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
 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
)