Ch 30: Sources of Magnetic FieldWorksheetSee all chapters

# Magnetic Field Produced by Straight Currents

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Sections
Magnetic Field Produced by Moving Charges
Magnetic Field Produced by Straight Currents
Magnetic Force Between Parallel Currents
Magnetic Force Between Two Moving Charges
Magnetic Field Produced by Loops and Solenoids
Toroidal Solenoids aka Toroids
Biot-Savart Law (Calculus)
Ampere's Law (Calculus)
Ampere's Law

Concept #1: Magnetic Field Produced by Straight Currents

Transcript

Example #1: Find Field due to Two Perpendicular Currents

Transcript

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.

Example #2: Find Zero Magnetic Field

Transcript

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.