In this problem were given two concentric but different size loops of current carrying wire and were asked to figure out what the direction and magnitude of the current needs to be in the outer wire in orderto have a zero net magnetic field at the center of both of the wires. So what is this looking like? Well, let's have a small inner wire. Let's make a little bit bigger A smaller inner wire here and ah were not asked to calculate things relative to the inner circle. So let's just pick a random direction for current will pick the direction that's conventionally positive, which is counterclockwise. So we'll have a loop with the current going counterclockwise here. Then we have a nether concentric loop that's bigger around this, and we need to then figure out Mrs Inner Loop current. What is the outer loop current? And then what is the direction? Right? So is it clockwise or counterclockwise? We're not sure. So we need to figure that out such that we want tohave right at the center here. Ultimately a net magnetic field of zero. So how do we figure this out? Well, we need to figure out if we need the outer loop to cancel the outer current loop to cancel the magnetic field from the inner current loop. Well, let's figure out what the magnetic field from the inner current loop would be and what direction it would be given the right direction of magnetic field would be at the center of the inner loop. Given the direction the current, we've defined the inner loop. Okay, so, by definition, a magnetic field from a, uh, just a one loop of wire, actually, any coil circular coil of wire. The man feel the center of that circular coil is given by n the number of loops and that coil times, or you're not times the current through that coil divided by two times are, which is the radius. So how far away we are from the center of the loop to the outside of the loop now four. The inner loop the magnetic field is center of the inner loop is going to be given by the number of turns in the inner loop multiplied by you not multiply the current by the in the inner loop divided by two times the radius of the inner loop Okay, so we have an expression for what the magnetic field would be. Well, what is the magnitude of the magnet field? What is the direction of this magnetic field? We follow the right hand rule. We have used our fingers to curve around the direction of the current, and we have our thumb That's gonna point in the direction of the resulting magnet field. And in this case, that is gonna be out. So the magnetic field from the inner coil of wire is definitely going to be directed outward at the center here. So what is the overall vector of the magnetic field at the center of the inner loop? Well, it's going to have a magnitude of the expression that I have listed here, and it is be going going to be going out of the screen. So we know that we need to cancel out this inner magnetic field by using the magnet UK by the outer, of course, the outer loop in this diagram. So what is the expression gonna be for the outer loop? So let's see what will be right here. What? We need the number of turns theater loop has not. When you know, the current in the outer loop, and then we need to know how big the outer loop is. Right? Okay, so we have the expression for the magnitude. So we know these magnitudes here have to be equal if they're able to add up and cancel each other out. But also, not only do they need to be equal, they needed the opposite in direction. So we have to have the magnetic field from the outer loop being into the screen. And what current direction do we need for that to happen? So if you want the direction of the magnetic field from the outer loop to be into the screen, we can put our thumb pointing down. They got some point it down into the screen and then rotate our fingers around. Follower wrist around and we'll see that the current needs to go, um, in a clockwise direction. So opposite of what we chose in order to cause this particular type of magnet fields so I can get rid of that. Perfect. So that's gonna be the direction that we need to have. The current needs to be going the opposite her in the outer loop needs to be going the opposite direction as the inner loop. Cool, cool, Cool. So we figured out that part of the problem we have to be going into the screen. The magnetic field must be going into the screen and therefore the outer loop current has been going. The opposite direction is the inner loop current. Now we need to figure out what that outer loop current needs to be in order to create such a magnetic field that canceled so that these two magnetic fields created by each of these clips canceled at the center. So that means that the magnitude of these magnetic fields have to be equal. So we're going to say the magnitude of the inner loop magnet field must be equal to the magnitude of the outer loop magnetic field. Let me do that. What do we get? Okay, so we have the number of turns on the inner loop times me not times the current in the inner loop divided by twice the radius of the inner loop and that must be equal to the number of turns in the outer loop multiplied by me not times the current in the outer loop divided by two times the radius of the outer loop. We'll see. We have a few things. Cancel across the city, quality arm units can cancel and our twos can cancel. And then they can solve for the current that we want to find because we know the dimensions of each the loops. And we know the current in the inner coil. So what does this look like? What's rearranged for I out and what we get is a nice low equation with some particular cool ratios. So that current in the outer loop ends up being equal to the ratio of the number of turns of the inner loop to the outer loop, multiplied by the ratio of the raid ei of the outer loop to the inner loop, and then multiply that by the current in the interview. So we plug all this stuff in. We have 140 turns in the interlude 180 turns in the outer loop. We have a radius of 0.23 meters in the outer loop, 0.15 meters in the inner loop, and the interlude has a current of 7.2 amps. So in the end, overall, what is the outer loop Current need to be in order to create a magnetic field center that cancels with the minute field created by the inner loop Will that current ends up being about 8.6 amps? And remember that current needs to be going in the direction opposite to the current in the inner loop in order to create a magnetic field that cancels with the magnetic field from the inner loop. So we have the answer of 8.6 amps for the magnitude and opposite the direction of the interlude current for the direction.