Circular Motion of Charges in Magnetic Fields

Hey guys. So, in this video, we're going to talk about how charges moving through a magnetic field actually experience circular motion, let's check it out. Alright, so first of all, remember, that the magnetic force on a moving charge is 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, it'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 going to 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, let's say, a positive charge q right here, that's moving this way so it moves with a constant speed but as soon as it gets right here it's going to now experience a force because it's moving inside of a magnetic field and we're going to use the right-hand rule to figure out that direction, okay? So, right hand, we're going to go into the plane, I want you do this with me, right? So, I want you to point away from your face and into the page because that's 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 going to do this, okay? 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 palm 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 to curve because it was moving this way but now it got tugged up a little bit. So, it's going to do something like this, okay? And, actually let me do a little bit differently, it's going to be something like this and then once it's over here, it's going to, again, what it's going to start doing now, your hand is like this, it's moving like this. So, now the force is going to go in this direction, okay? So, now you have a magnetic force that points this way. Now, it's going to curve a little bit more, the magnetic force is always going to point, it's always going to be perpendicular to the velocity, right? And the velocity vector is going to be like this, like this, tangentially, and the force is going to keep you in a loop, and you end up doing something like this, you end up doing something like this, okay? So, whenever you have a charge inside of a magnetic field it's going to move in circular motion, cool? So, that's that, and you're going to be able to write an equation for that. So, we have circular motion so we can write that F equals m, a, and we can say that this is a centripetal force. So, it's going to be m, a centripetal. Remember, the 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 the centripetal acceleration is our magnetic force, so I'm going to replace this with FB and it's going towards the center equals m, I'm going to replace a with V squared over R, and you'll see what we're going to get, this is a magnetic force on a charge, so this is going to be q, v, B, sine of theta but the angle is 90 degrees and the sine of 90 is simply 1, so that goes away and then you end up with this, and notice that the, this v here, one of the v's is going to cancel with this one and you're able to, by moving some stuff around, calculate or write an expression for R and R is going to be, if you move some stuff around, you move R to the other side and you move the q, V to the other side you get this R equals m, v divided by q, B, which is a huge equation in this chapter, it's going come back over and over again, okay? So, you may need to know how to derive this, depends on how picky 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 can work with it faster, okay? Super important equation, I have a silly trick to remember this equation, sometimes people get the letters confused, there's a lot of letters 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 of momentum, which is the top and then q, B is short for quarterback. So, momentum quarterback, it's a phrase that makes no sense but I just, it's just sticks, right? Momentum quarterback or at least for me it sticks and it's way from you to quickly remember this and if I forget you can just, I can always just go to F equals m, a 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 an electron, electron, so that means that q is going to be negative 1.6 times 10 to the negative 19 coulombs, enters the uniform field, B equals 0.2 tesla, directed perpendicular to its motion, perpendicular to its motion, meaning the angle is going to be 90 degrees therefore the sine of 90 will be 1, which means you don't have to worry about plugging a sign, okay? So, you measure the electron's deflection to have a circular arc of radius 0.3 centimeters, this is just the radius, right? Don't get thrown off by the word circular arc, what matters is that the radius of this, circular arc means that it's something like this and it's going like this and then it sort of bangs 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 wave, okay? 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 0.003 meters or 3 times 10 the negative 3 meters, and we want to know how fast must be electronically moving. So, what is V, okay? So, 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 R, 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. Alright, and we're just going to plug in all these numbers here to get the answer, radius is 0.003 q is 1.6 times 10 to the negative 19, and then B is 0.2, 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 a negative here? Well, and you might have noticed this, and this is a pattern in this chapter, none of these equations are going to have positives and negatives for charges, you're just always going to plug in as a positive. So, you can 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 is going to do it's going to affect how you use your right hand rule, okay? But you want the numbers to be all positive. So, you get the magnitude of the velocity as a positive number. So, you plug all of this you get spit, this is 1.1 times 10 to the 8. Remember, the velocity should always be less than the speed of light, this is less than the speed of light speed of light is 3 times 10 to the 8, this is less than that. So, it's a quick way to at least have a sanity check. So, we're good, that's a reasonable answer, right? So, what else do we have here? Actually that's it for this one.

So, let's keep talking about this real quick, we have two more points to make, if a charge moves perpendicular to a magnetic field, that's what we just discussed, right? Here, this is perpendicular to a magnetic field up here, you're going to get circular motion, okay? You're going to get circular motion, circular motion, if a charge moves parallel, perpendicular is 90 degrees, parallel is 0 degrees to the magnetic field. Remember, what happens is FB is q, v, b, sine of theta and if we're doing sine of 0 sine of 0 is 0. So, there is no force therefore, there is no force, which means this is going to move in a straight line, it's going to keep moving straight, okay? And, if you're moving at an aim, at an angle to the magnetic field. So, for example, if your V is this way but your B is this way. So, there's an angle here, you're going to have helical motion, helical 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. Alright, so this is helical motion, cool? You should note that conceptually, and the last point I wanna make is you may remember that the work done by any force is given by this equation f, Delta x, cosine of theta, where theta is the angle between the direction, the displacement, and the direction of the force, the two vectors of the equation, and the work done by the magnetic force in this case is going to be 0 because you're going, if you're going in a circle, right? Because you're going in a circle, it's going to be 0 because at any point your Delta x is tangential, let me get out of the way, at any point your Delta x is tangential and your force is centripetal, and this angle here is going to be 90 and if you write F, Delta x, cosine of 90, right? This equation use cosine not sine, this is going to be 0, everywhere you go when you go in a circle you're going to have this, okay? And by the way, you may remember this, the work done by any centripetal force is always 0, okay? So, that's where this stuff comes from. Alright, that's it for this one, let's keep going.