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Anderson Video - Electric Potential with Four Charges

Professor Anderson
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All right, let's draw a square And let's say that we have the following. We have a q1 right there a q2 a q3 and a q4. And each side here is going to be L of this square. And let's find the potential right at the center of the square. Okay, so we want to find V at the center and we'll give you some charges here okay. So q1 is in fact equal to q2 which is equal to q3 and those all have a value of 6 microcoulomb. And q4 has a value of minus 3 microcoulombs. And we'll say that L is equal to 2 centimeters. All right and we're looking for V right at the center. All right we need to figure out how to do this. One thing that we know about electric fields is they obey the principle of superposition right if I have two electric fields, I can add them up as vectors. It turns out that potential also satisfies the principle of superposition so if I want V at the center that is just V1 plus V2 plus V3 plus V4. And we know what V is for a point charge. So V1 becomes K Q1 divided by R where this distance here is R. V2 is K Q2 over R. B3 is K Q3 over R. And B4 is K Q4 over R. We know all the Q's. We can probably figure out what the R's are, but we also can call this thing just Q and we'll say that that Q is equal to 6 microcoulombs and therefore we can simplify this quite a bit, this first one just becomes KQ over R and then we have KQ over R and a third. And then the fourth one is the only one that's different. And so this whole thing becomes 3 KQ over R and we can even simplify this a little bit more right we have three of those KQ- no, let's just leave it like that for now. 3 KQ over R plus K Q4 over R. Okay, we know K We know the Q's We need to figure out R. What is R? Well R is the distance from the center to any one of these edges and so if I think about a triangle right here this side of the triangle is L over 2. This side of the triangle is L over 2. And so we have R squared equals L over 2 squared plus L over 2 squared which is 2 L squared over 4 which is L squared over 2. And so R in fact just becomes L over root two. Okay and now we have all those numbers and we can plug it in here and try it out. So let's do that. We've got a K over R out in front and then we have 3Q plus Q4 And let's make some room up here and we'll plug in some of these numbers. And you might have to remind me of these numbers but Okay, so we said that the potential at the center was K over R 3Q plus Q4. And we know that K is 8.99 times 10 to the 9. R we just said was L over root 2. L we said was 2 centimeters so that's 0.02 meters. Divide that by root 2. Q we said was six microcoulombs, right? 3 times 6 times 10 to the minus 6. And then Q4 we said was negative three microcoulombs. All right and now we can plug in all these numbers why don't you guys try it in your calculator and I'll see if I can approximate it here. So we've got 9 times 10 to the 9 We've got a 2 times ten to the minus two Root 2 is about 1.4. Put that right there. And then we have 18 minus 3 is 15 times 10 to the minus 6 so, let's see, what we get, we get 9 times 15 is 145-135. So 135, that 1.4 is going to come up top. We've got a times 10 to the 3 up top. And then we still have that 2 times 10 to the minus 2 down there And what do we get 135 times 1.4 That's got to be about 200. And 200 divided by 2 is 100 So I have a 100 there I have 3 there so that's a 10 to the 5 and then I have a 10 to the minus 2 so I'm gonna say this has got to be around 10 to the 7? Did anybody run in their calculator and get a number? Okay. So the actual answer is 9.53 times 10 to the 6 which is pretty close to our guess, right, because this is almost 10 and that would make this thing 10 to the 7. 9.53 times 10 to the 6 and we're asking about V, right V is volts. Okay, there is 9.53 times 10 to the 6 volts at the center of this square. Now, the nice thing about potential here is it's a scalar we don't have to worry about any direction. There was no Direction associated with our geometry. It was just what's the value there determined by the distance away from the point charge. Okay and then we just had to add them up. All right, questions about this? This would be the time to ask if you guys have any questions about what we just did. Okay, why don't we take a five-minute break and I'll see you guys back here in just a few.
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