When charged particles move in an electric field, work is being done by the field on the particles. In this special case the work can be expressed in terms of potential energy which is associated with something called electric potential. It is the potential that the particle has to do work, and since it is in an electric field, electric potential seems like the most logical name for it. It is actually just like potential energy in every respect. As you remember, work is an amount of for exerted over a distance, and force is given by Coulomb’s Law as:

F = (1/4(pi)(epsilon 0))(q(q')/(r^2))
(Equation 6-1)

And work is just that across a distance. Here is the work done on or by a particle moving from point a to point b:

W = ((q(q'))/(4(pi)(epsilon 0)))(1/r[a] - 1/r[b])
(Equation 6-32)

Let’s say we are talking about an electric field around a point charge, all spheres with a radius of r have the same electric field strength. What that means is that the force of the electric field is the same for two points as long as they are equidistant from the point. What that also means is that if the path of the charge being moved, having work done on it, was curvy or straight, the amount of work done would be the same. Work is another way of saying the change in potential energy, right? The change in potential energy is the same whether the path taken is curvy or straight.

Now instead of dealing directly with potential energy of a charged particle, it might be better to just deal with the more general concept of potential energy per unit charge, which is called simply potential. The potential at any point of an electrostatic field is defined as the potential energy per unit charge at the point and is represented by V:

V = U/q'
(Equation 6-33)

Incidentally for all you lovers of all things scalar, potential is scalar because both potential energy and charge are scalar. The potential of 1 J/C is called 1 Volt (V) and is named after the Italian scientist Alessandro Volta (1745-1827).

Now we want to put that equation in a form we will use:

W[ab]/q' = U[a]/q' - U[b]/q' = V[a] - V[b]
(Equation 6-34)

Va and Vb are the “potential at point a” and the “potential at point b,” respectively. The difference Va – Vb, called the potential of a with respect to b, is abbreviated Vab most of the time.

The potential at a point due to a collection of point charges is:

V = (1/4(pi)(epsilon 0))summation(q/r)
(Equation 6-35)

When you want to work with a collection of charges, the above equation is usually the one you want to use. But sometimes you have te E field and it is a lot easier just to use that because you know that force on the test charge is written as F = Eq’ and when substituted you get a very versatile equation:

V[ab] = integral(E dl) = integral (E cos (theta) dl)
(Equation 6-36)

The integrals on the right side are called the line integrals of E. They represent the process of dividing the path into small elements of dl, multiplying them each by the component of E parallel to dl, then summing them up.