 Suppose we'd have a very thin conducting shell of some radius R and let's say it is charged uniformly with some charge capital Q. The question we wanna try and solve in this video is find out what the electric field looks like everywhere. We will start by taking a random point outside first, then we'll look at what happens inside, but that's the goal. Figure out what the electric field looks like everywhere and start with outside at some point P. How do we calculate this? Well, the first thing that comes to my mind is I ask myself, I know the charge. Given the charge, how do I calculate electric field? Is there anything that I know about it? Well, yeah, we have studied Coulomb's law. Coulomb's law tells me that if there is a charge Q, then the electric field at some distance R is gonna be this formula, one by four pi epsilon naught Q divided by R squared. So can I just go ahead and substitute and solve this? No. And I want you to think a little bit about this. Why can't I use Coulomb's law directly and solve it in one step? Why can't I do that? Can you pause and think about it? Well, if you've tried, the main reason is because Coulomb's law only works when we're dealing with point charges. What does that mean, you mean? What does that mean? Well, what that means is if you have a charge which is so tiny, the charge is so tiny compared to the distance where you're calculating the electric field, then you can use this. You can say this is a point charge. But in our case, notice the charge is not, it's not just concentrated at one place. It's distributed over this sphere. And so it's not a small negligible sized charge. Now we're dealing with a charge which has a spherical distribution of radius R, right? So we cannot use Coulomb's law directly. I mean, think about it. What would the R over here represent? Would it be distance from this charge? Would it be distance from this charge? R is the distance from charge, right? Now if the charge is distributed, how do you figure out what R is? You can't. And so we cannot directly use Coulomb's law. So what do we do? Well, one way to solve this, using Coulomb's law, I mean, you can still solve it, but you can't do it directly. But one way to do that is you take this charge distribution and divide it into tiny, tiny pieces. What I mean is you can first calculate the electric field over here only due to discharge. Neglect everything else. Only due to discharge, calculate what the electric field is. A tiny charge over here. Then you calculate the electric field only due to discharge. Then you calculate the electric field only due to discharge and so on and so forth. You do that over the entire sphere, add them all up and you will get your electric field. Now does that sound tedious? Yes, it's gonna be tedious. There'll be a lot of integration involved in that, but it can be done. You can use Coulomb's law in all situations and calculate such problems. It can be done, but it's very tedious. And that's the thing about this. That's the thing I wanna discuss in this video. Is there a way to solve this problem without doing complicated integrals? The answer is yes. We have to get rid of Coulomb's law. Coulomb's law is not gonna be helpful for us. Let's get rid of the Coulomb's law. Yeah, Coulomb's law is not gonna be helpful, but you know what will be helpful for us? Gauss's law. Remember Gauss's law, which tells that if you take a closed surface, the flux through it will always equal total charge inside that surface divided by epsilon naught. And of course, if you're not familiar with this, we have discussed this in great detail in a previous video. You can go back and practice as well, but we can use Gauss's law to solve it over here. Now, looking at Gauss's law, you might say, hey, how can I use Gauss's law? Even there also, there is an integral, right? Isn't this also gonna be complicated? The whole point of using Gauss's law, and this will be true in any problem that we use Gauss's law, the whole point is you choose a closed surface such that the electric field everywhere on that surface is gonna be the same, at least in most cases. It's gonna be the same. Then that electric field can be pulled out of this integral, as we will see. And then the mathematics becomes very simple. You don't even actually have to integrate. That's the whole idea. So you can't do this for all cases. Only in certain special cases, you can choose such a closed surface. And we will see this is one of those special cases. And so the whole idea is, in such special cases, let's not use Coulomb's law. Let's use Gauss's law to solve the electric field. Okay, so let's begin. I'm pretty sure you're excited. So the first step would be to choose a closed surface, right? What kind of closed surface would I choose over here? Before answering that question, we need to first look at what the electric field looks like everywhere. Only then we can choose an appropriate closed surface. So let me get rid of this equation for a while and think a little bit about what would the electric field look like everywhere? Outside. Inside we'll come back a little bit later, but think about outside. Given that it's a spherical distribution, it's a spherically symmetric, meaning the charges are uniformly distributed, because of that, we will find that the electric field everywhere has to be along the radius. It's gonna be a radial field. And we will find that the field at a distance r, at any point, if you go any point at a distance r, you will get the field value to be exactly the same. So radial field whose value only depends upon the distance from the center. But of course, the question could come into your mind. How do I know this? Why should it be radial? And that's really important. I want to ask a lot of why questions over here, because a lot of details involved. So conceptually, this is very, very enriching. So if you had to convince this to someone, how would you convince them? Why would the field be radial? In fact, guess what? I want you to explore and interact with this derivation. So let's continue this in our article on our website, where you get to pause and wonder. If you're stuck, you can always get the solution. You get to try each step yourself. It'll be kind of rediscovering the whole derivation yourself. So it won't be just me talking and telling you what to do like in a normal video fashion. Yes, of course, this will take a little longer, but I believe your learning would be more solid and fun. And of course, if you're watching this on YouTube, then here's a link to continue on to our free website. So the next step would be for you to come up with an argument as to why this field is radial, should be radial.