Potential from a Line of Charge
In general whenever we have a distribution of charge we can integrate to find the potential the charge sets up at a particular point. In this process we split the charge distribution into tiny point charges dQ. Each of these produces a potential dV at some point a distance r away, where:
dV 
= 
k dQ
 
r


The net potential is then the integral over all these dV's. This is very similar to what we did to find the electric field from a charge distribution except that finding potential is much easier because it's a scalar.
Let's find the potential at the origin if a total charge Q is uniformly distributed over a line of length L. The line of charge lies along the +x axis, starting a distance d from the origin and going to d+L.
If we split the line up into pieces of width dx, the charge on each piece is dQ = λ dx,
where the charge per unit length is λ 
= 
Q
 
L


The contribution each piece makes to the potential is:
dV 
= 
k dQ
 
x


= 
k λ dx
 
x


= 
k Q
 
L

 1
 
x

 dx 
Integrating to find the potential gives:
V 
= 
∫ 
dV 
= 
kQ
 
L


∫ 

1
 
x


dx 
The integral of 
1
 
x


dx is ln(x), so we get: 
V 
= 
k Q
 
L 
 ( ln(d+L)  ln(d) ) 
= 
k Q
 
L 
 ln ( 
d + L
 
d 
 ) 