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

∫

d+L   d

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

)