Note the interesting trend: a zero-dimensional distribution (a point) produces a field that drops off as r-2, while a one-dimensional distribution (a line) produces a field that drops off as r-1. We have to be tempted to see if a two-dimensional distribution (a plane) will give us a field that drops off as r0 -- i.e. which does not drop off at all with the distance from the plane, but remains constant throughout space. This application of GAUSS' LAW is a straightforward analogy to the other two, and can be worked out easily by the reader. ;-)