Electric potential from multiple charges

- [Instructor] So imagine you had three charges sitting next to each other, but they're fixed in place. So somehow these charges are bolted down or secured in place, we're not gonna let'em move. But we do know the values of the charges. We've got a positive one microcoulomb charge, a positive five microcoulomb charge, and a negative two microcoulomb charge. So a question that's often asked when you have this type of scenario is if we know the distances between the charges, what's the total electric potential at some point, and let's choose this corner, this empty corner up here, this point P. So we want to know what's the electric potential at point P. Since we know where every charge is that's gonna be creating an electric potential at P, we can just use the formula for the electric potential created by a charge and that formula is V equals k, the electric constant times Q, the charge creating the electric potential divided by r which is the distance from the charge to the point where it's creating the electric potential. So notice we've got three charges here, all creating electric potential at point P. So what we're really finding is the total electric potential at point P. And to do that, we can just find the electric potential that each charge creates at point P, and then add them up. So in other words, this positive one microcoulomb charge is gonna create an electric potential value at point P, and we can use this formula to find what that value is. So we get the electric potential from the positive one microcoulomb charge, it's gonna equal k, which is always nine times 10 to the ninth, times the charge creating the electric potential which in this case is positive one microcoulombs. Micro means 10 to the negative six and the distance between this charge and the point we're considering to find the electric potential is gonna be four meters. So from here to there, we're shown is four meters. And we get a value 2250 joules per coulomb, is the unit for electric potential. But this is just the electric potential created at point P by this positive one microcoulomb charge. All the rest of these charges are also gonna create electric potential at point P. So if we want the total electric potential, we're gonna have to find the contribution from all these other charges at point P as well. So the electric potential from the positive five microcoulomb charge is gonna also be nine times 10 to the ninth, but this time, times the charge creating it would be the five microcoulombs and again, micro is 10 to the negative six, and now you gotta be careful. I'm not gonna use three meters or four meters for the distance in this formula. I've got to use distance from the charge to the point where it's creating the electric potential. And that's gonna be this distance right here. What is that gonna be? Well if you imagine this triangle, you got a four on this side, you'd have a three on this side, since this side is three. To find the length of this side, you can just do three squared plus four squared, take a square root, which is just the Pythagorean Theorem, and that's gonna be nine plus 16, is 25 and the square root of 25 is just five. So this is five meters from this charge to this point P. So we'll plug in five meters here. And if we plug this into the calculator, we get 9000 joules per coulomb. So we've got one more charge to go, this negative two microcoulombs is also gonna create its own electric potential at point P. So the electric potential created by the negative two microcoulomb charge will again be nine times 10 to the ninth. This time, times negative two microcoulombs. Again, it's micro, so 10 to the negative six, but notice we are plugging in the negative sign. Negative charges create negative electric potentials at points in space around them, just like positive charges create positive electric potential values at points in space around them. So you've got to include this negative, that's the bad news. You've gotta remember to include the negative. The good news is, these aren't vectors. Notice these are not gonna be vector quantities of electric potential. Electric potential is not a vector quantity. It's a scalar, so there's no direction. So I'm not gonna have to break this into components or worry about anything like that up here. These are all just numbers at this point in space. And to find the total, we're just gonna add all these up to get the total electric potential. But they won't add up right if you don't include this negative sign because the negative charges do create negative electric potentials. So what distance do we divide by is the distance between this charge and that point P, which we're shown over here is three meters, which if we solve, gives us negative 6000 joules per coulomb. So now we've got everything we need to find the total electric potential. Again, these are not vectors, so you can just literally add them all up to get the total electric potential. In other words, the total electric potential at point P will just be the values of all of the potentials created by each charge added up. So we'll have 2250 joules per coulomb plus 9000 joules per coulomb plus negative 6000 joules per coulomb. And we could put a parenthesis around this so it doesn't look so awkward. So if you take 2250 plus 9000 minus 6000, you get positive 5250 joules per coulomb. So that's our answer. Recapping to find the total electric potential at some point in space created by charges, you can use this formula to find the electric potential created by each charge at that point in space and then add all the electric potential values you found together to get the total electric potential at that point in space.