11.2 Electric Potential Energy

OK, today we're gonna look at electric potential energy and what that means and how to calculate it. But before we get to that, let's just quickly review the electric field and how the directions work. So, we know that if we have a negative charge, our electric field lines go inwards because we're assuming what would happen if we had a positive test charge. So, our electric field is inwards, remembering that. So, we based it on a positive, it's gonna get attracted, so it'll go inwards. And near the object, you can see our arrows are closer together, so it'll be a stronger electric field. And further away, it would be weaker. Then, a positive object will have electric field lines that go outwards. The positive test charge would get repelled or pushed upwards. So, that's that. So, we also know what happens if we have one of each. If we put one of each nearby, we're gonna get the electric fields sort of interacting with each other. We know the positive goes outwards, the negative goes inwards. We're gonna get an electric field that goes from one to the other and then the one that goes outwards at an angle and this one that goes inwards at an angle. First today, they're gonna tend to sort of connect, so we end up getting sort of these curvy electric field lines, something like that. And then, this one, they maybe do something like that, and so on. So, we start getting these curves, and then the ones that are out here would maybe curve outwards as well. And probably the only one that would not be curved would be the ones right at the very end. Oops, no. So, we'd get sort of electric field lines that look something like that. So, when we have two opposite charges, the electric fields tend to work together. And what's gonna happen is, the closer the charges are, the straighter those lines are going to be. So, right now, I've got them quite curved in between the two. If we got the charges super close together, we would tend to see that the electric field lines would be a little bit more parallel. Now, the idea is, well, how can we create a regular electric field? We don't want it to be where it's bending and different at different levels. We want it to be pretty consistent and uniform throughout. So, what we see is that when there are only two charges, we get quite a bit of bending going on. So, the idea is, what would happen if we have a whole bunch of charges? So, you can see in the diagram here, if we basically set up two plates, so we use two steel plates or whatever kind of metal, we'd put one with a positive charge as we load the other one with negative charges. We'd see that we pretty much get nice straight, even electric field lines going from the positive to the negative. The only bends would be right at the edges. We get sort of this upward and bow-like bend here from the positive to the negative. But we can pretty much ignore that because usually what we'll do is, if we're going to place a charge in there, we would put the charge somewhere in the middle. So, then the electric field would be pretty consistent and accurate throughout the whole thing. This whole idea of using charged plates leads us to electric potential energy. So, before we get to that, let's just quickly review what we did with gravitational potential energy. And you'll see that it works almost exactly the same way. So, let's go back to if we have a ball or an object above ground. We know it's a certain distance above ground, thanks. We have some sort of height. We knew that it had maximum potential energy at the top. We just went MGH, and that would be the maximum amount of energy, whatever it would have is potential. Then, we know that as it falls, and so if it's somewhere down around there, our potential will be a lot less. Right, because it's a little less height, so we'd have less potential now. But we know that our kinetic would be gained, right? We went from zero kinetic at the top to more kinetic as it falls. So, as it loses potential, it's going to be in kinetic. And then, if it was right at ground level, we basically say our potential is zero and our kinetic now would be max. All of the energy is converted from potential to kinetic. Okay, so the idea is sort of the energy switching from one to the other as it falls. And then the idea is, well, what happens when we want to pick up that ball and put it back to where it was? If we lift it back up, we're going against gravity, so we're gonna have to do some sort of work or change in energy to be able to move it back up. So, the idea is, we're gonna gain potential as we force it back up again, but our body is applying that energy to lift it back up. Once it's up there, it'll now have potential and it could do its thing all over again. So now the idea is, what happens if we do the same sort of thing in terms of electric potential energy? So, the logic works exactly the same, but the building is different. Instead of the ball above the ground, what we're gonna do is we're gonna think of two plates being separated. So, we'll have our positively charged plate there, and let's call our negatively charged plate down here. Okay, and since we typically use a positive for our Tesla point, we're gonna use the same logic. So let's suppose we have a test object right at the top, right? And if these don't have to be top and bottom, they could be sideways, whatever. But I want to draw it just like gravity, so it looks almost the same. So our positive test point is right at the top plate, so we know at this point then our potential will be maximum, right? And our kinetic would be zero, right? So before it moves, right when we first place it, and it's gonna be up near the positive plate, so it's not moving at a zero kinetic, but we know its potential is maximum because it's positive. It's gonna repel from the positive plate and it's gonna move towards the negative plate, so we know it's gonna, basically in essence, the way I drew this diagram, it's gonna fall. So instead of gravity being the thing that's gonna cause it to accelerate, it's gonna be the electric field that's gonna cause that thing to accelerate. But it works pretty much the same idea. So as we move further and further down, let's suppose it's somewhere halfway in the middle, we would probably get a potential of half as much, and our kinetic would be the other half. And then once we get right to the negative plate, our potential is now zero and our kinetic would be maximum. Exact same logic as we did for gravity. Just like gravity is different, it's the electric plates that are causing that object to move. And then once again, if we want to take that charge and force it back up to the top again, we're gonna have to do some sort of work or energy to move to gain the potential energy back again. How much work is it going to take for us to do that? Well, it's going to depend on what the charge is. So if this was just a small positive charge of say 1 Coulomb, it's going to take a certain amount of work to do it. But if we made the charge twice as large, we're gonna have to push twice as hard. If it was triple, we'd have to push three times. So the amount of charge will determine how much work has to be done. Alright, the second thing is, if our electric field is super strong, so gravity is basically the same all the time. When we lift up an object, we're pushing that up ball or whatever against gravity. But in this case, depending on how much we charge the plates, the electric field could be really weak, which would be easy to move the charge, or could be super strong, which would take a lot of work to do. So how strong that is would make a difference. And then also, the last thing is how far we move it. Just like gravity, if we lift up a long way, we're gonna have to do a lot of work. Same thing if our charged plates are separated a long way. It's gonna be more work than if we only took That charge and had to move it a small distance, right? So if the distance is small, the work or energy required will be less. But if the distance is large, it's going to be a lot more. Now, let's introduce a new term and formula: voltage. Electric potential energy or electric potential or electric potential difference can be used interchangeably. So, if we have a charge between positive and negative plates, for example, in a battery labeled nine volts, it means there is nine volts of energy between those plates. Therefore, it would take nine volts of energy to move a positive charge from one plate to the other. Let's dive into the details. Voltage is the change in potential energy per unit charge. For example, if we have a nine volt gap and one Coulomb charge, we would say that nine volts is equal to nine joules of energy per one Coulomb. Therefore, voltage is measured in joules per Coulomb or volts. Let's do a quick example. If we have a 3.4 Coulomb charge and it gains 2.63 joules as it moves through an electric field, the direction doesn't matter here. The change in energy, or kinetic energy, will be the amount of energy converted. So, in this case, the change in energy is 2.63 joules. To find the voltage, we divide this by the charge, 3.4 Coulombs. This gives us a voltage of 7.6 times 10 to the -3 volts. Now, the last part of the question asks whether this is an increase in potential or kinetic energy. It could be either, depending on the direction. However, in this case, the question states that the charge is moving on its own. It would be like a positive charge moving towards the negative plate. So, if it's moving on its own, we would say there is an increase in kinetic energy. If we were pushing that charge, let's suppose we drop stretch and it started at the bottom plate, and we're going to push it upwards, then we're going to move it from zero potential to kinetic energy. So, that would be an increase in potential energy. So, it's the same calculation, you just have to be careful of the direction and whether it's going to happen on its own or whether we're going to have to apply energy to make it happen. One last topic is that sometimes it's not going to be convenient to work with joules just because we're dealing with really small charges. So, our answers would be really big in terms of scientific notation, with lots of large negative exponents and things like that. What scientists have done is we call it the electron volt. An electron volt is a measure of energy that works for small objects like an electron, proton, things like that. We have to be careful because voltage, which is measured in joules per coulomb, is not the same as an electron volt. An electron volt is actually measuring energy, so it's similar to a joule but we say electron volt instead. The definition is that it's the amount of energy required to move one electron through one volt. So, if we had one coulomb of separation and we have one electron moving, one bit in one direction or the other, that's an electron volt. That's how much energy it's going to take to move one electron, one volt. So, if we actually figure out how much that works out to, we can use our formula we just had, which is V equals C over Q. What we're going to do then is just solve it for energy. This would be V times Q or Q times V. We do one electron times one volt. The charge on one electron is 1.6 x 10^(-19), so we do that times 1 volt. But remember, our volt was joule per coulomb, so you can see the coulombs would cancel. We end up with 1.6 x 10^(-19) which is the charge of an electron. So, an electron volt is the exact same thing as the charge of an electron, but it's in energy, not coulombs. Okay, so to convert back and forth, that's all you have to do is divide or multiply by 1.6 x 10^(-19). Let's do the second example. We're basically back to the parallel plate setup that we had in the previous one. So now it says, "How many electron volts would you need to move an alpha particle through 760 volts?" If we figure out the energy, and let's do it in electron volts, so our charge is Q, but remember it's an alpha particle which has a positive 2 charge. That would be the same thing as 2 times 1.6 x 10^(-19). Our volts is 760, so we multiply those together and we get 1520 electron volts. Okay, that's a pretty easy calculation, just multiply the two, and that's it. If you want to convert that to joules, then we would multiply that by the conversion factor that one electron volt equals 1.6 x 10^(-19) joules. So, if we have 1520 electron volts, oops, you can see that we would just have 2 times 1520 times 1.6 x 10^(-19). That would give us how many joules. And if we do that quick calculation, we see that this is the same thing as 2.4 x 10^(-16). So, it's a pretty small amount. That's why we tend to use electron volts for small energy measurements. Switched electron volts. When we're dealing with really small charges, it just makes the calculations a little bit easier to work with. And that's all for today.