12.5 Charges in a Magnetic Field

Ok, today we're going to look at what happens when there are moving charges inside a magnetic field. We could also have a wire inside, creating a magnetic field, and we're going to observe what happens in certain scenarios. So, we're going to be using our two part formula. Remember, we had the magnetic force as QVB, where Q is the charge, V is the velocity, and B is the magnetic field. We also had the magnetic force as ILB, where I is the current, L is the length of the wire, and B is the magnetic field. Both of these formulas will be used. Now, we're going to combine them with another formula. First, let's figure out what happens when we have any charge in a magnetic field. Here's a little bit of background information: you may have heard of the Northern Lights. We've discussed them before in class. The Northern Lights are just an example of charges from the Sun being trapped in our magnetic fields. They hit our atmosphere and get funneled into our North and South Pole. As they bump into particles in our atmosphere, they transfer their energy. Energy gives off light and that's how we see it. The light kind of dances, but we don't see the funneling so much because there are currents that blow and distort it a little bit. However, we do see streaks and zigzag lines. The reason why it funnels and we get this circular motion is if we use our right-hand rule. In the bottom diagram, we have an electron going to the right. So, point your thumb to the right. Our B is into the page, so your fingers would go downwards. This means our force is to the south. So, our force is going to be going south. In the second diagram, you can see that as soon as our electron enters the magnetic field, our force is downwards. This means it's going to move the electron downwards. But, as it moves, if you kind of turn your hand and move your thumb as it follows that curve, you can see that the force also goes inwards. So, really, what ends up happening is that the electron moves in a circular motion. What's happening is, as our electron goes around, the force keeps pulling it inwards. So basically, it's going in a circular pattern. Because it's going in a circle, we can say that our magnetic force is actually the exact same force as the centripetal force in physics. So we can use those two things to help us solve these questions. Here's a question with some numbers now, but it's based on the same thing. We have an electron going in a circle, so the magnetic field causes it to go in circles. That means our QVB (magnetic force) has to equal our centripetal force, which is MV²/R. And you can see that in the equation, we've got a V on both sides, so let's cancel that. So one of the Bs or the V² will cancel the other B. Therefore, we get QB = MV/AR. The question here is asking, what's our radius? So we can rearrange the equation to solve for R. The other side brings the QB down, so we get radius equals MV over QB. So all we have to do is plug that in our formula and solve, so we get our mass of electron, 9.1 times 10 to the power of -31. Our velocity is 2.38 times 10 to the power of 5, our charge of an electron is 1.6 times 10 to the power of -19, and our magnetic field is 1.3 times 10 to the power of -5. So, put all that in the formula, cut it out, and we get a radius of 0.12 meters. Okay, so pretty easy calculation. You just have to make your two forces equal to each other and solve it for whatever variable you want. So, the next example we're going to look at, there are a couple of different [Music] Examples: We're going to look at what happens if our charges are balanced by gravity, so they don't have to always be in circular motion. We can have our forces canceling out with each other and doing different things. So this first one I've got here, this first example is that we had an alpha particle being shot into a magnetic field at 90 degrees. So we have our magnetic field, and it says the magnetic force on the alpha particle is acting upwards. We have our alpha, let's draw that as a plus 2 charge. And it says the magnetic force is acting upward. So it says determine the speed at which it must be traveling so that the magnetic forces balance out the force of gravity pulling it down. So as we shoot this thing in there, we really don't care about the directions of the magnetic fields or anything else. We just basically have to see that we got magnetic pushing it up, gravity pulling down. So it's actually going to travel in that perfectly straight line. Those two forces have to be equal, so we just basically make them equal. So we'd have QV be. Again, equal to mass times gravity this time, and the question is asking for its velocity. See that mg divided by QB? All you have to do is plug in all those numbers. So, our mass, going off the particle, is 6.65 x 10^-27. Our gravity is 9.21, and our charge about the particle is 3.2 and RB. The question is 4.92 x 10^-9 or 4.92 x 10^-7. So, plug all of that in, print it out, and we get 0.42 meters per second. So, it doesn't have to go very fast. But if you think about it, the force of gravity is pretty weak. The mass times the gravity of them on and off the vertical is pretty small. So, we're not going to need a very Quick velocity at all in order to balance out the force down is example 3 on the bottom. Here is pretty much the same thing, except it says we have a wire. So now we have some sort of wire, and it says the current is going through the wire. Let me just draw it as an in and out sort of thing. So, we've got current going into the wire, and it says that the current is pushing the wire upwards. So we've got magnetic force pushing it up, and we've got gravity once again pulling it down. So it works out pretty much the exact same thing. The only thing that's different is when we use our formula this time, we're going to use IOB. So, that will be equal, and the question is, what's the mass of the wire in order to get these? things to balance out, we just have mass. It would be il be divided by gravity, so our I is already over on the side here. So, our current is 7.81 * 10^-2, our L is 0.298, and our B is 2.75 * 10^-3. And R Gean and twenty-one, so we'll just divide the vote vote by those out, and that will give us a mass of 9 * 10^-6 kilograms. Okay, so it's pretty small, but once again, the gravity's not going to be very much, so we're not gonna need much to push it upwards to balance it up. Okay, one last sort of example, and this one is where. We're going to shoot our electron in again, so let's start with our electron. We're going to be launching it horizontally, so we'll set this up in a way that the magnetic force and magnetic fields cancel out the electric forces. This time, we have a charge plate of 250 volts, with a distance of two centimeters between them. We're going to shoot the electron in, so let's suppose that the top plate is positive. Once our electron gets shot into this plate, you can see that in the middle, when it reaches a certain point, it will be attracted to the positive plate. This means our electric force will be going upwards. Additionally, we have a magnetic field. In this case, we would have a coil of wire or something around the battery. something like that, and it says the magnetic force is going downwards, right? If it's not going to deflect, if our electron is going to keep going perfectly straight through, then our two forces have to balance up. So in this case, we'd have magnetic force equal to electric force. That's really the only thing that's different. So first, let's go through a few of these. If our electric field is in the direction of the electric field, so our electric field always goes from positive to negative. So the electric field is going to be downwards. But we saw here that our electric force was upwards, in the direction of the magnetic field. So if our force is downwards and our E is to the right, that means our B then. Has to be into the page, letter B we'd have inwards, which would be the X, right? And then Part C is asking what's our velocity, so once again we just use our formulas. QV is put in this time for electric force, we just go Q times electric field, and you can see our Q's will cancel. So our velocity is a pretty easy calculation, it's just e divided by B. So all we have to do is do our electric field, which we don't know, right? But we do know that electric field equals voltage over distance, so they give us that. So if we go 50 divided by 2 centimeters, so 250 divided by 0.02, we get an answer of 12,500. And our B is 0.045, so divide. That and that gives us a velocity of two point eight, if we round it off, times 10 to the power of 5 meters per second. And that's it. So, we got our different types of questions. So basically, for each of these examples, our magnetic force will balance either with circular gravity, electric, some other cut the force, and you just solve it accordingly.