How to find electric fields and potentials for systems of point charges

One of the most essential and basic systems we study includes systems of stationary point charges. These bring out the essence of what charges do, which is produce a vector quantity, electric field, and a scalar quantity, electric potential.

We need to remember that finding electric fields includes us forgetting about the sign of a charge when calculating the fields. Being a vector, let the picture tell you whether we are dealing with positive or negative directions of components. But for potentials, which are scalars, we DO need to include the signs of charges numerically since positive charges produce positive voltage, and negative charges produce negative voltage, and we just add up the values.

How to do Gauss's law with NON-uniform charge density inside a non-conducting material

The vast majority of Gauss's law problems we do deal with uniform charge densities for non-conductors/insulators. These are the cases where there is charge inside the material, and therefore an electric field, and our job is to find the electric field inside. When the charge density, rho, is constant/uniform, the classic result is the field is linear with r. That is the result whether we have a sphere or a cylinder, and would be the same for gravity, electric fields, or magnetic fields (using Ampere's law).

But what about NON-uniform charge density, where rho depends on radius, r? What do we do with Gauss's law to find the electric fields inside these type of materials and objects? This video is an example of how to handle it. The gist is we need to set up an integral where we add the charges within skinny, hollow spheres of charge. Each little sphere has its own charge density value, and so the charge of each hollow shell is rho x dV. The trick is the dV = (4*pi*r^2)(dr), at least for a sphere. The dr is the small thickness of the hollow shell. The same idea holds for cylinders, where dV = (2*pi*L*r)(dr). I hope this helps!