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Ch 30: Sources of Magnetic FieldWorksheetSee all chapters

# Magnetic Field Produced by Loops and Solenoids

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Sections
Magnetic Field Produced by Moving Charges
Magnetic Field Produced by Straight Currents
Magnetic Force Between Parallel Currents
Magnetic Force Between Two Moving Charges
Magnetic Field Produced by Loops and Solenoids
Toroidal Solenoids aka Toroids
Biot-Savart Law (Calculus)
Ampere's Law (Calculus)

Concept #1: Magnetic Field Produced by Loops and Solenoids

Example #1: Find How Many Loops in a Solenoid

Transcript

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.

Practice: 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 or out of the page?

Example #2: Designing a Solenoid (Total Length of Wire)

Transcript

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.

Practice: 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 #3: Find Magnetic Field By Two Concentric Loops

Transcript

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.

Practice: 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.)