**Concept:** Magnetic Field Produced by Loops and Solenoids

Hey guys in the next video we're gonna talk about the magnetic field produced by loops in solenoids. Let's check it out.

Hi, so remember if you have a wire that is carrying current, it's going to produce a magnetic field away from itself. So it looks kinda like this, you have a wire with a current I, its gonna produce a magnetic field with strength B some distance away from itself, distance r. And you can find its magnitude by using the equation _not I / 2 Pi r where r is a distance from the wire. Okay. And this is important in fact remember this, but this only works for straight wires. In this video we're gonna deal with not straight wires but loops. So 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 gonna get a slightly different equation 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. So we use our thumb to indicate current. That what we've always done. And when you grab the wire you're fingers wore the the direction of the magnetic field around it. So because your magnetic field was curving you would curl or curve your fingers, okay. 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. Like its easier to see this, 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 round so its gonna be easier to use your fingers curved at your current. So you're gonna use your fingers as your current and guess what, what's gonna happen is that your magnetic field is gonna be straight so you're gonna use your thumb. So those two things flip. And you can think that in loops the right hand loop is backwards or you can also think that there's sort of this greater law this greater rule that you always want to curl your fingers. So whatever is curling in the problem is going to be represented by finger and the other thing is gonna be represented by your thumb. Okay. So let's look at single or multiple loops and the first thing I want to talk about is direction. So let's say that the current here is going in this direction, that's current which by the way is counterclockwise. So we're going to do, remember if its curling you're gonna use your fingers. And I'm going to curl my fingers in the direction of current, so current is going this way so I'm gonna go like this, okay. And if you do this and you should do this yourself, what you're gonna see is your thumb pointing at yourself which means its coming out of the page and that's the direction of B. So in the middle of this thing here in the center where every wire is inside actually you gonna have a magnetic field that is jumping out of the page towards you and you gonna 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 current you get current in the clockwise direction, its 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 my thumb is pointing away from me which means its going into the page which means on the outside of the wire you gonna have an out of the page magnetic fields everywhere. Cool. Now let's talk about the equation, the magnitude of the magnetic field of the loop that's produced by loop is gonna be _not I 2r and I almost forgot there's an N here 2r. So let's talk about a few things. First of all notice that there's no Pi in here, that is not a typo, not a typo, actually I want you to write not a typo because I don't want you to get confused that there is no Pi there, what happened, there is no Pi, okay. The other thing is in you have r big R which s 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's gonna be is the radius of that circle which is given to you or they will ask for it. Okay. And then third, what is this N here, N is going to be the number of loops. So you can have a single loop which could look like this, you got a straight wire and then you middle loop out of it or you can have multiple loops, let's say you do this, right. So in this case you have N=1 this N over here then in this case you have N=3. And the idea is that if you have one loop you produce a certain amount of magnetic field, let's say 10. But if you make three loops you gonna produce a magnetic field of straight 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 loop like this and its because you're trying to produce 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 end that goes in there. This equation is super important. There is one last point that I want to make here is that this equation is for the magnetic field B at the center of the loop. So its not at any point its always at the dead center so this has to be perfect circle, cool. Now, remember how it could make loops that were like this, you can have multiple loops. While if you make a loop that's really really really long, you actually gonna get a different equation. That's what we have here and I briefly mention this over here, okay. So if you have a very long loop, what's gonna happen is you gonna have a different equation. The equation now is going to be _not I N / L. Everything is the same, this L means this length over here, okay. So its 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, little n is big N over L, so you can see this as _not I and little n will replace those two 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 its kinda almost like a density, it has to do how tight this 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 N1 and 2, this N here is greater. Okay, cool. So that's the equation, what about the direction of the magnetic field? Well, remember when we talked about just now, which is if something curves we're gonna 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 gonna go in, and then its gonna go, in this picture its 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 gonna use our fingers for the current as well. Okay. So its just like this one, same thing. So but what you have to be careful, you have to be careful and look at this first loop right here. As soon as the current starts curving, is it first going backwards or is it going forward this way? Okay. So if the current starts here, notice that the cable goes into the page, so you have to get your thumb your fingers and curl into the page away from you and when you do this , looks like this, right. When you do this your thumb is to the left. So if the currents here, let's call this I1 is going to create a magnetic field that is going this way, B1, okay. But you could get a current in a different direction, what if the current is going here, let's call this I2, Notice, its kinda tricky to look, to know this, but if you look at the wire, it all really comes down to this little piece of the image guys. This is a really messed up, should be careful. The wire is going initially into the page and then it keeps doing this, right. So you gonna get your, you gonna, this is your current which is curling, you gonna get your fingers and you gonna go into the page and back and when you do that look what happens with your thumb. Your thumb is pointing right, which means you get a magnetic field that goes like this, B2, very very tricky, depends on the direction and depends on what this little image looks like. Is it coming at you then in or is it coming down and then back, right. So be very very careful with that. There's one other thing you should know, some questions will ask about the the total length of wire, total length of wire. What the heck is that? Well length L is just the side by side length. The total length of wire means what is the length of this entire thing here, right. And we're going to use, we're gonna say, we'll the length of a single loop, one loop has a length of circumference, the amount of wire we can make one loop is a circumference which is 2 Pi r, where r is the radius of this thing. But if you have any loops, the total length of wire will be 2 Pi r N. Okay. So I want you to remember this that the total amount of wire is 2 Pi r N. This does not have a variable, so I just write total wire equals 2 Pi r N, this is one of the other things you might get asked sometimes. Let's do an example, oops I got one last point to make here and this is actually really interesting point. Solenoid will produce magnetic fields that are very similar to magnets, magnets. Let me show you real quick, if you have a, let's say we have I1 over here, we have something like this, right, and this I1 and there is a the magnetic field B this way. Well guess what the magnetic field is gonna keep going and its gonna do this, right. And if you have a magnet, the magnetic field is always going from north to south on the outside. This means this is the north and this is the south. So it behaves very similar to a magnet. Anyway let's do this example here and in doing this example we'll kind of address this situation, where we have two fields at the same location. We already talked about this in the previous day. So a wire is twisted into 5 type loops 4 meters in radius. So you have a wire and you made one, two, three, four, five obviously I don't really have to draw it. But N equals 5 and the radius of these loops is 4 meters. A 3A current is run through the wire in the counterclockwise direction. Let's draw a frontal view of the wire. So the current is 3 in a counterclockwise, so imagine there's 5 wires here or 5 loops of wire here and its in the counterclockwise direction like that. I want to know the magnitude and direction of the field produced by the loop in its center. So right here the magnetic field is _not I / 2 big R, _not is 4 Pi times 10 to the negative 7. The current is 3 and then 2 times the radius, the radius is 4. So this 4 cancels with this 4. There's no Pi's to cancel unfortunately. So we gonna multiply this whole mess. And we oops we forgot the N, we forget the N over here, there a 5 over here, almost forgot that. And this is gonna be 7.5 times 10 to the negative 7 Tesla. Tiny amount of magnetic field. And its also asking for the direction. The direction is easy, I'm gonna grab the wire with my fingers crawling to the direction of I. And then my thumb pointed myself which means that this is the direction of the magnetic field in the center is out of the page. Cool. So that's how loops and so much work. Let's go to some more problems.

**Example:** Find How Many Loops in a Solenoid

Hey guys so let's check out this solenoid example. So 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=0.4T when a 3A current is run through it. So when a current I = 3A is run through it is there an equation that relates all these variables, of course there is and thatÕs the equation for the magnetic field through the center of the solenoid, B = _not I N over L weÕre looking for N so I can just move some stuff around, N=BL divided by _not I and B is 0.4 length is 2 _not is 4pi times 10 to the negative 7 and the current is 3A and if you multiply all of these, I have it here, youÕre gonna get, I havenÕt rounded, youÕre gonna get 2 times 10 to the 5th terms thatÕs the value for N which means by the way that youÕre gonna have 200,000 turns, thatÕs what you need to have to make this happen. Cool. LetÕs keep this one, letÕs keep going.

**Problem:** 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)?

**Example:** Designing a Solenoid (Total Length of Wire)

Hey guys let's check out this example. So here you are tasked with designing a solenoid that produces a magnetic field of this strength here, so B equals 0.03 Tesla at its center with a radius of 4 centimeters, so the radius is 0.04 meters and a length of 30 of 50 centimeters or 0.5 meters. And I want to know what is the minimum total length of 12 A wire, 12 A wire means that this wire is capable, can withstand currents of 12 A or more, if you try a higher current its just probably gonna burn the wire or its risky. But that means that we're gonna use a I with a current of 12 A. And I want to know the minimum total length you should buy to construct the solenoid. So now if you're going to the wire store and you gonna buy some wire, how much total length. Now remember total length is different from length, right. Length is just if you make a solenoid looks something like this. This is length here, sort of a side to side length. But the total length of wire is all these circumferences here, right, its 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 gonna get if you did that. Okay. L is this which is given to us, but we want to know total length. So I'm just gonna write total wire equals question mark. And you may remember the equation for this, 1 circumference is 2 Pi R, right 2 Pi R where R is the radius and we're given that. But if you have N loops then the total wire is 2 Pi R times N. Okay. So that's another equation that you need to know and that's what we're looking for here. Notice that that I have R, so that's good and 2 and Pi are constants so that's good but I don't have N. So before I can solve for this I'm gonna have to calculate N and to find N there's really only one of the equation that I can use which is the magnetic field equation. I'm given the magnetic field so we might wanna write the magnetic field equation. B for solenoid is remember _not I L over N, oops it's actually N over L, don't get it twisted, N over L or _not I little n because little n is big N over L. Okay. This is number of, this is turns per meter. Okay. This is a reminder. Cool. So I can find N using this equation very straight forward. So let's move some stuff out of the way, N is gonna be BL divided by _not I, B is 0.03, L is 0.5, _not is 4 Pi times 10 to the negative 7 and I is 12 A. Okay. And if you do this you get that N is 995 which some people will round that to a thousand. But let's just say 995 turns or loops, right. So there' 995 little winding. And now we can plug this N into here so that the total wire is 2 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. Hopefully made sense. Let's keep going.

**Problem:** 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?

**Example:** Find Magnetic Field By Two Concentric Loops

Hey guys, let's check out this example. So, here we have two wire loops that are concentrically arranged, meaning, concentric means one circle inside of the other, right? With a common middle. So, itÕs shown below and the interlayer has diameter 4. Now, real quick, in physics, remember, we almost never use diameter, we almost always use radius, I'm going to right away change this, instead of writing diameter 1, IÕm going to write radius 1, and radius is half of the diameter, so that's 2 meters, and a clockwise current of 5. So, that's the inner one here, which is blue, and itÕs got a clockwise current of 5 amps. So, I'm going to put here 5 amps. So, I1 is 5 amps and radius 1 is 2 meters, and then the red one is counter-clockwise, which is this way and it's got a current of 7 amps in thate direction, so I can write that I2 is 7 amps and R2 is the diameter, which is 6, it's actually going to bein 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 current, so 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 two 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 itÕs going to be a contribution of both, itÕs going to be a combination of both. So, let's find those two numbers, B1 and B2, and the equation is Mu naughtknot I divided by 2, big R, big R is the radius and we have all of these numbers, its I1 since its B1, and it's R1 since it's B1, right? So, one go with onceÕs, so this is 4 PI times 10 to the negative 7 and the current is a 5 and the radius here is a 2, okay? And, if you plug this into your calculator, you're going to get that this is 15.7, or actually I should say, 1.57 times 10 to the negative 6, okay? And you can do this with B2, it's very similar, just the numbers are a little bit different. So, instead of a 5 up here, you're going to 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 times 10 to the negative 6, okay? Now, let's talk about direction, to find direction you are going to use the right hand rule. So, first, let's look at the blue inner circle, the blue inner circle is not going in this direction, but it's actually going in this direction, right? It's going clockwise like this. If you do this, you're 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 ones in opposite directions, it must go in the opposite direction, so this is going to be out of the page and if you want to confirm, you can just use to get your hand and grab the outer wire goes this way, right? This way, and look my thumb is now pointing in 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 youthey're going to say: Hey, this guy is the bigger one, so it's the winner, this one wins, right? It's kind of a tug-of-war, ones pulling this way, the other ones pulling the other way, this one wins, so the net magnetic field is going to be winner minus looser. 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, 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 0.1, and you multiply here, we have to divide here, so it' fair. So, we're not actually changing the number, and this divided by 10 is 10 to the negative 7, by the way you can also have answer just 10 to negative 7, but that's that, so this is 1 times 10 to the negative 7 tesla, and in what direction? ItÕs going out of the page, because that was the winning direction of the two, okay? So, that's x, that's one way you couldto do it. Another way you could have done this, you could just assigned signs, then you could have said: Hey, into the page is like this, right? Away from you with my thumb and my fingers are currently 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 the same results anyway, cool? So, you can think of winner, the big one, minus the loser, the smallest one, and then the winner dictates the direction, the net direction, or you can just assign positives and negatives and then do the math, cool? ThatÕs it for this one, letÕs keep going.

**Problem:** 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 cm 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?

A wire in which there is a current of 3.09 A is to be formed into a circular loop of one turn. If the required value of the magnetic field at the center of the loop is 9.6 μT, what is the required radius? The permeability of free space is 1.25664 × 10^{−6} T · m/A.

1. 30.0092

2. 20.224

3. 29.2932

4. 50.2501

5. 32.3643

6. 42.8843

7. 55.4991

8. 19.6611

9. 16.7263

10. 33.25

Watch Solution

A large MRI magnet is a large solenoid (with superconducting wires having zero resistance) with diameter 0.60 m and length 1.8 m. During normal operation, the current through the windings is 110 A and the magnetic field magnitude is 1.3 T. How many total windings does the MRI magnet have?

(1) 16,900

(2) 9,400

(3) 13,000

(4) 15,500

(5) 14,700

Watch Solution

A circular current loop lies in the x-z plane. What is the direction of the magnetic field outside of the loop in the x-z plane?

A. the positive x-direction.

B. the negative x-direction.

C. the positive y-direction.

D. the negative y-direction.

E. depends on which part of the loop you are looking at.

Watch Solution

A long, straight wire carries a current of I _{1} = 8.0 A. A circular loop of wire lies immediately to the right of the straight wire, with the plane of the loop parallel to the wire. The loop has a radius of R = 0.03 m and carries a current of I_{2} = 2.0 A. Assuming that the thickness of the wires is negligible, find the magnitude and direction of the net magnetic field at the center of the loop.

Watch Solution

Two round concentric metal wires lie on a tabletop, one inside the other. The inner wire has a diameter of 20.0 cm and carries a clockwise current of 12.0 A, as viewed from above, and the outer wire has a diameter of 30.0 cm. What must be the magnitude and direction (as viewed from above) of the current in the outer wire so that the net magnetic field due to this combination of wires is zero at the comment center of the wires?

Watch Solution

A circular coil with 50 turns and a radius of 2 cm lies at the center of a larger coil with 100 turns and a radius of 5 cm. While the center of each loop coincides, the smaller loop's surface is rotated 30° away from the surface of the larger loop, as shown in the figure (from a top-down perspective). If the larger loop carries a current of 10 A, and the smaller loop carries a current of 15 A in the same direction, what is the magnitude of the magnetic field at the center of the loops?

Watch Solution

Point P is at the center of a circular loop of wire. Current *I *travels around the loop in a clockwise direction, as shown in the sketch. The magnetic field at point *P* is

A) zero

B) into the page

C) out of the page

D) to the left

E) to the right

F) toward the top of the page

G) toward the bottom of the page

Watch Solution

A long straight wire carries current *I*_{1} = 16.0 A and a circular loop of wire with radius *r* = 0.200 m carries current *I*_{2} = 22.0 A. The two currents are in the directions shown in the sketch. The edge of the loop is 0.0500 m from the wire. What are the magnitude and direction of the resultant magnetic field at point P, which is at the center of the loop?

Watch Solution

A wire carrying a current is shaped in the form of a circular loop of radius 4.0 mm. If the magnetic field strength at its center is 1.1 mT with no external magnetic fields contributing to it, what is the magnitude of the current that flows through the wire?

A) 14 A

B) 22 A

C) 17 A

D) 7.0 A

Watch Solution

A solenoid 3.0 cm long consists of 6658 loops of wire. If the magnetic field inside the solenoid is 2.0 T, what is the magnitude of the current that flows through it?

A) 0.14 A

B) 90 A

C) 3.0 A

D) 7.2 A

Watch Solution

Consider a solenoid that is very long compared to the radius. Of the following choices, the most effective way to increase the magnetic field in the interior of the solenoid is to do which of the following?

A) overwrap the entire solenoid with an additional layer of current-carrying wire

B) double its length, keeping the number of turns per unit length constant

C) reduce its radius by half, keeping the number of turns per unit length constant

D) replace the wire by a superconducting material

Watch Solution

A solenoid of 200 turns carrying a current of 2 A has a length of 25 cm. What is the magnitude of the magnetic field at the center of the solenoid?

A) 5 mT

B) 3 mT

C) 2 mT

D) 4 mT

Watch Solution