Professor Anderson

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So let's say we do the following, let's say we have this. Let's draw an x y z coordinate system x y z And now let's do this. We're going to take a positive q and we're going to send it flying in the y direction. And now it's also in a B field that is also pointing in the y direction. What is the force? What is the force on our charge? Okay, we can do the right hand rule again. So you put your finger straight in the direction of v, and then you have to curl them into the direction of B. But B is in the same direction as v so I can't curl my fingers that way because they won't be pointing towards B. And I can't do them this way because it won't be pointing towards B. And so none of those directions makes sense. And if none of the directions make sense, what does the force have to be? What is it 90 percent of the time when I ask you what it has to be? Zero. The force has to be zero, right? If the physics says you can't determine the direction, then no direction is preferred, it has to be zero. Okay, but how do we see that mathematically? Well, let's go back to our definition. F equals q v B sin theta. Then we add a direction which we determine from the right hand rule. Okay, we've got a q we've got a v we have a b but then we have the sine of the angle. And remember the angle here is between v and B. What's the angle between v and B? What do you think the angle is between v and B? Zero! There we go. What's the sine of zero? The sine of zero is zero. There is no force. Okay? If v is in the direction of B there's zero force on the particle. Alright. So, v is to the right still, but we're going to let our B be up everywhere. And now we do the right hand rule for qv cross B. So v, everybody hold your right hand up, v, finger straight, curl into the direction of B, that's telling me that my thumb is the direction of the force which is coming towards you guys. So there is a force in the- sorry, the dot, right? Coming towards you, which would be in our 3D picture in the negative or, I'm sorry, in the positive x direction. Okay. So that force will be coming towards you, what is its strength? Well, it's qvB sine theta n hat, we just said that it's going to be in the x direction. We know that the x direction means i hat. Okay, i j k x y z, so n hat becomes i hat. What is the angle? What is the angle between v and B? It's 90 degrees. Right? They're at a right angle. v is in the y-axis, B was in the z-axis, And so you get 90 degrees, and sine of 90 degrees is of course, one. So what's the force? It is qvB i-hat. It's coming out of the page and it is a strength of qvB. So if you are in some other arbitrary direction, then you have to include that theta. So if this is x y z and let's say that our particle is going to the right, but now our B field is up here at some angle theta, now what is the direction of the force? Well, again, we use our right hand so we do a q v cross B and that tells me that it's coming out towards you, right? Just like we had here, it's still coming out towards you, and the strength is going to be with the sine theta. Okay, sine theta is of course smaller than one so it's going to be a little bit less than this force. So if they're at a right angle you get the maximum amount of force, if they're parallel you get zero, if they're somewhere in between you get sine of theta.

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So let's say we do the following, let's say we have this. Let's draw an x y z coordinate system x y z And now let's do this. We're going to take a positive q and we're going to send it flying in the y direction. And now it's also in a B field that is also pointing in the y direction. What is the force? What is the force on our charge? Okay, we can do the right hand rule again. So you put your finger straight in the direction of v, and then you have to curl them into the direction of B. But B is in the same direction as v so I can't curl my fingers that way because they won't be pointing towards B. And I can't do them this way because it won't be pointing towards B. And so none of those directions makes sense. And if none of the directions make sense, what does the force have to be? What is it 90 percent of the time when I ask you what it has to be? Zero. The force has to be zero, right? If the physics says you can't determine the direction, then no direction is preferred, it has to be zero. Okay, but how do we see that mathematically? Well, let's go back to our definition. F equals q v B sin theta. Then we add a direction which we determine from the right hand rule. Okay, we've got a q we've got a v we have a b but then we have the sine of the angle. And remember the angle here is between v and B. What's the angle between v and B? What do you think the angle is between v and B? Zero! There we go. What's the sine of zero? The sine of zero is zero. There is no force. Okay? If v is in the direction of B there's zero force on the particle. Alright. So, v is to the right still, but we're going to let our B be up everywhere. And now we do the right hand rule for qv cross B. So v, everybody hold your right hand up, v, finger straight, curl into the direction of B, that's telling me that my thumb is the direction of the force which is coming towards you guys. So there is a force in the- sorry, the dot, right? Coming towards you, which would be in our 3D picture in the negative or, I'm sorry, in the positive x direction. Okay. So that force will be coming towards you, what is its strength? Well, it's qvB sine theta n hat, we just said that it's going to be in the x direction. We know that the x direction means i hat. Okay, i j k x y z, so n hat becomes i hat. What is the angle? What is the angle between v and B? It's 90 degrees. Right? They're at a right angle. v is in the y-axis, B was in the z-axis, And so you get 90 degrees, and sine of 90 degrees is of course, one. So what's the force? It is qvB i-hat. It's coming out of the page and it is a strength of qvB. So if you are in some other arbitrary direction, then you have to include that theta. So if this is x y z and let's say that our particle is going to the right, but now our B field is up here at some angle theta, now what is the direction of the force? Well, again, we use our right hand so we do a q v cross B and that tells me that it's coming out towards you, right? Just like we had here, it's still coming out towards you, and the strength is going to be with the sine theta. Okay, sine theta is of course smaller than one so it's going to be a little bit less than this force. So if they're at a right angle you get the maximum amount of force, if they're parallel you get zero, if they're somewhere in between you get sine of theta.