Work to Bring Two Charges From Infinity

by Patrick Ford
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Alright, guys. So let's work this one out together. This is actually a very common type of problem that you might see. It might pop up in homework or some tests or stuff like that. So we're basically asked to find how much work is done by the electric force in bringing a charge from infinitely far away to the origin of coordinate system. So let's go ahead and draw out with that coordinate system might look like. So it's gonna look like this. And basically, the idea is that you have a let's say, what is it? Five Cool in charge that's out here infinitely far away or very, very far away, and you're basically just gonna bring it towards the origin of this coordinate system. How much work does that actually take? What we're talking about? Basically like point charges. So let's go ahead and use our work formula for point charges. You know it's cake, one of her, our initial minus one of our final. The problem is, is that we actually don't have any other charge that's gonna interact with this five Coolum charge. It's basically just like bringing in something when it has nothing to interact with. So in other words, if there's no other charges, then there's gonna be no other forces that are acting on this charge. Which basically means that the work done is just gonna be zero. It doesn't take any work to move around a charge because there's nothing else to basically attract or repel it, which means there's nothing else around it to do work on it. Does that make sense? So basically, the if you just have one single charge and you're moving it around, it takes no amount of work. Now what does happen, though, is you have this second charge this not negative to Coolum charge that's floating away over here. And now, if you wanna bring this in towards a specific point now you have this charge that's at the center of the coordinate system, and you are gonna actually use this formula. So let's go ahead and see how this is gonna work out. So we've got basically, this is 3 m. This is 4 m. You're gonna bring it to a point right over here, in which the distance this our distance is basically gonna be the high pot news of this triangle right here. This is 3 m and this is 4 m. Then hopefully I should recognize this as a 345 triangle. That's gonna be 5 m over here. And by the way, this is actually gonna be the final distance once we take this charge from infinitely far away and then move it towards that specific location. So now the work done. So this is Let's let's say like this is gonna be with work done from the five Coolum charge. Now, the work that's done by the electric force on the negative to cool um, charge is gonna be K Q one. Start Que que que? So that's gonna be a big queue. Little Q one over our initial minus one over our final. Okay, so that's gonna be 8. 99 times 10 to the ninth. And now we've got the queue. The queue is just basically going to be that that one charge that that the center of the coordinate system, which is five cool homes Now we've actually have to incorporate the negative sign negative to cool homes. And now here's where we have to basically evaluate what are initial and final distances are Let's see, when you are initially infinitely far away So infinitely far away means that this initial distance right here is equal to infinity or just gonna be a really, really huge number. And what that means is that basically, this whole entire thing goes to zero because again, this our initial is just that infinity symbol. It just means it's a really, really, really big number. And then you have this minus one of our final in which that is equal to 5 m. So let's just go ahead and plug that in. So we're basically just gonna have zero for that first term and they're gonna have minus one over, and that's gonna be 5 m. So now we can go ahead and just plug this in, and we have the work that's done by the electric force because that's actually what we're trying to find here. How much work is done by the electric force. And that's just gonna equal. Let's see, I've got I've got positive 1.8 times, 10 to the 10th, and that's in jewels. Now, what I want to do is talk about basically the sign here because we actually got a positive sign, which means that this is work done by the electric force. And one way to think about this is being positive is the fact that if you were to just leave these things, you believe these two charges their natural state would be to try to attract one another because you have a positive and a negative charge you have, unlike charges. So basically, what happens is if you have this positive charge, it's gonna attract this negative charge from infinitely far away. And the fact the electric field, the fact that the electric force is gonna point in this direction and that the displacement is also gonna point in that direction means that you should have positive work that's done. If these were things were negative than you would have negative work because you actually have to force this charge in closer to the stationary charge, and that would actually be a negative work. That's one way to think about this. Let me know if you guys have any questions with this