# Uniformly Charged Cylinder - Potential at distance d?

1. Oct 12, 2011

### acedeno

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
V=∫kdq/r

3. 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)