Magnetic Field Produced by Loops and Solenoids - Online Tutor, Practice Problems & Exam Prep

The magnetic field produced by loops and solenoids can be determined using the right-hand rule and specific equations. For a single loop, the magnetic field strength \( B \) at the center is given by B = μ_0I2R, where \( R \) is the radius and \( I \) is the current. For multiple loops, the strength increases proportionally with the number of loops. In long solenoids, the equation modifies to B = μ_0IN1L, where \( N \) is the number of loops per unit length. Understanding these principles is crucial for applications in electromagnetism.

Magnetic Field Produced by Loops and Solenoids

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...

Find How Many Loops in a Solenoid

Hey, guys. So let's check out this solenoid example. Here, I want to know how many turns a solenoid is going to have. How many turns is the variable big N in solenoids? Not to be confused with little n, so big N is the number of turns, a 2 meter long solenoid meaning the length of the solenoid, the sort of sideways length, this L is 2 meters. In order to produce a 0.4 T magnetic field, b equals 0.4 T when a 3 amp current is run through it. So when a current I equals 3 amps is run through it, is there an equation that relates all these variables? Of course, there is, and that's the equation for the magnetic fields through the center of the solenoid, b = μ₀ I N / L, we're looking for N. So I can just move some stuff around, N = bL / (μ₀ I), and b is 0.4, length is 2, μ₀ is 4π times 10^-7 and the current is 3 amps, the current is 3 amps. And if you multiply all of this, I have it here, you're going to get, you're going to get, I have it rounded, you're going to get 2 times 10⁵ turns. That's the value for N, which means by the way that you're going to have 200,000 turns. That's what you need to have to make this happen. Cool, that's it for this one, let's keep going.

The single loop below has a radius of 10 cm and is perpendicular to the page (shown at a slight angle so you can better visualize it). If the magnetic field at the center is 10^-6 T directed left, what is the magnitude of the current? What is the direction of the current at the top of the wire:into the page (towards left) or out of the page (towards right)?

• A. 0.16 A, into the page
• B. 0.16 A, out of the page
• C. 16 A, into the page
• D. 16 A, out of the page

Designing a Solenoid (Total Length of Wire)

Hey guys, so let's check out this example. Here you're tasked with designing a solenoid that produces a magnetic field of 0.03 Tesla at its center with a radius of 4 centimeters. The radius is 0.04 meters and a length of 50 centimeters or 0.5 meters. We want to know the minimum total length of 12-amp wire. 12-amp wire means that this wire is capable of withstanding currents of 12 amps or more. If you try a higher current, it's just probably going to burn the wire or it's risky. That means we're going to use a current of 12 amps. I want to know the minimum total length you should buy to construct the solenoid. Imagine you're going to the wire store and you need to buy some wire, how much total length? Now remember, total length is different from length, right? Length is just if you make a solenoid it looks something like this. This is the length here, sort of like the side-to-side length. But the total length of wire is all these circumferences here, right? It's all of this. One way to think about this is if you get that solenoid that's all curled up and you pull all the way straight so that it doesn't curl anymore, what is the total length of wire you're going to get if you did that? Okay. L is this which is given to us, but we want to know the total length, so I'm just going to write total wire equals question mark. And you may remember the equation for this, one's circumference is 2πr, right? Where r is the radius and we're given that. But if you have n loops then the total wire is 2πrn. Okay? So that's another equation that you need to know, and that's what we're looking for here. Notice that I have r, so that's good, and 2 and pi are constant, so that's good, but I don't have n. So before I can solve for this, I'm going to have to calculate n. And to find n, there's really only one other equation that I can use which is the magnetic field equation. I'm given the magnetic field so we might want to write the magnetic field equation. B for the solenoid is, remember, μ0In over L, don’t get it twisted, n over L or μ0I little n because little n is big N over l. This is number of, this is turns per meter. Okay, just as a reminder. Cool, so I can find n using this equation, very straightforward. So let's move some stuff out of the way. n=BLμ0I, B=0.03,L=0.5,μ0=4π×10-7,I=12 amps. Okay? And if you do this, you get that n is 995 which some people would round to 1,000 but let’s just say 995, turns or loops, right? So there are 995 little windings and now we can plug this n into here so that's the total wire pi, the radius is 0.04 and then n is 995 and this gives you, this rounds to basically 250 meters of wire, okay? So that's it for this one, hope it made sense, let's get going.

A long wire having total resistance of 10 Ω is made into a solenoid with 20 turns of wire per centimeter. The wire is connected to a battery, which provides a current in order to produce a 0.04 T magnetic field through the center of the solenoid. What voltage must this battery have?

• A. 6.3×10⁻³ V
• B. 160 V
• C. 16,000 V
• D. 6.4×10⁸ V

Find Magnetic Field By Two Concentric Loops

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 current of 5 amps, so I'm gonna put here 5 amps. So I₁ is 5 amps and radius₁ 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 I₂ is 7 amps and r₂ is the diameter which is 6, it's actually going to be the radius which is going to 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 of wire with current going through it, it's going to 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 going to be a combination of both. So let's find those two numbers, b₁ and b₂. 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 π 10 - 7 2 and the current is a 5 and the radius here is a 2. Okay? And if you plug this into your calculator, you're gonna get that this 1.57 10 - 6 Okay? And you can do this with b₂, it's very similar, just the numbers are a little different. Instead of a 5 up here, you're gonna have a 7 and instead of a 2 over here, you're going to have a 3, okay? And if you do this, you get 1.67 10 - 6 , okay. Now let's talk about direction. To find direction, we're going to 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 want to confirm, if you want to 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 going to say hey, this guy is the bigger one so it's the winner, this one wins, right? It's kind of 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 × 10 - 6 minus 1.57 × 10 - 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 × 10 - 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 -7 . By the way, you could also have answered just 10 -7 but that's that. So this is 1 times 10 -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.

The two tightly wound solenoids below both have length 40 cm and current 5 A in the directions shown. The left solenoid has radius 20 cm and 50 m of total wire. The right solenoid has radius 0.5 m and 314 m of total wire. The thinner solenoid is placed entirely inside the wider one so their central axes perfectly overlap. Assume wires don't touch. What is the magnitude and direction of the magnetic field that is produced by a combination of the two solenoids at their central axis?

(Note:your worksheet may have a typo and say "0.5 cm"for the right solenoid's radius; it should be 0.5 m.)

• A. $2.20\times10^{-3}T$, left
• B. $9.42\times10^{-4}T$, right
• C. $9.42\times10^{-4}T$, left
• D. $2.20\times10^{-3}T$, right