Perhaps the best way to think about the meaning of electric potential, or voltage, is to make an analogy with gravity. When a mass is high above the ground we say that it has a lot of potential energy because if we drop it, it strikes the ground hard. We say that the potential energy due to height has been converted into kinetic energy. We can say the same things about electricity. If two positive charges are close together and we release them, they will repel each other and pick up speed, their electric potential energy having been converted into kinetic energy. In this analogy, the electrical quantity which corresponds to height is voltage.
Well, now we have to be more precise. The
definition of the voltage difference between two points, a and b,
in space is
where this integral is a path
integral, or line integral, starting at point a
and ending at point b. The mental picture that goes with this path
integral is this: Chop the path up into little segments of length ds
and make them into vectors 
by making them point along the path from a to b. At each
segment take the scalar product between 
and ,
then add all of these dot products together, starting at a and
ending at b. For electric fields made by electric charges, the
answer for the voltage difference is the same no matter what path is
chosen. Applying this definition to the field of a point charge, q,
and choosing the convention that the voltage is zero at infinity, we
obtain the formula for the voltage produced at any point in space by a
point charge:
where r is the distance from the point charge to the point at which it is desired to know V. This formula says that the voltage is large and positive near positive charges and large and negative near negative charges. It can be used to find the potential produced by rods, rings, etc., by integrating, just as we did with electric fields in Chapter 23.
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