So a reminder about how you get the electric field given a potential or potential function as the case may be, The electric field is defined to be the gradient of the potential with a minus sign in front of it, reminder of what a gradient is a gradient is in Cartesian coordinates will write that out. Uh The partial of some function with respect to X. For an X. Component. The same for the why and the same for the Z. Now we can simplify this. If we know that we just have one component, we can say something like E C. For example, the Z component of the electric field is minus the derivative of the potential with respect to the spatial coordinates. Z. So as an example of this, we have a potential function and it has shown in the graph, there is a portion that is quadratic and then parts that are linear. And just knowing that the electric field is the slope, the minus sign in front of it minus the slope of potential. It's probably good to go ahead and find the slope of the linear portions. And we see that in this case it's dropping on one side, in the positive sea region. It is going down with a slope of minus 10 volts per meter. And on the other side it is increasing with a positive slope of 10 volts per meter. And so if we demarcate our regions as one on the left, two in the middle and three on the right, We can quickly figure out in region one that easy is minus 10 volts per meter and we'll go ahead and figure out region too. Um but there we do need a derivative easy is minus um The derivative of minus five Z squared would be minus tendency. And that would mean that the uh constant there has to have units of volts per meter squared. Um So yeah, that gets a little tricky, but we'll assume that it has overall units of volts per meter. Yeah, but yeah, that constant has to have Some interesting units. And then in three our electric field in the Z direction is equal to a positive 10 volts per meter. Okay, now we are given a clue that this is a slab. And if you want to think about the way slabs or plates work, electric field is uniform outside a slab or a plate. The only difference is a plate is usually a conductor with a surface charge. Um whereas a slab has a finite thickness and some sort of perhaps volume charge density. So we can see we have the makings of a slab where our electric field is uniform and this is a slab that's we're going to pretend this infinite. So we'll pretend it has no edges, no real edges. It goes on forever. Okay, that's hard to draw. Hard to draw infinity. Anyway, let's make a Z axis. This is a slab and RC access points up and usually they like to center a slab With its equals to zero and going upwards from there. Yeah. Um but we see that what the electric field would look like is above the slab. It would be uniform and positive. This is Region one. And I should have used green arrows on my pardon? My bad. Um Yeah, we'll go ahead and use green arrows for the region one. Actually this is region three. I beg your pardon? It's above the slab with Z greater than we'll say greater than one. There's a where is it equal to one? Probably right at the border of the slab. But here we see that the slab extends from one up above two minus one below. Okay, and the electric field and goes uniform with the same strength down below. Um Now what does the electric field do in between Is it starts at zero at the origin and it grows proportionately as you get away from the origin. And to understand that we probably want to look at Galaxies law. So the second piece of the puzzle, his gases law, a reminder of how gases all works is it says that the electric flux through a closed surface is proportional to the enclosed charge. Now you almost never do the integral on the left and seldom even anything having to do with the integral on the right. Um For a slab we expect the electric field through a surface perpendicular to the electric field to remain constant. And let me kind of blow up the slab and show such a surface. So let's let's take a microscope magnifying glass and look inside the slab. Um What we basically have Is this the positions equal zero Where the electric field is zero. And I am going to draw a little calcium surface. So I'll draw it in blue. And this is an imaginary surface which uh is typically used to try to imagine an enclosed charge. Um The electric field. Uh huh. He was red again inside that slab. So the electric field should point perpendicular to that surface and there should be nothing going into the bottom bottom has zero electric field at that point. Um And so what we can say is the electric field Z component times the area of that surface has to equal the enclosed charge inside of that little surface. Um And if we are assured that there is a uniform density, uh sorry, I forgot to include the absolute zero. We'll do that. But if we are assuming that there is a constant uniform density, we can say that the enclosed charge is the density times the volume of that little surface. And yes, we do have to divide by absolutely not. But the volume of that little surface, it's a cylinder with area A. For the face, which is pi r squared and then Z for the length. Okay, so um the electric field then times the area is equal to row a. See And and don't forget the absolute or not. Um and fortunately that arbitrary area cancels, but the Z is not arbitrary. What this tells us is that the electric field inside the slab is a function of Z. And the constant in front of Z is rho times epsilon? Not. Yes. Now, up here we found that the constant was 10. After doing the derivative, we found that that constant was equal to 10. So by looking and compare to our region to, we found E C. Was 10 times see with 10 having units. Yes, it did have units. We won't worry too much, but that tells us that we must conclude that row over epsilon Is equal to 10. Yeah. And we could put in numbers absolute knots. Just a reminder is 8.85 times 10 to the -12. And it has some units Coolum squared over newtons metre squared. So yes, it does have some units. And we can see that Columns will come in there, that the 10 is not unit list either. So as long as we're using things in S. I. Units, the rose should have units of columns per cubic meter. Yeah,