I've noticed a pattern with the electric fields of charged objects of infinite dimensions.
A point charge, which can be considered a charge of 0 dimensions, has an electric field that goes as $r^{-2}$. An infinite line of charge, a 1 dimensional object, has an electric field that goes as $r^{-1}$. The infinite plane (2D) has a constant electric field, or a field proportional to $r^0$. And if one considers a sphere of electrical charge, you can make a Gaussian sphere inside it, and this electric field should grow as $r^1$.
I'm trying to figure out an explanation for this pattern. The original question that led me to consider this was the difference between the infinite line and the infinite plane: why does one depend on r but not the other? It's addressed briefly in Griffiths E&M, where I believe he says something along the lines of "you can't get away from an infinite plane". But by that reasoning, I don't understand how you can "get away" from an infinite line, either. The reasoning make some sense if you consider an infinite plank, a plane that is infinite in one dimension and finite in another, so that as you get further away, one dimension "shrinks" while the other continues to "appear the same". However, the "perfect" infinite line should only have one dimension to begin with, so it should "appear the same" no matter how far away or close you are...
I suppose my questions are these: why does this pattern between the dimensions of an object and its field exist? How do you "get away" from an infinite line?