Let’s say we have a magnetic field that’s coming out of the right side of the screen. And it’s not just along the screen, it’s actually three dimensional. So it’s going above the screen below but the direction of the magnetic field is from the right to the left. Let me just draw that. And I’m not going to draw a bunch of the field vector arrows because that’ll just take up a lot of valuable space on our, so that’s the vector magnetic field. It’s down here too. If I could I would draw it above your screen and below your screen but it’s coming from the right to the left. And then in that magnetic field I have an electric circuit. And I’ll draw the whole circuit right now, do that in a second. But let’s say, I have part of the electric circuit is a loop. And the loop looks like this, and I’m trying to draw it carefully. I think a careful drawing will be more useful in a second than an uncarefully drawn drawing. Let’s see, so it’s a loop. You can imagine taking a paper clip and putting it into this shape. Oh that’s good enough, I think. And I have a current going in this direction in this paper clip.
So, this is the positive, that’s the negative. So the current is going in the loop like that. Current is going in a loop like that. The current’s coming out of this end, it’s coming into this end. And let’s say that this loop of, it could be a paper clip or anything, let’s say that it can rotate and that’s important.
What’s going to happen? Well my magnetic field is coming in this direction, the current is going down here, up here. What’s going to be the net force of the magnetic field on this loop? Well, let’s try it out. And it turns out its going to be a different magnitude at different points of the current. All we’re worried about right now is direction and then maybe a little intuition of the magnitude.
So, we know that the force of the magnetic field is equal to the current times co the length vector cross the magnetic field, right? Well what would be the force of the magnetic field on this segment of wire? We could call this L, right? And that L goes in the same direction as the current. Let’s see, current is just a scale L is going down, magnetic field’s going to the left, cross product. Cross product I take my right hand, put my index finger in the direction of the current or in the direction of L because that’s the first term in the cross product, so that’s the index finger. So my index finger goes down, because that’s the direction of the current. And then my middle finger, and remember you have to do this with the right hand. If you do it with the left hand you’re going to get the opposite result. And now my middle finger is going to go on the direction of the field. So let me point my middle finger. My middle finger is going to go in the direction of the field, I keep having to look at my own hand. And then my other two fingers are just going to do what they need to do. So that’s my third finger, that’s my pinky. And then, what is my thumb going to do? Well my hand, that’s my hand. This is what my hand is doing, I’m pointing downward, right? And my palm is kind of pointing at my body. So what is my thumb doing? I know it’s hard to see. This is my middle finger right here, so my thumb is on the other side of this drawing, and my thumb is pointing downwards, I hope you see that. And you try it with your own hand. So my thumb is pointing downwards to the direction of the force, created by the magnetic field on this current is going to go downwards so let me draw that.
So the force vector, I’ll do it in this orangey-brown color. The force vector on this segment of the wire, is going to be going down. Now what about this segment of the wire? Well think about it. This segment of the wire, the L vector, it’s parallel to the magnetic field just in the opposite direction. And so when you take the cross product, remember the cross product is you’re multiplying the magnitudes of the vectors that are perpendicular to each other. But if this is the L vector right here, there’s no component of it that it’s perpendicular to the magnetic field. So the magnetic field and the current are in the same plane, they’re parallel. They’re not orthogonal at all. There’s no components of them that are at ninety degrees. So you take the cross product, you’re going to see that the net force on this segment of the wire is zero and likewise on this segment of the wire and this segment of the wire. Because they aren’t in any way perpendicular. No components of them are even perpendicular so fair enough. So all we know right now is the magnetic field is exerting in that downward force on this side of our paper clip or of our circuit.
Now what about this side? Well same thing. Take the cross product. If this is our L, L cross V. So take your index finger in the direction, so index finger goes like that. Your middle finger will go on the direction of the field. So your middle finger is going to look something like that. And then the other two fingers are going to be like that. And what is your thumb going to do? And this has to be the right hand to work. Your thumb is going to point straight up. This is like the heel of your thumb, and your thumb is going to, I don’t know if that is a good drawing of a thumb. But your thumb is essentially pointing out of the page, right, index finger in the direction of the current, middle finger in the direction of the field, thumb points out of the page. And do that with your own right hand and you’ll see that the net force of the magnetic field on this segment of wire is going to be upwards. Let me do that in a different color just to get some contrast going to be upwards.
So what’s going to happen? Assuming that this circuit can rotate, what’s going to happen? On this side, there’s a downward force, on this side there’s an upward force. So that magnetic field is actually exerting a torque on this wire right? Like if you viewed this little dotted line as our axis of rotation, the whole coil is actually going to rotate around that line. And so, there’s some force over here, along this whole line being applied downwards. And it’s actually perpendicular to our moment arm, if you remember what we had learned about torque. So it will actually exert all of that force, you know that force times this distance will be the torque applied on this side. And then likewise there’s a torque in the same direction, because here on the other side of the arm, is pushing upwards. So they’re not going to cancel out. They’re both going to reinforce and this whole coil is going to be turning in this direction. Right here is going to be moving up on your video screen, here is going to be moving down into your video screen.
Now what happens? So it’s going to start rotating. So the left hand side is going to go below the page. The right hand side is going to be above the page. I want to draw some perspective and so I’m just drawing it bigger. So maybe it looks like that. Maybe my circuit starts to look like that. And I’ll redraw my axis of rotation. So this is my axis of rotation. And either way I drew it, this part of the axis of rotation is still in the plane of our video. But this part of the coil is, you could imagine it popping out, I wish you’re three D glass is on, it’s popping out of your screen and this part is going into your screen. And the current is still going in the same direction. And the current is going that direction there, it’s going that direction there.
So using the same right hand tool, on this side of the wire, the magnetic field is going to be exerting a net downward force but the torque is actually going to be less, right. Because our moment arm distance is going to look like that, right. So it’s going to be going to the left and behind the page while the torque is still just into the page. So you would actually take the component of the torque as perpendicular. So there’s some component of the torque that’s actually perpendicular. But you could imagine the torque lessens even though the net magnetic force is the same, the component of that force is perpendicular to your moment arm. That lessens. So there’s still going to be some torque that’s going to be causing it to rotate downwards in that direction. We’re pushing up on the right hand side, we’re push down on the left hand side so the direction is going to be like that, pushing up on the right hand, down on the left hand. So you’re still going to be doing the same thing here. You’re going to be pushing up here but you’re going to be pushing up directly out of the page. But that’s not completely perpendicular to the moment arm. So the component that is perpendicular that’s actually creating rotational torque, that’s going to be a little less. And then you could imagine that all the way to the point, the coil is going to keep rotating at a smaller torque and at some point you’ll be looking at it head on.
So I could just draw it like a straight line, right. You could imagine this arm is on top and this arm is behind it. And at this point, what’s going to happen? All of the magnetic force of the top arm is going to be popping up out of your page. It’s not going to be providing any torque. Because it’s not perpendicular anymore to your moment arm. And likewise, on the bottom behind this, if you could visualize this, it would be exerting a net downward force. And that’s also not going to be helpful but maybe they have some angular momentum so that the wire will still rotate. But it won’t still rotate and what’s going to happen? And this is where I’ll leave you with a little bit of a conundrum.
Actually, I don’t want to go over the YouTube limit. So I’ll going to continue this in the next video and I’ll show you the conundrum.
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