Guys, so in this video, we're going to talk about the magnetic field produced by loops in solenoids. Let's check it out. Alright. So remember, if you have a wire that is carrying current, it's going to produce a magnetic field away from itself. It looks somewhat like this: You have a wire with a current I, it's going to produce a magnetic field with strength b, some distance away from itself, distance r. And you can find that magnitude by using the equation μπI/2πr, where r is a distance from the wire. Okay? And this is important and you gotta remember this, but this only works for straight wires. In this video, what we're going to deal with is not straight wires, but loops. Imagine you have a long straight wire, you can actually turn that wire to look like a loop, something like this. Right? And in this situation, you're going to get slightly different equations, but also the right hand rule will be a little bit different. Let's check it out.

So, in a straight wire, you may remember that our current is obviously if you have a straight wire, you have a straight current. We use our thumb to indicate current. That's what we've always done. And when you grab the wire, your fingers were the direction of the magnetic field around it. So because your magnetic field was curving, you would curl or curve your fingers. So b was curved, so you would curl. You use your curled fingers to represent b because it makes more sense. It's better visually to curl your fingers than curling your thumb. It's easier to see this way. Right? So that's why we did it that way, but now when you are in a loop, in a wire loop, your current is going to be going around, so it's going to be easier to use your fingers curved as your currents. So you're going to use your fingers as your currents, and guess what? What's going to happen is that your magnetic field is going to be straight, so you're going to use your thumb. So those two things flip, and you can think that in loops, the right hand rule is backwards or you can also think that there's sort of this greater rule that you always want to curl your fingers. So whatever is curling in a problem is going to be represented by your finger and then the other thing is going to be your thumb.

Okay? So let's look at single or multiple loops. And the first thing I want to talk about is direction. Let's say that the current here is going in this direction. That's current, which by the way is counterclockwise. So what we're going to do is remember, if it's curling, you're going to use your fingers, and I'm going to curl my fingers in the direction of current. So current's going this way, so I'm going to go like this. Okay? And if you do this, and you should do this yourself, you're going to see that your thumb is pointing at yourself, which means it's coming out of the page and that's the direction of b. So in the middle of this thing here, in the center or everywhere inside actually, you're going to have a magnetic field that is jumping out of the page towards you, and you're going to have a magnetic field outside of the loop that is going into the page, which is a reverse direction. Right? And obviously, as you would expect, if you're going in the clockwise direction of currents, you get current in the clockwise direction. It's just the opposite, and you can do the same thing with your fingers, and you can now curl them to the right like this clockwise. And notice that my thumb is pointing away from me, which means it's going into the page, which means on the outside of the wire, you're going to have an out of the page magnetic field everywhere. Cool? Now let's talk about the equation.

The magnitude of the magnetic field, of a loop that's produced by a loop, is going to be muνI2r, and I almost forgot there's an n here for 2ρ. So let's talk about a few things. First of all, notice that there is no pi in here. That is not a typo, not a typo. I actually want you to write not a typo because I don't want you to get confused and think that there's a pi there. What happened? There is no pi. Okay? The other thing is that you have r, big r, which is the radius and not little r, which is a distance. Okay? Little r is a distance. I wrote it here. You have the radius. So what this number is going to be, is the radius of that circle, which is given to you or they will ask for it. Right? And third, what is this n here? n is going to be the number of loops. So you can have a single loop, which looks, could look like this. You got a straight wire, then you made a little loop out of it, or you can have multiple loops. Let's say you do this. Right? So in this case here, you have n equals 1, this n over here. And in this case, you have n equals 3. And the idea is that if you have one loop, you produce a certain amount of magnetic fields. Let's say 10. Well, if you make 3 loops, you're going to produce a magnetic fields of strength 30. Triple that. Okay? In fact, that's why sometimes you see, you might see some electric devices that have a ton of wires tightly loops like this, and it's because you're trying to produce a stronger magnetic field. It's like magic. You loop more, you get a stronger magnetic field. Cool? So that's it for these points that that n just goes in there. This equation is super important. There's one last point that I want to make here is that this equation is for the magnetic field b at the center at the center of the loop. So it's not at any point. It's always at the dead center. So this has to be a perfect circle. Cool? Now remember how we could make loops that were like this, you can have multiple loops. Well, if you make a loop that's really long, you're actually going to get a different equation. That's what we have here. And I briefly mentioned this over here.

Okay? So if you have a very long loop, what's going to happen is you're going to have a different equation. The equation now is going to be μνIn/ℓ, n/ℓ. Everything is the same. This ℓ means this length over here. Okay? It's this length over here. So it's similar equation but different. Alright? Now what you see sometimes, another version of this equation gets n/ℓ and changes into little n. Little n is big n over ℓ. So you can also see this equation like this. Let's move this out of the way. You can also see this as μνI and little n will replace those 2 guys like this. Okay? This is another version of the equation. So you can see either one of these two versions. n is the number of loops per meter. This is loops per meter. And it's kind of almost like a density. It has to do with how tight these things are. So for example, this has 4 loops, not very tight. This has 4 loops super tight. Okay? So the n here is greater than the n here. Let's say n 1 and 2, this n here is greater. Okay. Cool. So that's the equation. What about the direction of the magnetic field? Well, remember what we talked about just now, which is if something curves, we're going to use our fingers. What's curving here? Well, let's say the current is entering here. Let's say the current is entering this side. Notice what the current does. The current is going to go in and then it's going to go in this picture, it's going into the page and back into the page and back into the page and back. So the current is what's curving. So we're going to use our fingers for current as well. Okay? So it's just like this one. Same thing. But what you have to be careful...