Electric Field of a Finite Cylinder

1. The problem statement, all variables and given/known data
Derive expressions for electric field produced along the axis of radial symmetry for an H km thick cylindrical slab of radius R with charge distributed around the volume. Then, give the electric field on the vertical axis for four of these cylindrical slabs.

2. Relevant equations
Obviously start with Coloumb’s Law (q/4*pi*ε0*r2). Must integrate from there.

3. The attempt at a solution
As this isn’t for an infinite cylinder, we can’t use a Gaussian surface. Knowing that q = ρV where rho is the charge density and V = ∏R2, I’ve come up with:

ρ/4ε0 ∫∫∫ R2h2/r2

However, I’m not sure how to integrate through the heights of the cylinder in the case if the charge is not found on the axis. I know this is a vague attempt at the answer up to this point, but I’m honestly just not sure how to do the height part of the integration. Any help is appreciated.


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