Main Question or Discussion Point
Can anyone explain this to me?
there are a lot of different scaler functions of that distance.First you must think what are the possible vector fields caused by a single point source (charge).
All information you have is the location of 2 points: the source and the point where you want to evaluate the vector. The problem has cylindric symmetry around the axis connecting the two points, so vector MUST be directed parallel to this axis: either straigth away or towards the source.
The magnitude of the vector must be independent of the choise of coordinate system, so it can only depend on the distance between the two points..
the reason why that surface integral evaluates to a constant that is proportional to the enclosed point charge is because the magnitude of that field vector is inversly proportional to the square of the distance and proportional to the charge. why it's proportional to the amount of charge is because if you had two different charges sitting virtually next to each other, the total force you get from both should come from adding the force you get due to each charge. why it's inverse-square with distance is because of this concept of flux and flux density and where we model the field as being proportional to the flux density (or, with judicious choice of units, the same as flux density). and because the surface area of a sphere is [itex]4 \pi r^2[/itex].Under this circumstances the integral of the electric field over a sphere is not difficult to evaluate.