Alright, guys. Let's work this one out together. We have a long straight wire on a horizontal surface in the xy plane, and it carries a constantly increasing current in the plus y direction. So now we have a square loop, and we need to figure out what the direction is of the induced current. So that should automatically tell you this is going to be a Lenz's law problem. So the first thing I like to do in these kinds of problems where I'm given sort of like a 3D perspective is just draw out sort of what's going on here. So we have this axis right here, which we're told is the plus y direction, and we're told directly on the side of it, on the right, that is the plus x direction. So in other words, sort of I have a straight line like this, and this is going to be my x direction. Alright? So it's kind of weird. So that means that this direction was sort of like the z-axis. So I've got some three-dimensional stuff going on here. So what I always like to do in these kinds of situations is see if I can change my perspective a little bit and sort of draw, like, a parallel or a different diagram. So if I were to view this current from the side, then basically what would happen is, if I were to view along the axis of the current, then that means that the current would be going away from me like this, and then that means the square loop would be sort of on the right like that. So I'm looking sort of, like, perfectly on the side on the axis like that. Okay. So we need to find out what the direction of the induced current is. We need to use Lenz's law, but there are a couple of things we need to figure out first. So we need to figure out what the direction is of the magnetic field through the square loop, and then we need to figure out how the magnetic flux is changing. So in other words, what's delta phi b? That'll tell us what the direction is of the induced magnetic field. Okay?
So we need to do is first we need to figure out what the magnetic field is. So for a straight current-carrying wire, we know that there is some relationship between the current in that wire and the magnetic field that it generates. Now you don't have to necessarily remember this, but remember that the magnetic field for a straight current-carrying wire is μ0I/2πr. Okay. So the direction of our magnetic field is going to be given by our right-hand rule. So now what happens is our thumb points in the direction of the straight thing, in this case the current, and our fingers will curl in the direction of the magnetic field. Alright. So get out your right hands, and what we're going to do is we're going to point our thumbs in the direction of that current. So in other words, our thumbs are going to point in this direction. But remember that we're sort of viewing this from a weird perspective. So what I like to do in this situation is we're going to use the side view. So in other words, we're going to be looking at what's going on in this diagram right here. So what happens is we need to point our thumbs in the direction of the current, in this case, and what you should see is that you should be pointing your thumb into the page away from you, and your fingers will be curling in the direction of the magnetic field from that wire. So what that means is that our magnetic field lines actually curl clockwise. So what that means in this sort of diagram is that in the first diagram on the right, I have magnetic field lines that are going like this. But on the side view, now what happens magnetic field lines that are sort of curling around like this. And so we know that they're given by our right-hand rule, they're going to go clockwise or sorry, they're going to go clockwise like, wait, no. That's right. Yeah. So clockwise like that. So what that means is that the magnetic field is going to be pointing in this direction through the square loop. So in other words, the magnetic field through the square loop points downwards. So that is the direction of b. So that's our first question. The magnetic field through the square loop points downwards. So now what we need to do is to figure out what the change in the magnetic flux is. So remember that the change in the magnetic flux is going to be given by three variables, b, a, and the cosine of theta. So what is changing as this current is constantly increasing in the wire? Well, remember that we said that the relationship between I and b is that as the current increases, the magnetic field also increases. So what happens is the changing variable in this case and our magnetic flux equation is going to be b. Right? Because the area of the square loop is not changing. It's always just a constant right there. And the direction of it, so there's the cosine of theta, is always going to be, again, straight and not changing. So what happens is our b field is changing. So as the current knows positive magnetic flux change and a downward-pointing b, magnetic field, then we can figure out the direction of the induced magnetic field using Lenz's law. Lenz's law says that it's going to do whatever the opposite is of that change in magnetic flux. So if it's downwards and it's increasing, then that means that Lenz's law wants to counteract that change by producing a magnetic field that sort of fights that. So in other words, our magnetic field is going to point upwards like that. So now we find out what the direction is of the induced currents by using our right-hand rule again. But in this case, we have to use a different right-hand rule. So before, we had our thumb pointing in our direction of the current because that was the straight thing, but now the straight thing here is our magnetic field. So now what happens is our right thumb is going to point in the direction of the magnetic field, and our fingers will give us the direction of the induced current. So be careful when you're using these right-hand rules because sometimes in the same problem, you might be using two different right-hand rules to solve it. Okay? So now what we're going to do is on our page, right? So just follow exactly what we are doing. We're going to take our right hands, and we're going to point in the direction of the magnetic field. So what happens is if we have a magnetic flux, or sorry, if we have an induced magnetic field that points upwards like this, then that means that our fingers will curl in the direction of that induced current. So if we're looking at it from the side view, so we're still looking at the side view right here, then what happens is our induced currents is going to point sort of in the counterclockwise direction. So Or sorry, that's going to be counterclockwise. So in other words, if we're looking at it from this perspective in the first diagram like this, it might be helpful to visualize this as your thumb pointing, so like it's quite like upwards, and our induced magnetic, or sorry, our induced current will point like this. So that means that in our first diagram, our induced current is going to be pointing in this direction. So that's the direction of our induced current, and that actually happens to be the counterclockwise direction. So that's going to be counterclockwise. Okay? So let me know if you guys have any questions with this, and I'll see you guys in the next one.