Potential Difference

The electric potential at a point a which is a distance d from a charge q₁ is given by V_a = Ed. From the notes on the electric field we have E = kq₁/d², so V_a = kq₁/d. The potential difference between two points, a and b, in a field is denoted by V_ab = V_a-V_b.

The moving of a charge carrier in an electric field changes the carrier's potential energy because a force has been applied. If a charge q₂ is moved a distance d against the field, then the potential energy is increased. If q₂ is released in a field, then work is done to q₂ (W = Fd) and the magnitude of the force is propotional to the electric field (F = q₂E).

We know that W = Fd and, according to Coulomb's law, F = kq₁q₂/d². So

W = (kq1q2/d2)d
  = (q2kq1/d2)d
  = q2Ed. 

This work is measured in joules. From this we get W/q₂=Ed. So, electric potential is work per coulomb. This is measured in joules per coulomb (volt) or newtons per coulomb.

References

Fischer-Cripps. A.C., The Electronics Companion. Institute of Physics, 2005.