29. Sources of Magnetic Field
Magnetic Field Produced by Loops and Solenoids
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Hey guys, so let's check out this example. So here we have 2 wire loops that are concentrically arranged. Meaning concentric means one circle inside of the other, right, with a common middle. So it's shown below and the inner wire has diameter 4. Now real quick in physics remember we almost never use diameter, we almost always use radius. So I'm gonna right away change this, instead of writing diameter 1 I'm gonna write radius 1 and radius is half of the diameter so that's 2 meters, and a clockwise currents of 5. So that's the inner one here, which is blue but it's got a clockwise currents of 5 amps, so I'm gonna put here 5 amps. So I 1 is 5 amps and radius 1 is 2 meters. And then the red one is counterclockwise counterclockwise which is this way and it's got a current of 7 amps in that direction. So I can write that I 2 is 7 amps and r 2 is the diameter which is 6, It's actually gonna be the radius which is gonna be 3, half of the diameter, okay? And what we're looking for is the net magnetic field at the center. Remember when you have a current, when you have a loop current, so you have a loop of wire with current going through it, it's gonna produce a magnetic field through the center of the ring either in or out, right? And we have 2 rings with the same common center, so both rings or both loops will be contributing, to this here which is why we're talking about the net magnetic field because it's gonna be a contribution of both, it's gonna be a combination of both. So let's find those two numbers, b 1 and b 2. And the equation is mu naught I divided by 2 big R where big R is the radius and we have all of these numbers. It's I one since it's b one and it's r one since it's b one, right? So ones go with ones. So this is 4 pi times 10 to the negative 7 and the current is a 5 and the the radius here is a 2. Okay? And if you plug this into your calculator, you're gonna get that this is 15.7 or actually I should say, 1.57 times 10 to the negative 6 times 10 to the negative 6. Okay? And you can do this with b 2, it's very similar, just the numbers are a little bit different. So instead of a 5 up here, you're gonna have a a 7 and instead of a 2 over here, you're gonna have a 3, okay? And if you do this, you get 1.67 times 10 to times 10 to the negative 6, okay. Now let's talk about direction. Let's talk about direction. To find direction, we're gonna use the right hand rule. So first let's look at the blue inner circle. The blue inner circle is not going this direction but it's actually going in this direction, right? It's going clockwise like this. If you do this your thumb points away from you which is into the page, which means that the first one, the inner one is going to go into the page. And the other one is in the opposite direction so it must go in the opposite direction. So this is going to be out of the page. And if you wanna confirm, if you wanna confirm you can just use again your hand and grab the outer wire goes this way, right? This way and look, my thumb is now pointing my face which is out of the page and towards me. Because these guys are going in different directions, we can't just add the magnitudes, in fact we have to subtract. And the way to do this is you start with the bigger one and then you're gonna say hey, this guy is the bigger one so it's the winner, this one wins, right? It's kinda like a tug of war, one's pulling this way the other one's pulling the other way, this one wins. So the net magnetic field is going to be winner minus loser. So 1.67 times 10 to the negative 6 minus 1.57 times 10 to the negative 6. This is actually just a matter of subtracting this minus this because it's got the same power of 10. So this is going to be 0.1 times 10 to the negative 6, 0.1 times 10 to the negative 6. But we can multiply this by 10 and then we have to divide this by 10. We multiply this by 10 so we get 1 times instead of point one. And if we multiply here, we have to divide here so it's fair, so it's so we're not actually changing the number. And this divided by 10 is 1, 10 to the negative 7. By the way, you could also have answer just 10 to negative 7 but that's that. So this is 1 times 10 to negative 7 Tesla And in what direction it's going out of the page because that was the winning direction of the 2. Okay? So that's it. That's one way you can do it. Another way you could have done this, you could have just assigned signs and you could have said hey, into the page into the page is like this, right? Away Away from you with my thumb and my fingers are curling in the clockwise direction, clockwise is usually negative, so we can say that into the page is negative and out of the page is positive, right? So then you would have done this with numbers and you would have gotten you can direction. Or you can just assign positives and negatives and then do the math. Cool, that's it for this one, let's keep going.
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