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Potential at Center of Charges in a Square

Patrick Ford
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Alright, guys, we're gonna work this one out together, so we'll have the square of charges right here. They're not all equal charges, but we're supposed to do is find the potential at the center of this arrangement right here. So just right off the bat potential, which formula is that potential is the formula with one Q and one are so the words, the potential is K times the producing charge at some distance right here. And we're supposed to find out what the what is the total potential at this point right here. This is our point of interest. So basically, at the center of this arrangement right here, all we need to do is figure out what the distances is. The distances are to each one of these charges that are on the corners. So one of things we could do is sort of break this up into a little triangle, like a little mini triangle. And basically, we've cut this thing in half. So we know that this distance, which is half, is gonna be 2.5 millimeters. This is also gonna be 2.5 millimeters. So that means we can actually figure out by using the high pot news in the Pythagorean theorem that this is basically 3.54 millimeters. Right? And that's just something we get for the Pythagorean theorem. So we know that this our distance right here, which is 3.54 millimeters, is going to be 3.54 times 10 to the minus three. That's gonna be in meters. So basically, what the idea is is that all of these charges are gonna be producing potentials at the center. And these charges, these potentials that they produce are gonna be scaler, which means that we can just add them up together. So the total potential So that's one thing that you could possibly right is just v. Total is gonna be v the potential from the two nano Coolum charge, plus the potential from the 1.5 Nano cool in charge of the negative 1.5 and then so on and so forth, right? You could basically just out of all of those potentials together. So that means that the potentials here v total um, you would get by using cake you over our and the only thing that's changing is what? Just depending on the producing charge that you're putting inside of that equation. But if you add all these things up together, what ends up happening is that this K and this are end up being greatest common factors. So one of the shortcuts that you can use is say that this is gonna be Cavor are and then basically, just add up all of the charges that contribute a potential Q one Q two Q three and then so on and so forth. Right, Because basically, just you could just add those things up together. You don't have to keep sticking k over our into your calculator over and over. So what we can do is that the total potential is gonna be 99 times 10 to the ninth. Now, we've got the distance between all of these charges, and center is always going to be 3.54 times 10 to the minus three. And now we just add the charges up together individually. And it doesn't matter which order we add them in because Edition is just community. It doesn't matter which order you do it in. So that means I could do to nano columns minus three Nano columns. Let me see. Minus three narrow columns and then plus one nano Cool. Um and then we've got minus 1.5 narrow columns. Let me scoot down. So basically, what happens here is if you close off this parentheses, this to the negative three and the plus one are all gonna cancel out to zero on. All you have to do is just do one multiplication here 8.9 times 10 to the ninth, this K divided by r and then times this negative 1.5 nano columns, which is 10 of the negative nine, by the way. So don't forget to stick this in as 10 of minus nine, and you should get that the potential is equal to negative 3.81 times, 10 to the third. And that's gonna be in volts. So it's negative, by the way. And I just want to point out the fact that you could Onley do the shortcut here because all of these distances that we calculated to the charges are gonna be the same. So a lot of times, this is one way that symmetry can work out on here advantage. So if you know how to cancel these things out and take these shortcuts, then it's gonna be a great tool for you for you to use. Let me know if you guys have any questions in the comments, and I'll see you the next one.