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Circular Motion of Charges in Magnetic Fields

Patrick Ford
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Hey, guys. So in this video, we're gonna talk about how charges moving through a magnetic fields actually experienced circular motion. Let's check it out. Alright. So first of all, remember that the magnetic force on a moving charges always perpendicular to its velocity. So you can remember this from the right hand rule. The force is always perpendicular to the velocity. This is the velocity. The force is always the palm of your hand. So what's going out this way? And even if it's this way, right, this is always going to be a 90 degree angle. Okay, So because of this, you're gonna have circular motion, let me show you. So let's imagine that there's a magnetic field inside of this square and you have a charge Q of, let's say a positive charge, Q Right here. That's moving this way. So it moves with the constant speed. But as soon as it gets right here, it's going to now experience a force because it's moving inside of magnetic field and we're gonna use the right hand rule to figure out that direction. Okay, so right hand, we're gonna go into the plane I want you to do this with me, right? So I want you to point away from your face and into the page, because that's what the little X's mean. Now notice my hands like this, and I want my thumb to actually go in the other direction. So I'm gonna do this, OK, So please follow me and do this yourself. And when you do this, you're pointing away from yourself with your thumb to your right, your palm should be pointing up and your poem pointing up means that the direction of the force is upward. Okay, so there will be a force here. The magnetic force will be up. What that means is that this thing will start the curve because it was moving this way. But now it got tugged up a little bit. So it's gonna do something like this. Okay. And actually, me do a little bit differently. It's gonna do something like this. Um, and then once it's over here, it's going to again what it's going to start doing now your hands like this, it's moving like this. So now the force is gonna go in this direction. Okay, so now you have a magnetic force that points this way. Now it's gonna curve a little bit more. The magnetic force is always going to point. Um, it's always going to be perpendicular to the velocity. Right? And the velocity vector is gonna be like this, like this tangentially, and the force is going to keep you in a loop, okay? And you end up doing something like this, you end up doing something like this. Okay, so whenever you have a charge inside off a magnetic fields, it's going to move in circular motion. Cool. So that's that. And you're gonna be able to write an equation for that. So we have circular motion so we can write that ethical that may, um And we can say that this is a centripetal force, so it's gonna be m a centripetal. Remember, this centripetal acceleration is V squared over R, where r is the radius of the circle. So R is radius of the circular motion. And the force here that's responsible for a centripetal acceleration is our magnetic force. Someone to replace this with F B. And it's going towards the center equals m. I'm gonna replace a with the square over our and you'll see what we're gonna get. This is the magnetic force on a charge. So this is gonna be Q v b sign of data, but the angle is 90 degrees in a sign of 90 is simply one. So that goes away. And then you end up with this and notice that this v here, one of the V's he's gonna cancel with this one, and you're able to buy moving some stuff around, calculates or write an expression for our and our is going to be. If you move some stuff around, you move are to the other side and you move the Q B to the other side and you get this r equals M V divided by cube, which is a huge equation in this chapter, it's gonna come back over and over again. Okay, so you may need to know how to derive. This depends on how picking your professor is about this kind of stuff. But even if you remember how to derive the whole thing, you also should memorize this equation so you could work with it faster. Okay. Super important equation. I have ah, silly trick to remember this equation. Sometimes people get the letters confuses a lot of letters. Um, M V, if you remember, is momentum. Momentum. P equals M. V. So I think of this. The way I remember this is I think off mo Mentum, which is a top. And then Q B is short for quarterback. So momentum quarterback, Um, it's afraid that makes no sense, but I just It just sticks, right? Momentum, quarterback, or at least for me, it sticks, and it's a way for me to quickly remember this. And if I forget, you can just I can always just goto because they may and sort of re derive it. Okay, so that's it for that. Let's try a quick example here. So it says in an experiment in Electron Electron. So that means that Q is going to be negative 1.6 times 10 to the negative. 19 columns enters the uniforms fields B equals 0.2. Tesla directed perpendicular to its motion perpendicular to its motion. Um, meaning the angle is going to be 90 degrees. Therefore, the sign of 90 will be one, which means you don't have to worry about plugging a sign. Okay, eso you measure the electrons deflection toe have a circular arc of radius 0.3 centimeters. This is just the radius, right? Don't get thrown off by the word circular Ark. What matters that the radius is This circular arc means that it's something like this and it's going like this. And then it's sort of bends a little bit right. And then it keeps going. And in this little arc, if you make it into a big circle, will have that radius. OK, but long story short, it just means that that's what you use as the radius. Notice that this is 0.3 centimeters. So it's 0.3 m or three times to the negative 3 m and we wanna know how fast must the election be moving? So what is V ok eso to solve this? We're going to use this equation right here because that's the equation that ties all these variables together. And we're looking for V. So r equals momentum quarterback, right? And we're looking for V. So v equals our Q B over m. Gotta move some stuff around. By the way, Mass is going to be the mass of the electron, which is 9.1 times 10 to the negative 31. All right. And we're just gonna plug in all these numbers here to get the answer. Radius is 0.3 Q is 1.6 times 10 to the negative 19 on then B is zero point to the mass is 9.1 times 10 to the negative 31. And if you plug all this stuff, you're going to get the answer now real quick. You might be wondering, Why didn't I plug in the negative here? Well, and you might have noticed this and this is a pattern in this chapter. Um, none of these equations were gonna have positives and negatives for charges. You're just always going to plug. It is a positive. So you could think of it as this Q. Being the absolute value of Q same thing with all these numbers, by the way, are just always absolute values. What positive and negative or one direction versus another direction going is going to do is it's going to affect how you use the right hand rule. Okay, but you want the numbers to be all positive, so you get the magnitude of the velocity a za positive number. So if you plug all of this, you get that. This is 1.1 times 10 to the eighth. Remember, the velocities should always be less than the speed of light. This is less than the speed of light. Speed of light is three times 10 to the eighth. This is less than that. So it's a quick way to at least have a sanity check. Um, so we're good. That's a That's a reasonable answer, right? So what else do we have here? Um, actually, that's it for this one. So let's keep talking about this real quick. We have two more points to make. Um, if a charge moves perpendicular to a magnetic field, that's what we just discussed. Right? Um, here. This is perpendicular to a magnetic. Fields appear you're going to get circular motion. Okay, You're gonna get circular motion. Um, circular promotion. Um, if a charge moves parallel, perpendicular is 90 degrees parallel is zero degrees to the magnetic field. Remember, what happens is FB is Q v b sign of data. And if we're doing sign of zero sign of 00 so there is no force. Therefore, there is no force, which means this is going to move in a straight line. It's gonna keep moving straight. Okay? And if you're moving at an angle on an angle to the magnetic fields. So, for example, if your V is this way, but your B is this way. So there's an angle here. You're going tohave Hill ical Motion hill. Ical motion. Okay, which looks like this you are moving in this direction while also spinning. Okay, You're moving and spinning, which is going to look like this. Okay, you're moving like this. All right, so this is he locomotion. Cool. You should note that conceptual in the last point I wanna make is you may remember that the work done by any forces given by this equation f Delta X cosine of data where data is the angle between the direction, the displacement in the direction of the force, the two factors in the equation and the work done by the magnetic force in this case, um, is going to be zero because you're going. If you're going in a circle because you're going in a circle, it's going to be zero, because at any point. You're Delta X is tangential. Let me get out of the way at any point. You're Delta X is tangential and your forces centripetal. And this angle here is going to be 90. And if you write F Delta X CO sign of right this equation use co sign not sign. Um, this is going to be zero. Everywhere you go, when you go in a circle, you're going tohave this. Okay? And by the way, you may remember this. The work done by any any centripetal force is always zero. Okay, so that's where this stuff comes from. All right, that's it for this one. Let's keep going.
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