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# Electric potential at a point in space

## Video transcript

- Let's talk about Electric Potential V. This is confusing, this is one of, if not the most confusing ideas in all of physics. For one it sounds just like Electric Potential energy but it's not, this is different, it's related to Electric Potential energy but the Electric Potential V is different from Electric Potential energy. That was a poor choice of naming. And the other reason it's confusing is that Electric Potential V is a number, just a number, that's it, associated with points in space. So it's abstract, this is an abstract idea, you can't go hold Electric Potential in your hand, it's a number, an abstract number at every point in space. Here is points in space, I just put circles around empty spots on the screen, there's like nothing here, I just put circles here, these are circles around empty points in space, just so we can talk about them explicitly. And, well, if there was no charge around, if you literally had an empty universe, the V value at every point in this universe would be zero. It'd be zero there, it'd be zero over here, the number associated with every point in space would be zero. That'd be boring, and useless. How do we make it so that the V value, the Electric Potential value, is not zero? We just stick a charge in here, just stick a big ol' positive Q at some point in space over here, take a big ol' charge and we'll stick it right there. Now, points in space around this charge will have a V value that's non-zero, and they'll be big if you're near this Q, so the V values around here are gonna be really big, and then the V values way out here will be smaller, the further way you go the smaller it gets. And why do we care? Who cares? The reason we care is this. The units of Electric Potential are Joules per Coulomb, so Electric Potential has units of Joules per Coulomb, that gives you a hint of why you should care, we care about Joules, Joules are energy so something about energy, that's useful, you can get work out of that or can turn it to Kinetic energy. And Joules per Coulomb, that let's you know, alright, well, if this point over here happened to have, say 100 Joules per Coulomb, let's say the V value at this point in space happened to be 100 Joules per Coulomb, what that means is, remember there's nothing there, but if there was something there, if we happened to take, say we had a positive two Coulomb charge and we took that charge and we put it there at that empty point in space, before we put it there the V value was 100, when we stick it here, who cares? Why do we care about this 100 value? 'Cause look at it, it's 100 Joules per Coulomb, that's what V, the Electric Potential, is telling us. So if it's 100 Joules per Coulomb and I stick two Coulombs there, how many Joules of energy do you think it's gonna have? It'll have 200. And that's the key, that's why we care about Electric Potential 'cause it let's us find Electric Potential energy, either PE, sometimes people write Electric Potential energy as U. So the formula is just Q, you take the Q that you sticked at that point in space, in this case it was two Coulombs. Take whatever Q there is, multiply it by the value of the Electric Potential and that tells you how many Joules there would be for the charges in that region, so this Electric Potential energy is between these two charges here, the charge that created the V and the charge that you sticked at that point, and the V is a quick way to figure out how much Potential energy, Electric Potential energy there will be. So in other words, in this case, since I have two Coulombs there, I take my two Coulombs and I multiply by 100 Joules per Coulomb 'cause that's the V value, and I get that there are 200 Joules of Potential energy now stored between these charges. So that's why we care about Electric Potential V, it's a way to figure out the Electric Potential energy for a charge that's placed at that point in space that has that V value. But, how do you get this V value? If I hadn't given you the 100 Joules per Coulomb we wouldn't have been able to figure this out, we need a way to figure out the V value at points in space based on the charges creating them, 'cause charges create the V value. There's a formula for it, and the formula says that the V, Electric Potential, created by point charges equals K, K is the Electric constant 9 times 10 to the ninth, and it has units of Newton meter squared per Coulomb squared, that's always K. You take that K and you multiply by the charge that's creating the V value, so in this case is this Q, this positive Q here, whatever Q it is creating the V value that you wanna find, and that's key, if you plug in five Coulombs here you're finding the V created by that five Coulombs, if you plug in negative three Coulombs you'll find the V created by the negative three Coulombs, sometimes there's problems with multiple charges in it, like this one, and this Q gotta be the charge creating this V, not the charge you placed at that point in space, and I'll put the two Coulombs up here, but the charge creating the V value that I wanna find. And then you divide by the distance, so I divide by the distance between this charge and the point in space that I wanna figure out the V value at. Some people call this the radius, I don't like calling it radius, makes this sound like there has to be a circle, it doesn't really have to be a circle, this r would be the distance from this point of charge creating this V value to the point in space where I wanna determine the V value, that's r. So this is r. So how do we determine this? Let me just give you some numbers, let's say the charge we stuck here was one nanoCoulomb, nano is 10 to the negative ninth, so let's say that was one nanoCoulomb. And let's say the distance from this charge to this point in space was, let's say it was nine centimeters. And I wanna know what's the V value, well I can solve for it now, we got our formula, the V would equal, alright my K is 9, always, times 10 to the ninth, and it's Newtons meter squared per Coulomb squared, and then I multiply by my charge, and I told you that the charge here was 10 to the negative ninth Coulombs, and my distance, I divide by the r value and the r value is nine centimeters, but be careful, everything's gotta be in terms of meters, kilograms and seconds when you're doing physics with constants. Look at this is in terms of meters, so I've got to use meters here, so nine centimeters is .09 meters. And if I multiply all this out what you'll get is, 10 to the negative ninth cancels this 10 to the ninth, the powers are 10, these just go away, and then I have nine divided by .09, that gets equal 100. So I chose this so that we got the same answer down there. Okay, 100 Joules per Coulomb, you might be like, where the Joules comes from? And how is this Joules per Coulomb? Well let's look at it, if we took, look at, one of these meters cancels one of these meters, and one of these Coulombs cancels one of those Coulombs, what are we left with? We're left with Newton times meter over Coulomb, but Newton times meter, that's force times distance, that's Joules, that's where we get Joules per Coulomb. So this really does give us the number of Joules there would be at a point in space per Coulomb of charge that you put there. And it works for any point, you pick any point, if I picked a point twice as close, it's half as far away, let's say some point over here, let's say this r value here was only 4.5 centimeters, well I'm dividing this by r, so if the r is half as big this point over here will have a V value of 200 Joules per Coulomb and the closer I get, if I went even closer, if I went to a point that was three centimeters away, well this is a third as much as this other distance, so if I'm only dividing by a third as much distance as you get three times the result 'cause r is not squared, it's just r. So at this point, we'll have a V value of 300 Joules per Coulomb. This tells me, if I wanted to get a charge that have a whole bunch of Potential energy, I should put that thing nearby, I should stick it over here, this will give me a lot of Potential energy. Not quite as much, even less, the further I put my charge the less Potential energy it will have. There will be no Potential energy until there is a charge, there'll just be Electric Potential. But once you place another charge in that region to go with the first one, then you'll have Electric Potential energy and this will be a way to find it, Q times the V that you get out of this calculation. You gotta be careful though, sometimes people get sloppy, and V looks, you know, we use V for Electric Potential and we use V for Voltage, what's the difference? Are they the same? Hmmm, not quite. Sometimes you can treat them as the same and you don't get into trouble, but sometimes you do and messes you up. Voltage is a, technically a change in Electric Potential between two points, this is the difference in Electric Potential between two points in space, so it's got the same units 'cause the change in Electric Potential still gonna have units of Joule per Coulomb, it's just, when it's a change in we give this a new title, we call the Joule per Coulomb unit a Volt. So Joules per Coulomb are Volts, but the word Voltage specifically refers to a difference in Electric Potential, what am I talking about? Well, look at, this point is 300 Joules per Coulomb, this point over here 100 Joules per Coulomb, so the delta V, if I were to take delta V between these two points right here and I ask, what's the difference in V? Well the difference in V is 200, 200 Joules per Coulomb, that means the Voltage between those two points in space is 200 Volts, that's what it means. So, when you're talking about a difference in Electric Potential between two points in space we call it a Voltage, when you're talking about just the Electric Potential value at one point in space we call it the Electric Potential, and that's how they're related.