Alright, guys, welcome back. We're going to work out this example together. We're supposed to figure out what the surface charge density is on the inner and outer surfaces. Now, before you watch this, just make sure that you've seen the last video on how to figure out what the electric fields are at different points throughout this conducting surface. Because you need to know what happens with the charges that are set up on the inside walls, things like that. Just make sure you watch the last video and you're good to go. So we do know from the last video that due to Gauss's Law, there's an electric field inside of here. The electric field is zero within the conductor, and that exists again on the outside. And this is because whatever charge exists over here gets set up and basically distributed around the inner walls of the inside of this conductor.
Here, we know that if this is three Coulombs, then there has to be a negative three Coulombs that gets set up on the inner walls. And then on top of that, there has to be a positive three Coulombs that gets set up on the outer walls because of charge conservation, right? So I'm just going to use plus signs everywhere. So basically what we're supposed to do is figure out what the surface charge density is on these inner surfaces. Now, that surface charge density is given by the letter Sigma and that Sigma has an equation. It's the total amount of charge divided by the total amount of area.
I want to be very careful here because this Q is not the enclosed charge, like what we're using in Gaussian surfaces. So Q is not the Q enclosed. What we're basically doing is we're just figuring out, "Okay, if all of these charges here get distributed on the surface of this tiny little thin shell right here on the inside walls of this conductor, what is the total amount of charge divided by the total amount of area here?" So just don't make the mistake and say that this is the total amount of enclosed charge. That's different.
Σ = Q A , where Q is the total amount of charge and A is the area.
Sigma on the inside wall is going to be the total amount of charge that's on the inside wall, in other words, the negative three Coulombs, divided by the area of the inside wall. Now, the area of the inside wall here is just equal to 4 π R 2 inside squared, and we actually have what the surfaces are. We know this is three centimeters, and this is five centimeters on the outer wall. In other words, A _ inside is just 4 π , and now we have to convert that; we have to square that. So the area of the inside wall is going to be 1.13 × 10 - 2 in square meters.
So basically, I can go ahead and just plug that in here and figure out what the surface charge density is. So Sigma on the inside is going to be Q, which is negative three Coulombs, divided by 1.13 × 10 - 2 , and my sigma inside equals, let's see, negative 265 Coulombs per meter squared, right? So that's basically how you figure out what the surface charge density of the inside wall is.
To do the same thing on the outside wall is just going to be the exact same process. So, to figure out what Sigma out is, that's just going to be Q divided by A of the outside wall. So this A _ outside is just going to be 4 π , and then now we just use instead of using three centimeters, we're just going to use five centimeters. I'm going to plug that in. So I've got 0.5, and then we're going to square that. So that means that this outside area is just equal to 3.14 × 10 - 2 , and that's just going to go straight in this equation right here.
Okay, so that means that the outside surface charge density is just the let's see, we've got three Coulombs distribute on the outside divided by the area, which is 3.14 × 10 - 2 , and that is equal to 95 Coulombs per meter squared. Notice how this surface charge density ends up being positive, and this one ends up being negative because that actually has to do with the amount of charge that's divided by this area here. So those charges actually could be negative or positive. Alright, let me know if you guys have any questions with this, I'll see you in the next one.