Anderson Video - Force on a Wire in a Magnetic Field

Professor Anderson
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Uh, let's talk a little bit about wires carrying current. So let's say I have a wire here and it's going to carry some current. Okay? Let's say that current I is to the right. And, let's take this wire and let's put it in a magnetic field. Okay? And let's see if there is a force on it All right. How do we do this? Well, let's say this is our wire and let's put a magnetic field everywhere pointing up. Is there a force on that wire? Well, let's go back to the magnetic force for a second. Magnetic force is the following: F equals qVB sine theta. And then, we have to worry about the direction from the right-hand rule. So, we certainly have a B. We probably have some angle, theta. What about this right here, q and V? Do I have a charge moving? Well, we have a current, right? Current is charge moving. We know what current is: it's delta q, in some amount of time delta t. So if I think about a wire carrying current, I can consider that is consisting of charges moving at speed v. Charge q moves at v through the wire. That is current. Okay? So, F, we could really say, is the following: Let's just call this thing delta qVB sine theta n hat. The delta q just means some amount of charge. Okay? We're going to look at some chunk of charge. But, what if I divide by a delta t? I can't just divide by a delta t, arbitrarily, without changing the value. But if I divide by a delta t, I can multiply by a delta t and then I'll get the same thing. Okay? So, I haven't changed the relationship, it's the exact same relationship. But, now we can write delta q over delta t as I. What about V delta t? What is V delta t? Well if the charge is moving along at V and it does that for some amount of time, delt t, then, that corresponds to some length of wire . The length of the wire is, L and that is just v delta t, right? Speed times time gets me distance. So, this v delta t just becomes L. It's some length of this wire. We have an I. We have an L. And then, we have magnetic field B. And then, we have sine theta n hat. So, a wire carrying current, if it is sitting in a B field, it will feel a force. It will feel a force, given by that: ILB sine theta and then the n-hat -- again is determined by the right-hand rule. So, let's say we do this and let's see if we can figure out the direction. And let's simplify our wire a little bit. Let's say our wire is this. Okay? We don't have to draw it all 3D. It's just a single line. Now, let's again say that the B field is everywhere pointing up. Okay? What is the force? Well, we know the magnitude. It's ILB sine theta. The direction is going to be the right-hand rule. Theta, now, is the angle between I and B. Okay? I becomes like our qV. B is still the magnetic field, of course. So, in this problem, what is the angle between those two? Well, I is to the right. B is up. So, the angle between them is 90 degrees, which is just one. But now, we have got to figure out the direction. Remember, the direction is always going to be perpendicular to I and perpendicular to b. So, you only have two choices here. It's either out of the screen or it's into the screen. And, the way we do it is that we again use the right-hand rule. But, the fingers go in the direction of I. So, hold up your right hand. Put your fingers in the direction of I. B we said is everywhere going up. So, I cross B gets me something that is coming out of the page. Right? My thumb is coming out of the screen towards you. And so, there is a force here which looks like that. I cross B is coming out of the screen. So if you want to write this direction, you could say out of the screen or out of your computer monitor. Okay? If you increase the length of the wire, you increase the total force. If you increase the current, you increase the total force. If you increase B, you increase the total force. So, let's ask you this question. Let's say I have current I going to the left. And now, I have B going up. What direction is the force on that wire? Into the screen or out of the screen? Okay? Into the screen. Let's see that: put my fingers in the direction of I, rotate my fingers bent into the direction of B. My thumb is pointing into the screen -- into your computer monitor. And so, the force is in fact into the screen. All right. What about this? Let's say I draw current going up, B field going up. What's the direction of the force on that? Or, what is the force on that? Remember, I -- fingers straight in the direction. Your fingers straight in the direction of I. Now, you curl your fingers into the direction of B. But wait a minute. I can't curl them in any direction, right? That doesn't work. That doesn't work because B is in the same direction. So, the force is zero. And that's because theta is, of course, zero. You get sine of zero which is zero. How about this? Let's say I have I going down but B is going up. What's the force there? Well, I put my fingers in the direction of I and now I curl it in the direction of B. But, I got to come back all the way around. Should I go that way, or should I go this way? I don't know. So, again, it's zero. That's because this is 180 degrees. Sine of 180 degrees is equal to zero. Okay? So, just like we had with qV. Anytime the current is parallel or anti-parallel to the magnetic field, there's no force. All right. But, this looks like a section of wire. This looks like a section of wire. These two are sections of wire. Can we just put them all together into one loop?
Uh, let's talk a little bit about wires carrying current. So let's say I have a wire here and it's going to carry some current. Okay? Let's say that current I is to the right. And, let's take this wire and let's put it in a magnetic field. Okay? And let's see if there is a force on it All right. How do we do this? Well, let's say this is our wire and let's put a magnetic field everywhere pointing up. Is there a force on that wire? Well, let's go back to the magnetic force for a second. Magnetic force is the following: F equals qVB sine theta. And then, we have to worry about the direction from the right-hand rule. So, we certainly have a B. We probably have some angle, theta. What about this right here, q and V? Do I have a charge moving? Well, we have a current, right? Current is charge moving. We know what current is: it's delta q, in some amount of time delta t. So if I think about a wire carrying current, I can consider that is consisting of charges moving at speed v. Charge q moves at v through the wire. That is current. Okay? So, F, we could really say, is the following: Let's just call this thing delta qVB sine theta n hat. The delta q just means some amount of charge. Okay? We're going to look at some chunk of charge. But, what if I divide by a delta t? I can't just divide by a delta t, arbitrarily, without changing the value. But if I divide by a delta t, I can multiply by a delta t and then I'll get the same thing. Okay? So, I haven't changed the relationship, it's the exact same relationship. But, now we can write delta q over delta t as I. What about V delta t? What is V delta t? Well if the charge is moving along at V and it does that for some amount of time, delt t, then, that corresponds to some length of wire . The length of the wire is, L and that is just v delta t, right? Speed times time gets me distance. So, this v delta t just becomes L. It's some length of this wire. We have an I. We have an L. And then, we have magnetic field B. And then, we have sine theta n hat. So, a wire carrying current, if it is sitting in a B field, it will feel a force. It will feel a force, given by that: ILB sine theta and then the n-hat -- again is determined by the right-hand rule. So, let's say we do this and let's see if we can figure out the direction. And let's simplify our wire a little bit. Let's say our wire is this. Okay? We don't have to draw it all 3D. It's just a single line. Now, let's again say that the B field is everywhere pointing up. Okay? What is the force? Well, we know the magnitude. It's ILB sine theta. The direction is going to be the right-hand rule. Theta, now, is the angle between I and B. Okay? I becomes like our qV. B is still the magnetic field, of course. So, in this problem, what is the angle between those two? Well, I is to the right. B is up. So, the angle between them is 90 degrees, which is just one. But now, we have got to figure out the direction. Remember, the direction is always going to be perpendicular to I and perpendicular to b. So, you only have two choices here. It's either out of the screen or it's into the screen. And, the way we do it is that we again use the right-hand rule. But, the fingers go in the direction of I. So, hold up your right hand. Put your fingers in the direction of I. B we said is everywhere going up. So, I cross B gets me something that is coming out of the page. Right? My thumb is coming out of the screen towards you. And so, there is a force here which looks like that. I cross B is coming out of the screen. So if you want to write this direction, you could say out of the screen or out of your computer monitor. Okay? If you increase the length of the wire, you increase the total force. If you increase the current, you increase the total force. If you increase B, you increase the total force. So, let's ask you this question. Let's say I have current I going to the left. And now, I have B going up. What direction is the force on that wire? Into the screen or out of the screen? Okay? Into the screen. Let's see that: put my fingers in the direction of I, rotate my fingers bent into the direction of B. My thumb is pointing into the screen -- into your computer monitor. And so, the force is in fact into the screen. All right. What about this? Let's say I draw current going up, B field going up. What's the direction of the force on that? Or, what is the force on that? Remember, I -- fingers straight in the direction. Your fingers straight in the direction of I. Now, you curl your fingers into the direction of B. But wait a minute. I can't curl them in any direction, right? That doesn't work. That doesn't work because B is in the same direction. So, the force is zero. And that's because theta is, of course, zero. You get sine of zero which is zero. How about this? Let's say I have I going down but B is going up. What's the force there? Well, I put my fingers in the direction of I and now I curl it in the direction of B. But, I got to come back all the way around. Should I go that way, or should I go this way? I don't know. So, again, it's zero. That's because this is 180 degrees. Sine of 180 degrees is equal to zero. Okay? So, just like we had with qV. Anytime the current is parallel or anti-parallel to the magnetic field, there's no force. All right. But, this looks like a section of wire. This looks like a section of wire. These two are sections of wire. Can we just put them all together into one loop?