Electric Field of a Finite Cylinder

[wxguy28]

Homework Statement

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.

Homework Equations

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

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.

 

[wxguy28]

I have found the E-field for a disk of charge, that being (2∏ρ/ε₀)(1-[r/√r²+R²])

However there is no height dependence here. Is it has simple as integrating over some dh term from h1 to h2?

 

[skimpyman]

God damn it, I am dealing with the same problem.

 

[haruspex]

I have found the E-field for a disk of charge, that being (2∏ρ/ε₀)(1-[r/√r²+R²])

However there is no height dependence here. Is it has simple as integrating over some dh term from h1 to h2?

Yes, but be careful with r. How are you defining that?

Do you want the field inside the slab, outside the slab, or both?