Magnetic Field Produced by Straight Currents

by Patrick Ford
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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