Electric field (at any point) due to a finite cylinder

[crazygirl89]

Homework Statement

I'm not even attempting the graph yet, but I'm having trouble figuring out how to do this problem for a finite cylinder. All I've found in my notes is finite spheres and infinite cylinders.

Homework Equations

E=∫[ρdv]/4∏εR²] $\hat{R}$

The Attempt at a Solution

Ok here's what I've got so far:

E=(1C/(m^3))/(4∏ε) * ∫(ρ,∅,z)/[((1cm)^2)*√(ρ²+∅²+z²)]

Here's a few of my random thoughts on why this isn't done yet:
Is ∅ even needed in the integral? shouldn't ρ and z cover any point inside or outside of the cylinder?
Where the heck does the length l of the cylinder come in?
And finally, the dreaded graph: I think I could figure this one out if ∅ isn't relevant, but if it is, can someone describe how I'd draw that on the graph?


Thanks for all your help/time guys. I searched around on the forums and the net but I could only find people needing help with infinitely ling cylinders or needing the field at a point on the z-axis, not anywhere.

 

[ShayanJ]

The volume element in cylindrical coordinates is $\rho d\rho d\phi dz$ and the distance is calculated by $d^2=\rho^2+z^2$.

 

[crazygirl89]

ok soo..

Ok so this is what I've got now, let me know if I'm understanding you correctly:

E=[(1C/m³)(1cm)] / [4πε(1cm²)] * ∫(ρ,∅,z)/(ρ²+z²) *dρd∅dz

Is that correct? And do I still need the (ρ,∅,z) in the numerator? I guess I'm just confused about whether there should be a ∅ at all in my final equation or not.

 

[haruspex]

You don't seem to have posted the original question. (Graph? What graph?).
Is it a uniformly charged solid cylinder?
You seem to have ρ meaning two things: charge density and radius in the XY plane. I suggest switching to r for the latter. The volume element is then, as Shyan posted, rdrdϕdz. You seem to have dropped the initial 'r'.
The field will clearly depend on ϕ. Might be easier to work with the potential first, since that won't depend on ϕ, then differentiate.