Electric force inside a cylinder(my friends and I can't solve this

[catsonmars]

Electric force inside a cylinder(my friends and I can't solve this:(

Homework Statement

A right circular cylinder has radius R and length L, and a nonuniform volume charge density ρ($\hat{r}$). If the z axis is chosen to coincide with the axis of the cylinder, the charge density is ρ($\hat{r}$) = ρ(z$\hat{z}$) = ρ(o)+βz This means that the charge density varies linearly along the length of the cylinder. Find the force on a point charge q placed at the center of the cylinder.

Note ρ(o) is the initial density or "ρ not". Also my picture has been cut off the line on the far right is capital L

Homework Equations

The total force on the charge is the sum of the froces from each charge density. Note that the charge densities are not all equal. The porblem states that if were to look at the charge density as we move along the z axis it would change linearly.

F=∫dF
where dF is

The individual force equations can be written as.
dF=[κ(dq^2)/r(12)^2]$\hat{r}$(12)/r(12)
NOTE:r(12) is read as r subscript 12.

where r is the distance from the charge density to the center charge and is written as

r=(h^2+l^2)^1/2

The charge density is ρ=dq/dV
Solve for dq and substitute in ρ=ρ(o)+βz

dq=(ρ(o)+βz)(dV)

The Attempt at a Solution

Use these equations to solve for dF to find the total force. This is where I start getting confused. First before I begin. The problem makes it sound as if the charge density is exclusively on the z axis, is that correct. If that is the case I am just integrating the density from 0 to L. This feels wrong though. Also this would make my picture wrong! Here is what I have so far.dF=[κ(ρ(o)+βz)(dV)]$\hat{r(12)}$/r^3

substitute in r=(h^2+l^2)^1/2

dF=[κ(ρ(o)+βz)(dV)]$\hat{r(12)}$/(h^2+l^2)^3/2

Now integrate both sides.

∫dF=κ∫[(ρ(o)+βz)(dV)]$\hat{r(12)}$/(h^2+l^2)^3/2

I am integrating over F in the first integral and in the second one I am integrating from 0 to L.
If my picture is right I feel like I am missing two more integrals so I could integrate over the volume. Where am I going wrong or what am I missing?

 

[Simon Bridge]

First before I begin. The problem makes it sound as if the charge density is exclusively on the z axis, is that correct.

No. The problem states that the charge density varies only in the z direction - implies it is uniform in the x-y directions.

You need to slice the cylinder into disks thickness dz.
1st find the expression for the force due to the disk of charge at position z.