The electric potential at a point a which is a distance d from a charge q1 is given by Va = Ed. From the notes on the electric field we have E = kq1/d2, so Va = kq1/d. The potential difference between two points, a and b, in a field is denoted by Vab = Va-Vb.
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.
Fischer-Cripps. A.C., The Electronics Companion. Institute of Physics, 2005.
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