A straight circular plastic cylinder of length L and radius R (where
R ≪ L)
is irradiated with a beam of protons so that there is a total excess charge Q distributed uniformly throughout the cylinder. Find the electric field inside the cylinder, a distance r from the center of the cylinder far from the ends, where r < R.
Gauss's Law where ∫Edl = q/e0 where q is the charge inside a certain area. Volume of sphere: 4/3πr^3 and volume of a cylinder: 2πr^2L with r being radius of cylinder and L being the length of the cylinder.
The Attempt at a Solution
I decided to make my gaussian surface a sphere which was located inside the cylinder. The charge inside that sphere would then be Q(Asphere/Acylinder). I then divided that by e0. I set that equal to the electric field multiplied by the area of the sphere. Like it is supposed to - the surface area of the spheres cancel leaving me with E = Q/(e0*Acylinder). The surface area of the cylinder is given as 2piRL + 2piR^2 so my final answer is Q/(e0*(2piRL + 2piR^2)) but apparently that is not correct. I have a feeling I messed up my charge distribution somehow because I believe that I have to find the charge inside the sphere as a function of volume instead of surface area.