 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 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 options 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. The goal is to figure out which one of this is going to be and why. The why part is also super important. And before you start thinking, let me give you a little bit of a direction as to how we could even start thinking about it. Well, we can start by using something that we already know about conductors in electrostatics. 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'll talk 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 because there are so many, maybe the electric field just cancel out everywhere. How do I know if electric fields exist or not? Well, to calculate electric field from charges, one of the things that we can use is Coulomb's law. But it gets so messy because Coulomb's law 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 even 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, 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 part of the conductor. And I'm just gonna 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. Therefore, this is a 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 zero, I wouldn't have gotten any flux. And therefore, I know now there is electric field is present. And therefore, this is a 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 wouldn't be any electric field. How do I check? Well, again, choose a Gaussian surface. I'm gonna 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's 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 leaves us with 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 electric field 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 it a little deeper. We'll have to do some more analysis. Something we'll do in future videos.