Potential energy of a system of 3 charges

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 and we're just going to call q1 as my first charge let me first bring q1 from infinity and place it over here and i'm going to ask myself how much work i did over there there'll be some work done let's 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 would 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 represent the total potential energy of this 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 uh 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 going to give it an push in the opposite direction exactly the you know 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 does 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 chart so let me name the first one and let's look at this one what do you think do you think i 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 in the 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 going to do that we have a faster way of doing this because we've already done all the hard work in 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 k q by r so k into q that's this charge divided by r the resistance r 1 2. now you may ask why why why am i talking about potential over here because remember potential this number represents how much work uh 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 being 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 you have to do well it's going to 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 chart so let me doing this let me bring this now again one way is i can calculate the work done using the integral of force and distance i don't want to do that and we can use potential concepts and again i want you to pause the video and think about what with the work done w3 in moving q3 from inferior to this point all right again we can use the concept of potentials if i know 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 will 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's 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 k q by r so k q 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 k q2 divided by this distance that's r23 okay and this now what does this number represent that's the workman bringing one coulomb from infinity to this point so what is the work done 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 and we're done we've done all the hard work literally we did 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 k q1 q2 by r12 i'll add the color a little bit later okay plus this would be k q 1 q 3 by r 1 3 plus k q 2 q 3 by r 2 3 and there we have it that's our expression for the total potential energy now if you look at this expression it's something something very beautiful has come out if you look at the first term k q 1 q 2 by r 1 2 that's actually the potential energy of these two charges alone so let me mark that so this is u 1 2 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 k q one q three by r one three is the potential energy of these two charges alone so this would be 1 u 1 3 this one so it is this one and if you look at this one q 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 you could have more charges we could just keep on increasing that so just consider each pair add 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