, perpendicular, perpendicular again. Remember, it means 90 degrees. They intersect at zero common zero. So this point right here, where they cross is zero comma zero. And remember, this is always exposition common white position, the vertical wires, A current of two amps up. So this current right here is we're gonna call it I one is to not to am's to micro amps. And then this current here is I two, which is three micro amps to the left. And we want to know what is the net magnetic fields at a point p located in this position here. So I'm gonna actually extend this blue hire a little bit so we can draw it better. Negative four. Negative. Eight. Negative. Nine would be somewhere over here. Negative four means you are far away from the y axis. So you're four to the left and negative nine. Of course, means that you are nine down. So you over here. P negative four negative nine centimeters. Okay. And the reason it says net magnetic field is because they're two wires. Therefore, there will be two magnetic fields produced here, and we want to know what is the sort of combination of those two magnetic fields. Okay, so the magnetic fields were looking for It will be a combination of magnetic field, be one which comes from current one and B two, which comes from current, too. And we'll figure out a way to combine that. You. So the equation is B equals, Remember, Ma? Right? Mu not I divided by two pi r But if I'm looking for B one, it's I won in our one where r is the distance. And for B two is ma to two pi r two. And now we're just gonna plug all the numbers. So Munitz is four pi times 10 to the negative seven. The currents, for one, is true times 10 to the negative six because it's micro divided by two pi. The distance, everything straightforward. The distance is the part that you have to pause a little bit and make sure that you get the right number. So this be one is coming from. I one. So we look, we have to look at is the parallel distance or the shortest distance between. Why're one in point P And the shortest distance is going to be right here. So this is going to be the distance, which is a four. Now I know it's negative four, but the gap, the distance between those two points, it's just gonna be a positive four. And that's four centimeters, 0.4 m. Okay. And if you do this, you will notice that the pie cancels There is a four here that's gonna cancel with this four. So this becomes a one. This becomes a 0.1 and then the two is also canceled. So you're left with 10 to the seven times 10 to the negative seven, 10 to the negative six divided by 10 to the negative to down here. And you can combine all this and you get 10 to the negative 11 so or one times 10 to the negative. 11 Tesla. Okay, so that's what we get there. Um, let's calculate them. And then we're gonna talk about direction to see how we can combine. Uh, something I'm gonna do here. Four pi 10 10. A native of seven, the current is three micro amps and two pi. The distance is going to be, um we're talking about B two was So we're looking at I two in the distance here is nine. And again that distance Just a positive number. Um, so it's gonna be 0.9 m because it's nine centimeters, And if you do this, I have it here. You get 0.67 times 10 to the negative. 11 tesla. Now, how they combine depends on your just under directions. So what we're gonna do is we're gonna grab wire the first wire in the direction of current, so we're gonna put a current going up. And if I grab it, notice that how much fingers And you have to do this yourself. Don't look at me. Um or I guess look at me and do it yourself is well to make sure that you get this right. Because sometimes it looks weird in the camera, right? So you want to grab the wire, and when you grab the wire, your fingers, you're going to be going into the page, which is away from you. Right? So, my fingers, you're going towards you. But you need to look at your fingers How they're going to go away from you on this side, right with my thumb up. But when they come back around because P is over here, right when they come back around, they're coming towards view. So they're coming out of the page. Okay, So why Air one is gonna produce, um, is gonna produce a be one on this side everywhere. Right? That's into the page. And then a B one right here, everywhere that is out of the page. So at this point, B one is going to be out of the page out of page. Now, I want you to do the same thing for I want you to do the same thing for B two. What is the direction of B two positivity if you have to, I'm gonna keep rolling here. I'm gonna grab this wire right with my thumb to the right, with my thumb to the right, And this is a little bit easier to see on the top of the wire. My fingers are going into the page away from my face. It's towards you because it's my fingers right and I'm facing you. But if you do it yourself, it's into the page. And then when he comes back around because peas below here, right when he comes back around, it comes towards me, it comes towards me. So be be to hear everywhere. On top of this line, B two is going to be into the page. But everywhere over here, below this line, B two is going to be out of the page. Peas on this side. So be, too, is below the line. So be, too is also out of the page out of the page. And because both of these are in the same direction, I could just Adam and I could just say that the net magnetic field is the addition of these two guys here. So it's going to be 1.67 times 10 to the negative 11 Tesla, and that is the final answer. Cool. That's it for this one. Let's keep going
3
example

Find Zero Magnetic Field

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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.
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