 Suppose we have three charges kept like this. Our goal in this video is to figure out what the potential energy of this system is going to be. So what do we even mean by potential energy over here? Well, imagine at the beginning, all these charges were very far away from each other. We could say infinitely far away from each other. Then we can ask ourselves, if I were to bring these charges from infinity to these points, how much work would I have to do? How much work I would have to do? That work done gets stored as potential energy. So I basically have to calculate the work done in assembling these charges from infinity to these points. So let's go ahead and do that. So let me make some space and ask this question. How do I calculate that work done in assembling these charges from infinity? Well, we can do it step by step. Here's what I mean. First, we'll imagine an empty universe. There are no charges kept as of now anywhere. And ask myself, and let's say I bring the first charge. You can take any of these charges as your first charge. I'm just gonna call Q1 as my first charge. Let me first bring Q1 from infinity and place it over here. And I'm gonna ask myself how much work I did over there. There'll be some work done. Let's call that work done as, let me write that over here. Let's call that work done as W1. Then, keeping that charge over there, let's bring in the second charge from infinity to this point. Again, I'll have to do some amount of work. Let's call that work done W2. Let me bring this down a little bit. Okay. And then finally, we'll now have two charges kept over there. And then let's think about bringing the third charge in. And then the work done in bringing the third charge will be W3. And this now represents the total work done in assembling the charges. The beauty of electric fields is it doesn't matter how you do that work. It is independent of the path that you choose to do that work. You could have brought this charge first. You could have brought these three charges together. You could have done any ways you want. The total work done would not change. And that's why we choose a path which is the easiest for us, bringing one charge at a time. And so that this total work done would now represents the total potential energy of the system. So now we have to figure out what is the total work done? So let's do that. Let's focus on the first one. So let me dim the other two. All right, so let's start with the first one. How much work would I have to do in moving the charge Q1 from infinity to this point? Remember, there are no other charges in this universe. So can you pause the video right now and think about how much work W1 I would have to do? All right, because there are no other charges in this universe, nobody is attracting or repelling my charge Q1. And as a result, I would have to do zero work. Now at first, this was really hard for me to digest. I used to ask myself, I'm bringing the charge from infinity to this point. I have to make it move, right? So shouldn't there be some work done? Well, think of it this way. Imagine that when it's at infinity, I give it a very slight, very tiny push. And as a result of that push, the charge starts moving. So I did a very tiny positive work in the beginning. And then finally, when the charge comes to this point, I'm gonna give it a push in the opposite direction, exactly the same amount of push in the opposite direction to stop it. And in doing so, I did a little bit of negative work. And so the total work done becomes zero. It's tiny positive, tiny negative. And so in the entire journey, I didn't do like network done was zero. So is that, hopefully that helps, that convinces me that, yeah, indeed I'll have to do exactly zero amount of work. So the work done in bringing the first charge is zero. All right, now let's think about the work done in bringing the second charge. So let me name the first one and let's look at this one. What do you think? Do you think I'll have to do some work over here? Well, yeah. If you imagine that Q1 and Q2 are both, let's say positive, just to keep things simple, then you can imagine as I bring the Q2, oh, I'm being repelled by Q1. And over here, we'll imagine that Q1 is fixed in place. Somehow we have nailed it somewhere, okay? I know it's somewhere in space, but somehow we've nailed it. And as I bring Q2, Q1 is going to repel me. So I have to overcome that repulsion. And so clearly I have to do some work. The question now is, how much work do I have to do? So again, can you pause the video and think about this? All right, one way to answer this question is to go back to the definition of work. Work done is equal to force times distance, but then we see that the force keeps changing. As I come closer, force becomes larger and then I have to do an integral. Oh no, I'm not gonna do that. We have a faster way of doing this because we've already done all the hard work in the previous videos. So if you remember, we can bring back the concept of potential. We know how to calculate potential at any point. So let's calculate the potential at this point due to this charge, because this charge is placed, this charge is not yet placed over here. So what is the potential at this point? Let me call that point V2. What is the potential due to a point charge? We know the formula. It is KQ by R. So K into Q, that's this charge, divided by R, the distance R12. Now you may ask, why am I talking about potential over here? Because remember, potential, this number, represents how much work I have to do in moving one coulomb from infinity to this point. That's the meaning of potential. So if this number is 10, then in bringing one coulomb from infinity to this point, I have to do 10 joules of work, which means I know how much work I have to do in moving one coulomb. So now the question is, how much work I have to do in bringing Q2 coulombs of charge? This is for one coulomb. So for Q2 coulombs, how much work do I have to do? Well, it's gonna be Q2 times this number. Therefore, potentials are so important. So the work done in moving this charge from infinity to this point would be Q2. Let me write that over here. Yeah, Q2 times this number. So let me just copy paste that. So I'm just gonna copy this and paste it over here. Q2 times that number. Let me put a bracket over here. So that's that. All right, now let's talk about the work done in moving the third and the final charge. So let me do this and let me bring this now. Again, one way is I can calculate the work done using an integral of force and distance. I don't wanna do that. And we can use potential concepts. And again, I want you to pause the video and think about what would be the work done, W3 in moving Q3 from infinity to this point. All right, again, we can use the concept of potentials. If I can calculate what the potential at this point is, that represents the work done in bringing one coulomb of charge. Then I just multiply by Q3 and that'll be the total work done. So what is the potential at this point? Let me call that potential as V3. What would that potential be? Well, that would be the potential due to these two charges at this point. And we can, that'll be the potential due to this charge at this point plus the potential due to this charge at this point. So the potential due to this charge at this point is going to be, again, KQ by R. So KQ1, and the distance is R13. So you just have to be mindful of which charge you're looking at and what's the distance. And then there'll be potential due to this one, will be Q2, KQ2, divided by this distance, that's R23. Okay, and this now, what does this number represent? That's the work done in bringing one coulomb from infinity to this point. So what is the work done in bringing Q3 coulombs from infinity to this point? It'll be Q3 times this number. So this is going to be plus Q3 times this number. So again, let me copy this whole thing, copy and paste it over here. And there we go. That's our total work done. We're done, we've done all the hard work. Literally, we do all the hard work. And now we just have to simplify this. So let me make some space. All right, so if we now simplify, we get the total potential energy will be KQ1, Q2 by R12. I'll add the color a little bit later, okay? Plus this would be KQ1, Q3 by R13 plus KQ2, Q3 by R23. And there we have it. That's our expression for the total potential energy. Now if you look at this expression, it's something very beautiful has come out. If you look at the first term, KQ1, Q2 by R12, that's actually the potential energy of these two charges alone. So let me mark that. So this is U12, potential energy of just these two charges alone. Similarly, so this one, okay? This one, sorry, this one. Okay, similarly, if you look at these two, KQ1, Q3 by R13 is the potential energy of these two charges alone. Ooh, so this would be one U13, this one. So it is this one. And if you look at this one, Q2, Q3, that is the potential energy that this expression is the potential energy of the system of these two charges alone. Potential energy of system of these two charges alone. So what's interesting is that the total potential energy is the sum of the potential energy of two charges taken in pairs. And if we could have more charges, we could just keep on increasing that. So just consider each pair at their potential energy and then sum them up, that becomes your total potential energy. Beautiful, isn't it? Of course, this is a very neat way to remember this potential energy formula, but if you ever forget any of this, we go back to our basics and from basics we will be able to always derive it.