Positive electric charge is distributed in the shape of an infinitely long insulating cylinder of radius R and volumetric charge density ?=?0(1-r/R).
(a) Find an expression for the linear charge density
(b) What two things does symmetry tell you about the electric field due to this charge cylinder in all regions where the field is not zero?
(c) Use Gauss's Law along with geometric properties of the electric field based on symmetry to find the electric field E(r) inside and outside the cylinder as a function of distance r from the cylinder's axis. Be sure to draw and label carefully the Gaussian bubble you use, and show how the flux integral evaluates on various surfaces of this bubble. Express your results first in terms of ? and then in terms of ? .
(d) Draw a graph of E(r) vs. r from r=0 to r=3R. How does your result compare with E(r)=2k?/r=?/2pi? for a long 1d line of charge?
(e) Draw electric field lines whereever the electric field is not zero. Show them on Side and End views of the cylinder.