hello class professor anderson here let's talk about gauss's law in a little more detail so how do we write gauss's law mathematically the way we write it is the following integral of e dot d a equals q enclosed over epsilon not now this maybe is introducing some math that you haven't seen before but let's talk you through it integral we understand how to do an integral what does this little closed circle mean that means it is a closed surface okay so you need a closed surface for this particular integral that surface could be a sphere it could be a cube it could be a cylindrical surface but it has to be a closed surface e of course is the electric field but it is the electric field at the surface of that closed surface okay dot product we know what that is okay and we know how to deal with that mathematically d a is a surface area element for that particular closed surface so we need to dot the electric field at the surface with the little surface area element and then we have to integrate over the entire closed surface what about the stuff on the right side q enclosed that is our good old charge enclosed within that surface so if it's a closed surface it's everything inside of it so for a sphere it would be all the charges that are inside that volume of the sphere if it's a cylinder it would be inside the cylinder what about epsilon nut that is our good old permittivity of free space okay so this is gauss's law in mathematical form what is the goal of applying gauss's law the goal is the following we want to one exploit the symmetry of the charge distribution what that means is if it's a point charge we want to use spherical symmetry if it's a line charge we want to use cylindrical symmetry and if it's a planar charge we want to use cartesian symmetry okay the second thing is once once we exploit that symmetry we want to stick in the e field at the surface and dot it with the d a such that the dot product disappears now what do we mean by that what we mean is that e and d a have to be either parallel or perpendicular if they are parallel then e dotted with d a remember dot product we would become e d a cosine theta theta would be zero and so that cosine term goes away just becomes one or if they are perpendicular to each other then e dotted with da we'd get a cosine of 90 degrees and that term would be 0. so either way it's going to simplify our math okay and then we use gauss's law to calculate now i want you to look at what i wrote there i said we use gauss's law to calculate e but i didn't put a vector sign on top of it okay what we are calculating here is the magnitude of the e field gauss's law tells us the magnitude of the e field but it doesn't tell us the direction of the e field the direction of the e field has to come from the symmetry of the problem back here in step one and when we do a couple examples you see exactly what i mean by that okay so this is sort of the goal of gauss's law use symmetry to simplify this equation such that you can calculate an e field in magnitude all right and let's see how that works