Why charges reside on surface of conductors

let's explore how charges get distributed when we add them to a conductor so let's start by looking at a spherical you know imagine this is a metallic sphere which has a hole inside of it so there's a cavity inside and let's say right now this is neutral now imagine i'm going to pull some electrons from inside the metal don't ask me how i do that imagine i just pull them off and as a result because i'm pulling electrons out it's going to leave behind some positive charges so now this metal is this conductor is positively charged and here's the thing these positive charges are all going to start repelling each other and because conductors have free electrons because charges can move freely inside a conductor these charges would want to go as far away from each other as possible you can imagine the way that happens is electrons from somewhere over here get attracted by the positive charge and they start moving as a result they they leave positive charge behind somewhere over here and more electrons can swoop in and so on and so forth which means that these charges are going to start moving these positive charges will keep moving but eventually they will stop right so the question now is when these positive charges have stopped moving in other words we have reached electrostatic condition what would the new arrangement of these charges look like and i have a few options for you to pick from one of the option is these charges distribute themselves throughout the entire conductor you'll find them in the inner surface you'll find them throughout the bulk of the conductor you'll also find them on the outer surface charges will not be inside the cavity because there's nothing inside the cavity you can imagine it's vacuum there's nothing over here so you can't have charges so this is one option one of the ways in which charges could rearrange themselves but another option could be maybe all the charges for some reason they only redistribute themselves on the outer surface of the conductor that's another option or maybe the charges distribute themselves on the outer surface and the inner surface of the conductor and the goal is to figure out which one of this is going to be and why the y part is also super important and before you start thinking let me give a little bit of a direction as to how we could even how could we even start thinking about it or we can start by using something that we already know about conductors in electrostatics uh inside the conductor electric field must always be zero at electrostatic condition why should this be the case because if electric field was not zero everywhere inside the conductor then electrons would move due to that electric field and we would not be in electrostatic condition and we've talked more about this in great detail in our previous videos on electrostatic shielding and faraday cages so if you need more refresher feel free to go back and check that out so what we could do is figure out in which of these cases electric field through the conductor stays zero so why don't you pause the video now and give it a shot all right now even at this point if you had asked me i would still say it's a little hard because i would ask myself how would i know if there's an electric field that exists everywhere for example there are positive charges but but because there are so many maybe the electric fields just cancel out everywhere how do i know if electric fields exist or not well to calculate electric field from up from charges one of the things that we can use is coulomb's law but it gets so messy because only works for one charge and there are so many charges over here but guess what there's another law we can use to think about electric fields remember gauss's law our old friend gauss can come back and help us now gauss's law states that if you consider any closed surface the flux through it the electric flux through it must equal the total charge inside that surface divided by epsilon naught so here's the thing i can just choose a closed surface imaginary surface and check whether there's any charge enclosed if there is an enclosed charge i know that there should be some flux and therefore i know there should be some electric field okay so i can use this law and figure out whether electric field exists in these three options or not so again if you didn't you know if you couldn't do it earlier maybe now would be a good time to see if you can try this on your own all right let's do this so let's look at the first one because i want to check whether there's an electric field somewhere inside the conductor the in the conducting part because that's where the electric field has to be zero let me choose a gaussian surface that goes through this this part of the conductor and i'm just going to choose a nice spherical gaussian surface so here's my nice sphere and now i ask myself is there charge enclosed by this gaussian surface yes there is a charge q inside is positive you can see all the charges are enclosing that which means i do have a positive flux and therefore there must be some electric field there must be electric field everywhere over here and therefore this is the wrong option because from gauss law this tells me that the electric field inside is not zero because there is some flux if electric field everywhere was 0 i wouldn't have gotten any flux and therefore i know now there is electric field is present and therefore this is the wrong option that leaves us with two options again can you pause right now and see which of the two options would be right all right let's look at this one this seems reasonable there are no charges over here so maybe there won't be any electric field how do i check well again choose a gaussian surface i'm going to choose a gaussian surface again just like before and now i ask myself is there charge enclosed by this gaussian surface and the answer is yes the charges on the inner surface are enclosed by this gaussian surface which means again there is a non-zero flux there is a positive flux meaning there must be a non-zero electric field which means even this is wrong i mean so this is so amazing without gauss's law i wouldn't be able to guess this but gauss's law makes it so easy for me i know that this is also wrong so charges can neither be present in the bulk of the conductor nor can they be present in the inner surface of the conductor which uses only one option the charges can only reside on the outer surface of a conductor beautiful isn't it and we can just go ahead and check it now let me draw a gaussian surface if i now draw a gaussian surface is there any charge enclosed no there isn't and so charge enclosed is zero and therefore flux is also zero and because it's symmetrical we can now say electric field everywhere is also zero so this is the only one that satisfies that condition and therefore this is how charges will redistribute themselves so this means whenever we're dealing with conductors and this by the way only works for conductors if this was an insulating material charges would just stay there they would not redistribute because they're not free to move but in conductors charges will always always redistribute themselves on the outer surface because this is the only way electrified inside the conductor can stay zero one last question i have is how would these charges distribute themselves on the outer surface would they be uniformly distributed or maybe because they were initially charged over here maybe there is more charges over here and there could be less charges over here what do you think well because this is a sphere it has spherical symmetry therefore the charges must also be symmetrically distributed and so because of its spherical shape they have to be uniformly uniformly distributed on the outer surface but what if it was not a sphere what if it had some bumps over here what if it had some irregular shape then it won't be uniformly distributed and then if you ask me how would it get distributed again we'll have to dig into a little deeper we have to do some more analysis something we'll do in future videos