Magnetic Field Produced by Straight Currents

Hey guys so in this video we're gonna talk about how current in a wire will produce a magnetic field. Let's check it out. So you may remember that if you have a moving charge, that moving charge will produce a new field away from itself. So let's draw a little charge here, here and it's moving so it's got a v and that charge will produce a magnetic field up here or over here, right. In a number of different places away from itself. So and the magnitude of that magnetic field if you remember is mew not q v sine of theta divided by 4 pi r squared. But this is old news, that's why I said remember, right. And what we wanna talk about now is how just like moving charges will produce a new field well current will also produce a new field because current 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 lots a little q's here that have v's. Right, lots of little q's with v's. That's what current is. So if a q moving with v produces a B field, than an I will also produce a B field. Okay, so current 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 to have a magnetic field somewhere over here and any number of distances. The magnitude of that equation looks a little cleaner than this one, a little nicer. It is that B equals mew not I divided by 2 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 really really important, you're gonna see this all the time. I should know that this only works for a very very long wire and in most of the problems you're just going to assume that the little wire is very long even if it's not said, even if it's not stated. If you have a short wire, you're gonna get a different equation than this one, it's a little bit more complicated but most of you are gonna have to deal with that. Remember mew not is a constant and this r over here is a distance, little r is always the distance and never a radius. In fact if you have a wire it would be this right here r. Some people even us a letter d or sometimes a to make this equation. Cool. So that's how you find the magnitude and what about the direction. The direction comes from the right hand rule, and were gonna grab the wire with our thumb in the direction of current 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 of which the charges are 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 current is going to the left we're gonna grab it like this, but if the current is 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 by convention positive. Cool. And then if you have 2 fields in the same location, and the fields are 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 one. In 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 the magnetic field produced by current in a very long wire if the current is oriented up. Meaning if you have a wire and the current's 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 it into the page 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 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 the direction of the magnetic field, well we're gonna grab the wire with my thumb pointing the direction of the current. 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 here so you can follow a little bit better. So this is super important, here is what we're going to do. This is going up which looks like this, this is the direction of the current so we wanna grab with my right hand always, I'm gonna grab it and then my hand should look like this. Please do this, please do this so that you can see. So what you're gonna do here is notice that my fingers are curling into the page here, into the page here, right. They're going like that direction away from my face into the page and when they come back around they come 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. Whenever your thumb is up you're always gonna get that 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 the cable is gonna be into the page and anywhere here is going to be out of the page. Okay. So you can draw a bunch of little x's 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 dot and the bottom of the wire will be either an x or a dot. Figure out real quick, pause the video 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 paused the video and I'm gonna do it over here. So my thing is gonna be my current and into trhe right, I'm gonna grab it with my right hand which looks like this, right. Notice that my fingers are going into the page up 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. Cool. 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 this 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 point it at the page which means my hand is gonna grab it like this. I'm actually gonna get rid of the pen, 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. Trying to get the angle here right, so I'm gonna go in a clockwise direction. Please do this yourself, it looks like you gotta do it to make sure you're getting this right. So that means that actually the direction of the current is going, I'm sorry the direction of the 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 than the magnetic field would flip direction as well. Okay. So in this case you would have your thumb pointing at you and you can't see it from my hand but if you look at your thumb, your hand with your thumb pointing at you like this, right. Looking really silly right now, you're gonna se that B goes counter clockwise. Hopefully you got that. Let's look at example 2 which is a computational example, it says 2 wires are shown below 4 meters away from each other. So this distance here is 4 meters and I wanna know what is the magnitude of the direction of the magnetic field that is produced at the center between the 2. The center between the 2 is somewhere over here, let's call this point p and it is a distance of 2 meters away from both of them and what I wanna know is essentially what is the magnetic field is the point p. I want the magnitude and the direction of the magnetic field. This is really the net magnetic field, that's gonna be a combination of 2 because this guy produces a magnetic field and this guy produces a magnetic field. They both produce a magnetic field there so really this is you could think of this as a net magnetic field so it's gonna be the magnetic field do to the 1st current plus the magnetic field do to the second current. I1 and I2. 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 1 over here, it's got a dot so it means it's coming towards you and that means that the now it's coming towards you, current is my thumb towards me which means that the direction of the magnetic field is gonna be counter clockwise, okay so it's gonna create a magnetic field right at that point that's gonna be counter clockwise. So this is gonna be B 1 is counter clockwise and then B 2 is an x so it's going into the page. If it's going into the page it's going to be look at my fingers over here. If it's going into the page it's going to be clockwise. So please do this yourself, clockwise which means it's gonna go in this sort of direction, B 2 is going to be clockwise. They're going opposite directions and what we set up here is that if they're going opposite directions we're gonna subtract them. And that's because you're 1 is, you can think of 1 as being positive the other one is being negative and if you're adding them you have to subtract. So let's calculate B 1 and B 2 and then we'll figure out how to combine B 1 and B 2. So the equation is mew not I and by the way 1 divided by 2 pi r. One cute way to remember this is it spells moi so it's totally silly but B equals moi or if you wanna get fancy and French it's mou which is me I think, I don't know. So moi and maybe by making weird sounds I help you remember these equations, that's worth it. So B equals moi 2 pi r. Mew is 4 pi times 10 to the negative 7, the I if it's B 1 it's I 1 and r 1 so it's 3 amps divided by 2 pi. The distance is not 4, distance from this guy to the middle is 2 so look what's gonna happen here. The 4 cancels here, the pi cancels here so you're left with 3 times 10 to the negative 7 tesla. Because it's a magnetic field. What about this guy. B 2 is moi. Hopefully you don't have to say that out loud like me, people will think you're weird. 4 pi times 10 to the negative 7 the current is 5 divided by 2 pi and the distance is 2. Just plugging in the numbers, same thing happens the 4 cancels the pi cancels and now you're left with 5 times 10 to the negative 7 tesla. Now we want to combine the 2 and this is super important. Up here I wrote that you just add them but I really meant that you combine them okay which means you might have to actually subtract so you can think well B 2, B 2 is this clockwise B 1 is this counter clockwise, B 2 has the bigger magnitude so it wins, right and wins and then now you can just subtract and say B net is gonna be winner minus looser so 5 minus 3, big minus small which is 2. 2 times 10 to the negative 7 tesla and if this is the winning direction that's the final direction you get clockwise. Okay, that's one of the ways you can do this and that's the way I prefer just look at the big, if they're different just look at the big number, subtract the 2. Now another thing you should know is that you can assign a sign to these thing and you may remember from rotation that counter clockwise is actually the positive direction and clockwise is the negative direction so what you could have also have done is you could have said well B 1 means counter clockwise which is positive so it's positive 3 times 10 to the negative 7, B 2 is clockwise which makes it negative 5 times 10 to the negative 7 and then you can actually here just add the 2 and say that the net one is gonna be the addition of the 2. So you have a minus 7 with a plus 3 which gives you minus 2 times 10 to the negative 7 and minus means it is clockwise, okay. So you could have just look at the big one and said big minus small or you could have actually assigned signs to them and than based on right the sign coming from the direction and then the final sign also gives the direction. I wanna show you both because different professors different textbooks different way. Different people just prefer different things and they might understand differently. That's a lot of difference in one sentence. Alright guys, let's keep going.

Hey guys, so in this example we have two perpendicular wires and we want to find the magnetic field that they produce at a point, letÕs check it out. So it says two very long perpendicular, perpendicular again remember, it means 90 degrees, they intersect at (0,0). So this point right here where they cross is (0,0), and remember this is always (x position, y position). The vertical wire has a current of 2 A up, so this current right here is weÕre gonna call it I1 is 2 not 2 A, 2 _A, and then this current here is I2 which is 3 _A to the left. And we want to know whatÕs the net magnetic field at a point P located in this position here. So IÕm actually gonna extend this blue bar a little bit so we can draw it better, -4, -8, -9 would be somewhere over here, -4 means you are 4 away from the y-axis, so youÕre 4 to the left and -9 of course means that youÕre 9 down. So youÕre over here P(-4, -9) centimeters. Okay. And the reason it says net magnetic field is because there are two wires therefore there would be two magnetic fields produced here and we want to know whatÕs the result of combination of those two magnetic fields. Okay. So the magnetic field weÕre looking for would be a combination of magnetic field B1 which comes from current 1 and B2 which comes from current 2 and weÕll figure out when we combine the two. So the equation is B equals remember mua, right, _not I divided by 2 Pi r, but if IÕm looking for B1 it's I1 and r1 where r is the distance and for B2 is mua, 2, 2 Pi r2. And now weÕre just gonna plug all the numbers. So _not is 4 Pi times 10 to the negative 7, the current for 1 is 2 times 10 to the negative 6 because micro divided by 2 Pi (the distance), everything straight forward, the distance is the part here you have to pause a little bit and make sure you get the right number. So this B1 is coming from I1 so we have to look at the parallel distance or the shortest distance between wire 1 and point P. And the shortest distance is going to be right here. So this is going to be the distance which is a 4. Now I know it's a negative 4 but the gap, the distance between those 2 points is just going to be a positive 4. And thatÕs 4 centimeters, its 0.04 meters. Okay. And if you do this you will notice that the Pi cancels. There is a 4 here thatÕs gonna cancel with this 4, so this becomes a 1, this becomes a 0.01 and then the 2s also cancel, so youÕre left with 10 to the 7 times, negative 7, 10 to the negative 6 divided by 10 to the negative 2 down here and you can combine all this and you get 10 to the negative 11 or 1 times 10 to the negative 11 Tesla. Okay. So as we get there, letÕs calculate, weÕre going to talk about directions and see how we can combine, same thing IÕm going to do here 4 Pi 10 to the negative 7, the current is 3 _A and 2 Pi the distance is going to be, weÕre talking about B2, so weÕre looking at I2 and the distance here is 9, and again that distance is just a positive number, so its gonna be 0.09 meters because it's 9 centimeters. And if you do this, I have here, you get 0.67 times 10 to the negative 11 Tesla. Now, how they combine depends on your distance, on the directions. So what weÕre gonna do weÕre gonna grab wire, the first wire in the direction of car, so weÕll put a car going up and if I grab it notice how my fingers, and you have to do this yourself, donÕt look at me, or I guess look at me and do it yourself as well, to make sure that you get his right, because sometimes it looks weird in the camera. Right. So youÕre gonna grab the wire, and when you grab the wire, your fingers are going to be going into the page which is away from you, right, so I think itÕs going towards you but you need to look at your fingers how they are going away from you, on this side, right, with my thumb up, but when they come back around, cause P is over here, right, when they come back around, theyÕre coming towards you, so theyÕre coming out of the page. Okay. So wire 1 is gonna produce a B1 on this side everywhere right, thatÕs into the page. And then a B1 right here, everywhere that is out of the page. So at this point, B1 is going to be out of the page. Now I want you to do the same thing for, I want you to do the same thing for B2. What is the direction of B2, pause it if you have to, IÕm going to keep rolling here, IÕm gonna grab this wire, right, with my thumb to the right, and just look at here to see, on the top of the wire my fingers are going into the page, away from my face, itÕs towards you, because itÕs my fingers, right, and IÕm facing you, but if you do it yourself, its into the page and then when it comes back around because P is below here, right, when it comes back around it comes towards me. So B2 here, everywhere on top of this line, B2 is going to be into the page, but everywhere over here below this line, B2 is going to be out of the page. P is on this side, so B2 is below the line so B2 is also out of the page. And because both of these are the same direction, I can just add them and I can just say that the net magnetic field is the addition of these two guys here. So it's going to be 1.67 time 10 to the negative 11 Tesla. And that is the final answer. Cool. We took up this one. LetÕs keep going.

Hey guys in this example we want to find where between two wires the magnetic field is zero, let's check it out. So we got two horizontal wires that are 6 meters away, they're both horizontal which means they're parallel. I'm gonna make the first one red and the second one blue and it says here the currents are 4A at the bottom, I'm gonna call that I2, 4A and 5A at the top and they're both going to the right. I1 equals 5A to look like this and their distance 6 meters away from each other and we will know 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 that will be produced by these currents. So if you have a wire and it points to the right you need to 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 going to be pointing the 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 they come back around here towards me. Okay. So what that means is that I1 will generate an into the plane field at below it right so over here this is B1 due to wire 1 and then on top of the wire it's going to be towards me so I see a dot coming towards my face so that's gonna be B1. And by the way this extends out anywhere below the wire is going to have an XB field so if you keep going here this is also gonna be XB1 XB1. Just like how here keeps going this is a dots B1 dots B1. Okay. So it sort of a separation above and below the wire. Now if you do the same thing for wire 2, wire 2 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 due to B2, X, i'm sorry, X direction of B2 due to this current and on top of the wire you gonna have a dots on top of the wire its coming towards you so this is gonna be B2 magnetic field due to current 2 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 noticed 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, 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 field produced by those two wires are going into the page which means that these guys can never be zero. Okay. The magnetic field will never be zero here, you can only get zero in the middle because that's where we have different directions we have opposite directions so Bnet 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 that there's 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 cancel 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 distance A and I want to know what is A and you can call this distance B or we can call it we can also just call it r1 or r2 right, and this is r1 and what we're looking for for is r2 and by the way keep in mind that r1 plus r2 equals 6 meters, okay, equals 6 meters.Cool. So what do we do now, well, if we want the magnetic field to be zero this means that the magnitude of B1 equals the magnitude of B2. Two things with the 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, its mua _not I divided by 2 Pi r, so we'll have this twice, now obviously the first case here were looking at B1 so this is current 1 and distance 1, current 2 and distance 2. Okay. These guys are just constants by the way so they get cancelled out which is nice, so you end up with I1/r1 = I2/r2 and we are looking for these numbers, we are looking for r2. Okay. Now if you notice I can quickly replace the I's, I's are 5 and 4, r2 is what I'm looking for, what about r1, so the problem with r1, is r2 is my variable that's what I'm looking for but I don't have r1 so what I have to do is I have to write an expression for r1, and if you look at r1 here, I can rewrite r1 as 6 minus r2 so 6 minus r2, and the good news here, is that here you have two unknowns. two variables and that's bad news with just one equation. Okay. Here you have one unknown which is good news. Now you can actually solve this. So now this is just an algebra problem, we have to cross multiply in getting our r2 solve for. So if I cross multiply 5r2 equals 4 times 6 minus r2 and I can expand this 5r2 equals 24 minus 4r2, I'm looking for r2 so I'm gonna move it over to the other side, 5 plus 4r2 is 9r2 equals 24, so r2 is 24 divided by 9 which is 2.7 meters. Okay. And that is the final answer. Would that means that this distance here is 2.7 meters? If the whole thing is 6 by the way it means that this is 3.3 meters. And it should make sense that the wire is a little closer to I2 than to I1 because the magnetic field comes from this equation. Notice that these two guys are constant, so they don't really matter right now. So the stronger my I, the stronger my B and the stronger my r the weaker my B. So the bottom one has a weaker I, has a smaller I, so to compensate for that, you also have to have a smaller r, so that its a little stronger, so the smaller I makes it weaker, but then the smaller r means that its closer which makes it stronger. Okay. So anyway weaker I means you want it to be closer which means you have a smaller r. Okay. And that would a bounce.That's it for this one, let's keep going.