Uniformly Charged Cylinder - Potential at distance d?

[acedeno]

Homework Statement

Consider a uniformly charged solid cylinder of radius R, length L and charge density ρ. Find the
potential at a distance d (> L/2) from the centre of the object, along the axis of the cylinder.

Homework Equations

V=∫kdq/r

The Attempt at a Solution

For me, it makes most sense to express this integral in cylindrical coordinates, seeing as the object is a cylinder. Also, since the axis of which the cylinder is on is not specified. I chose the z axis on a (x,y,z).

-stuff used for integration, respectively.
s[0,R]
Ø[0,2π]
z[0,L]

dq=sdsdØdz


I'm not too good at expressing notation on the computer so this is the basics of what i tried:

1st attempt: I know V(z)=∫E.dl , so, I tried to solve for E by using
E=∫kdq/r^2
E=∫∫∫(kρ/z^2)sdsdØdz

after I finished this integral, I lost confidence when doing the integral for V(z) because It didn't seem right to integrate over the same limits of integrations seeing as dl would be expressed as sdsdØdz - please tell me if I'm wrong.

2nd attempt: I just started with V=∫kdq/r.
V=∫∫∫(kρ/z)sdsdØdz
giving me
V(z)=kρπ*ln(z)*R^2

- I'm not sure which method is correct, if either. Help would be greatly appreciated.
- Also, since were just looking for the function with respect to a distance and because of the way the question was stated, I felt that it was okay to express r as just z rather than (L/2 + z)

 

[AvroArrow]

as the question did not state that anything other than the distance from the centre of the cylinder was given.