Find Field due to Two Perpendicular Currents

by Patrick Ford
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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