Ch 26: Magnetic Fields and ForcesWorksheetSee all chapters
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Ch 01: Units & Vectors
Ch 02: 1D Motion (Kinematics)
Ch 03: 2D Motion (Projectile Motion)
Ch 04: Intro to Forces (Dynamics)
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Ch 26: Magnetic Fields and Forces
Ch 27: Sources of Magnetic Field
Ch 28: Induction and Inductance
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Ch 30: Electromagnetic Waves
Ch 31: Geometric Optics
Ch 32: Wave Optics
Ch 34: Special Relativity
Ch 35: Particle-Wave Duality
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Ch 38: Quantum Mechanics

Concept #1: Magnetic Force on Current-Carrying Wire

Transcript

Hey guys so in this video we're going to talk about forces acting on current carrying wires meaning if you have a wire and charge is moving through it, it has a current and if that wire is inside of a magnetic field it would feel a force. Let's check it out. So remember charges can move in space and we've done this if you have a little charge moving through space and it walks into a magnetic field it will feel magnetic force. Well chargers can also move inside of a wire. And if you have charges inside of a wire you have a current. You can think of a wire as simply a way to sort of restrict have a chance you can simply a way to restrict the path of charges so that they're not going into crazy directions they have to move along the direction of the wire but in a way a wire is simply charges moving just the same. So remember if charges are moving in space they will be producing a new magnetic field and we did this, the equation is mew not q v sin of theta over 4 pi r square right, just like how a charge moving freely through space will produce a new magnetic field away from it self, a charge moving in a wire, in other words if you have a current carrying wire, that is also going to produce a new magnetic field. Okay, and the equation for that is mew not I divided by 2 pi r. So it looks somewhat similar to this but it's not exactly the same. Were not gonna talk about this now we'll talk about this later but I wanna make the point that they produce a charge, they produce a field away, the wire will produce a field away from itself. Remember that the charges do not only produce a field away from themselves but they are also going to feel a force if they are in the presence of an existing magnetic field, and this is a really important theme of magnetism is that moving charges produce a field and if they are in an existing field they will also feel a force and just like how this happens for charges and space this is also going to happen for charges moving in a wire, okay. So if a charge is moving through space it will feel a force even by q v b sine of theta, right so for example if you have a magnetic field this way in a charge moving this way you can use your right hand rule and figure out that the force will be will be into the field, okay. Similarly if a charge is moving not through free space but it's moving inside of a wire it is a current and that wire will feel a force as well so charge moving in an existing field feels the force and a wire in an existing field also feels the force because it's the same thing. Here it has to be moving and here it has to have a current so it's a current carrying wire, right. Just like here, current carrying wire. And that force is going to be given by B I L sine of theta, and this is the big equation that we're going to talk about. I wanted to do a big summary of all the different situations but this is what we're gonna talk about for the next few videos. Now most textbooks and professors write this equation in a different order, I like to write it this way just because there's so much crap to remember in this chapter, this spells Bil which is easier to remember than some other combination, okay. So F equals Bil sine of theta, I has a direction which is the direction of the wire and B has a direction and theta is the angle between those two directions. The directions of the forces will be given always by the right hand rule. Now remember there's sort of this exeption that if you have a negative charge, If you have a negative charge that's moving freely than you're going to use the left hand rule, but in currents because of sine convention you're always going to use the right hand rule everytime. When you have a current, you don't have a negative current so you're always going to use the right hand rule no matter what. Alright, so the last thing we wanna talk about here is that this force will cause the wire to bend slightly, okay. So cause the wire to bend slightly. So let's say here we have a wire here and the current is up and here we have a wire between these two magnets and the current is moving down. Now you may remember that magnetic fields go from positive to negative on the outside so positive to negative looks like this. But more importantly we wanna know what is the field around here. Well it goes from positive, I'm sorry, from north to south so it's gonna look like this. So as far as the wire's concerned you have B field lines going this way or I guess this way. Okay, so what is the direction of the magnetic force this wire will experience. Well right hand rule, right, and guess what. Current is the direction of the motion so it's gonna be your thumb and your B is going to be your fingers. So here I have my B is going to go to the right and I want my current to go up so this is already in the direction that I want, right, this mathes the picture. So if I do this, notice that my palm is away from me, right, so please do this, it's away from you which means that it's into the page. So this force and the force from the wire will be into the page, everywhere the wire is being pushed into the page. Okay. We can't calculate this because it's with magnets, but we're just talking about the direction. Here you also have north to south so the magnetic field will look the same, and what do you think will be the direction of the force. Well hopefully you're thinking well for fliping the current you must flip the force and that's correct. So right hand rule again, this is going to the right but now my current's down so the only way that I can keep these going to the right with my current going down is if I do this., right, and if I do this and you sort of bring it close to your face, right, this is weird but hopefully you see that your palm is facing you which means it's coming out of the page, so F B is coming out of the page everywhere. So F B is out of page. So that's direction of the force. Let's do an exampe here. There are two parts. So it says here a 2 meter long wire that's the lenght of the wire L equals 2 is passed through a constant magnetic field as shown. So there's a little peace of wire here and you might be wondering how's a current on a little piece of wire that's floating. Don't worry about that for now, I'll talk about it later, just assume that's possible, which it is. Okay. So there's a wire there and somehow it's connected to a batery but don't worry about it. So the piece of the wire there is 2 meters, it sits through a constant magnetic field, this magnetic field is going into the page and you can tell by the little x's and we wanna know for part a if the wire experiences a force of 3 Newtons and has a current of 4, what is the strenght of the wire. What is the strengt of the field so what is B. So is there an equasion that ties these guys together so that you can find B and it's the equation we talked about earlier F equals Bil. Bil sine of theta. Okay. We're looking for B. B equals F B divided by I L sine of theta. Remember the angle, theta is the angle between the B and the I. I is either to the right or to the left which we don't know but it's either one of those and the direction of the field is into the page so the current is going this way or this way, right, this way or this way and the field is going into the page so they make 90 degrees of each other. It's either this which is 90 degrees right here or it's this, right, which is also 90 degrees. So this angle here is 90 degrees. And by the way generaly speaking if you see a weird angle it's probably 90, right. So let's plug these numbers in. F is 3, I is 4, L is 2 this is 3 over 8 and that is 0.375 tesla. That's part a. Part b has to do with direction. It says if the wire experiences a downward force, so if the wire experiences down force what must the direction of the current be. So here's the wire and it's experiencing a force F B that's down. Right hand rule, the force is down so I want my palm to be down and I want my 4 fingers for field to be going into the page which into the page is away from you, right. So my fingers like this, L on this force down. Notice what happens with my thumb, which is it's going left and that's it. That tells me that the current must be going left so the direction of the current is left. That's all there is to this. Hopefully it makes sense, let's keep going.

Example #1: Find Force on Current-Carrying Wire at an Angle

Transcript

Hey guys. So in this example we have a wire that sits on a magnetic field and it has a current and we wanna know what is the force on that wire in 3 different scenarios. So let's see. The wire has a length of 2 meters L equals 2, the magnetic field strength is 3 B equals 3 and it's directed in the negative y x the magnetic field. And we wanna know what is the magnitude of the magnetic force which in this case because it's in a wire it's going to be given by Bil sine of theta. B I L sine of theta. By the way we're also given that the current is 4. Okay. Current is 4 amps. That's a really ugly 4. 4 amps. Cool. So letÕs get to it. So the magnetic field I like to draw multiple lines, letÕs just do 2, is going down, right. Right there. And the wire is also or the current rather is flowing in the negative y xÕs which means that the wire goes down this way and the current is going down in this direction here. So the equation is F B I sine of theta and in these questions the tricky part is the angle because IÕm given B I and L so itÕs just plug and chug. Now the angle is going to be the angle between B and I. TheyÕre both going down so theyÕre parallel to each other so the angle is 0 and the sine of 0 is 0. The sine of 0, 0 which means there is no force here because theyÕre going parallel to each other. Remember in current we use the right hand rule always and this is a reminder, you can think of this as a reminder that youÕre supposed to have a 90 degree angle between your B and your I, right, 90 degree angle between your B and your I and here theyÕre like that and thatÕs not how itÕs supposed to be. In this situation you have maximum force, if you have an angle like this you have less than maximum force but you still got some force and as you keep going this way the force goes from max, max 90 degrees to smaller smaller smaller to get here and you get 0. Okay. So the angle is 0 because theyÕre parallel or if the angle is 180 which is anti parallel opposite directions there will be no force. Cool. So no force on this one here. What about here. So B again is down and the current is in the positive x xÕs which means the wire is horizontal and the current is going in this direction. The angle between these 2 guys is 90 degrees so F B is Bil sine of 90, sine of 90 is just 1 so really we only have Bil. So 3 I is 4 L is 2 so this is 24 Newtons. What about the direction of the force, what is the direction of the force. Well right hand rule cause itÕs a wire, B is going down, I is going to the right you got this, right you gotta do this okay away from me B is down so my palm is pointing at your face, gotta do this yourself. ItÕs not my palm itÕs your palm. Your palm is pointing away from you which means itÕs going toward your page so itÕs into the plane. Okay, were into the page. Into the page. Make sure you master your right hand rule. Into the page is direction and the magnitude is 24. Okay so this one is a little bit more complicated cause itÕs got an angle and here we have the B this way and we have a wire in the direction that makes 53 with the y xÕs so here is the positive y xÕs is up here. This makes 53. Now this is a little tricky because itÕs a big, itÕs not totally clear whether itÕs 53 with the y xÕs this way or 53 with the y xÕs this way but what you will see is that it actually doesnÕt matter when it comes to calculating this, okay. So weÕre gonna think of this as 2 possibilities the current could be going this way or it could be going that way so weÕre gonna calculate that. So the equation is F B equals Bil sine of theta and the angle is the angle between the direction of the current and the direction of B. So B is pointing down so if you want what you can do is draw B over here, okay. And what is the difference between these guys. So if you go counterclockwise here this is 90 and then this here is a 37 so 37 plus 90 is 37 plus 90 is 127 degrees. 127 degrees or you can go another or you can go in this direction here which will be negative 127 degrees or you can go all the way positive and you can say that itÕs 90 plus another 90 plus 53 which is 180 plus 53, 180 plus 53 which is 233. 233. Did I get that right. Yea 223 away from this and the sine of all these numbers will be the same, they may have different signs one might be a positive or negative but you were just looking for the magnitude of these things so you can pick or choose. IÕm gonna write that B equals 3 4 I is 4 L is 2 sine of 127 positive. Again whatever we get here just think of this as a maximum value because weÕre just looking for the magnitude of this force and if you do this you get that the answer is 19.2 Newtons. So thatÕs the force and it will work whether you go this way or this way. What about the direction. Right hand rule, B is down so letÕs do this and I want this guy to be going in this direction okay I want this guy to go in this direction so in this case here where i 1 right if you were going that way you would be going down and you would have the I like this, right, which means my palm is pointing towards me which means itÕs away from the page so for in the case of i 1 if it was going in that direction and by the way question your test would tell you exactly which, right, one I just want to talk about both cases here. So in this case you would be what di we say it was we said it was out of the page. What about for i 2 what if you were actually talking about this direction. Well if you try to do B down and this I here you canÕt put this thumb all the way over here, right, not without breaking it. DonÕt do that, youÕre just messing yourself up and youÕre breaking the right hand rule. So what can we do, well you have to do this, right, you have to do this and now your fingers are still down following B and this guy is up like that so it looks all kinds of weird but my palm is away from me which means itÕs going into the page. So in this case the direction of those 2 wires actually made a difference in terms of direction, okay. DoesnÕt make a difference in terms of the force but it does in the terms, not the magnitude but it does make a difference for the direction. Cool, thatÕs it for this one, letÕs keep going.

Practice: A 5-m current-carrying wire (red line) is ran through a 4 T magnetic field (blue lines), as shown. The angle shown is 30°. What must the magnitude and direction of the current in the wire be when it feels a 3 N force directed into the page?

Additional Problems
A 1m, straight wire carrying a 1.5 mA current in the presence of a perpendicular, 2mT magnetic field will feel a force due to that magnetic field. Because of this force, the wire should deform, as shown in the following figure. However, this deformation causes stretching in the wire (obviously the arc is a further distance than 1m), and if the wire is elastic, then there will be some elastic force that resists stretching. If we can treat the wire like we treat a spring, and say the elastic force is proportional to the vertical displacement, Fel = -kΔy, how much could the wire stretch if k = 300 N/m? Assume that the wire is still essentially 1m long after stretching.
A straight segment of wire is in a region of uniform magnetic field. If the current in the wire is toward the right and the direction of the magnetic field is into the page, as shown in the sketch, the direction of the force on the wire is A) into the page B) out of the page C) to the left D) to the right E) toward the top of the page F) toward the bottom of the page
A long wire carrying 4.50 A of current makes two 90° bends, as shown in the figure. The bent part of the wire passes through a uniform 0.500 T magnetic field directed as shown in the figure and confined to a limited region of space. What is the magnitude of the net force acting on the wire?
A straight wire segment carries current towards the left in a magnetic field that is directed into the page. What is the direction of the force that the magnetic field exerts on the wire? (a) to the left (b) to the right (c) toward the top of the page (d) toward the bottom of the page (e) into the page (f) out of the page
A wire that is lying in the plane of the page carries current  I in the direction toward the top of the page, as shown in the sketch. The wire is in a uniform magnetic field (produced by a large electromagnet) that is directed into the page. The direction of the force that the magnetic field exerts on the wire is A) toward the top of the page B) toward the bottom of the page C) to the right D) to the left E) out of the page F) into the page G) no direction since the force is zero
A 5 meter length of wire carrying a current of 5 A lies on a horizontal table with a rectangular top of dimensions 0.300 m × 0.400 m. The ends of the wire are attached to opposite ends of a diagonal of the rectangle. A vertical magnetic field of 0.30 T is present. What magnetic force acts on this segment of wire?  A) 1.1 N B) zero C) 7.5 N D) 0.75 N E) The force cannot be determined without knowing the shape of the length of wire.
A wire in the shape of an "M" lies in the plane of the paper. It carries a current of 2.0 A, flowing from A to E. It is placed in a uniform magnetic field of 0.65 T in the same plane, directed as shown in Figure 1. What is the magnitude and direction of the force acting on section BC of this wire?A) 0.090 N perpendicular into the pageB) 0.060 N perpendicular out of the pageC) 0 ND) 0.090 N perpendicular out of the pageE) none of the above