[Music] want to talk a little bit about how ga's law and our concepts of potential that we're developing here related to each other and also start to kind of develop in your minds an intuition for for what potential looks like when I'm around a charge so let me start off with just point charge again and what I'm going to do is is draw a gaussian surface around it sort of a a shell that doesn't really exist it's in my imagination but the reason I'm doing that is cuz on that shell I know that the electric field everywhere is going to be equal to the same thing it's spherically symmetric so I'm going to just to to visualize better I'm going to cut through the middle of this and kind of turn sideways and take a cross-sectional slice if you will so I've got the the charge sitting there in the middle and the dotted ring there represents that sphere that I've cut into everywhere on that of radius R the field the same so the flux is going to be the field times the area 4 Pi r^ 2 for a sphere and it's equal to the charge en Clos so from that I can solve for e subr is equal to KQ over r^2 or Q over 4 Pi Epsilon KN r^ 2 whatever your your personal flavor is and if I take this and integrate from Infinity into R I can figure out what the potential is Q over 4 Pi epsilonr we did this on another lesson so I'm not not going to repeat it entirely I just wanted to walk you through it again to bring up another idea here what if I take this shell this uh this this uh Point charge and I'm going to plot the the the potential for it just to show you what it looks like it goes as one over R so it's a real sharp Spike there that goes to Infinity but I'm going to take it now and I'm going to kind of blow it up put it onto a shell instead of being at a single point Shell's got a radius R so I can do ga's law all over again for a a gaussian surface the sphere that's outside that that red shell of charge and everything's the same about it Q is V is still going to be Q over 4 Pi epsilonr I haven't changed anything about the potential because I've spread the charge out over a surface now on the other hand though if I'm inside the sphere if I take my gussian service and bring it inside notice that things are a little different in here because here there's no charge inside my gussian sphere inside that black dotted line so therefore the field in there is zero there's no field inside this shell which means when I integrate the potential I integrate up to the surface from Infinity to R Big R and then I have to integrate inside from Big R to little r but that integral is kind of meaningless because there's no field in there so what I get is the V inside is equal to Q over 4 Pi Epsilon Big R notice that's all constant so what I've got then is a potential that rises up to the surface of the sphere and then is flat throughout it so the potential is constant inside this sphere let me take this one step further what if I take this sphere and instead of making it a hollow sphere I'm going to make this a solid metal ball and I'm going to put the charge all over in inside this thing well what's going to happen is those charges are all going to repel each other so they're all going to force themselves to the outside edge so what I'll have is a sphere where the charge has been shoved outward to the the surface in other words it winds up being effectively mathematically the same thing that I just had walk through it all again and what I've got is the potential outside is going to be Q over 4 Pi Epsilon R and the potential inside is going to be Q over 4 Pi Epsilon Big R so in other words the potential inside this sphere is still a constant and there's a couple thoughts about this I want you to have in mind first off take a look at that graph there realize that it drops away as I go to Infinity but it also Rises up to where the charge is and then is flat across the middle where the field is zero the relationship between field and potential is something I'm going kind of come back to in in another lesson or two here but I just want to have that sort of in your sights right now the other thing about this is if e is zero inside the conductor what that means V is constant inside there once the charge all kind of ran to the surfaces the electric field went away but because of that it developed a potential inside there that's a constant just like it was for the hollow ball of charge well let's go a step or two further with this basically what we're saying here is if I have a conductor all the charge is going to go to the surface of the conductor so it's all going to flee to the outside edges because all those positive charges dislike each other so much the V on the surface and the E on the surface are things we know now Q over 4 Pi Epsilon KN big r^ 2 for e and Q over 4 Pi epsilonr for the V so there's a relationship at the surface of this sphere between the potential the voltage in the electric field the electric field is equal to V over R which means that the smaller the spere is the larger the field will be it's going to if if I hold this thing at a constant voltage basically what happens is if I make it smaller and smaller the electric field will be larger and larger and what this means is a sharp corner is going to have a very large electric field which is why sparks jump from sharp sharp Corners it's actually the nature of how lightning rods work in any case the point I want you walking away with here is that the relationship between field and potential is something we're getting close to to standing a little bit better uh and that in this particular case of a spherical charge there's a a relationship between the field strength and the curvature the radius of that that sphere that means that the field is going to get larger when the curvature is tighter when the radius is smaller