Electric Potential Due to a System of Point Charges

For any collection of point charges $q_1, q_2, \ldots, q_n$, located at distances $r_1, r_2, \ldots, r_n$ from a point $P$, the net electric potential at $P$ is found by the superposition principle, applied to a scalar quantity:

Since each charge's own sign is carried directly into the sum (a positive $q_i$ contributes a positive term, a negative $q_i$ a negative term), this net potential can come out positive, negative, or exactly zero, depending on the particular values and signs of the charges involved and their distances from $P$ -- unlike, say, the magnitude of a resultant vector sum, which can never itself be "negative."

The great practical advantage of this rule, compared with finding the NET ELECTRIC FIELD due to the same collection of charges, is that potential is a plain algebraic (scalar) sum, requiring no attention whatsoever to direction: each term $kq_i/r_i$ is simply a number, positive or negative, and the numbers are added exactly the way ordinary numbers are added. Finding the net field at the very same point $P$, by contrast, would require resolving each individual field vector into components along a common set of axes and adding component by component -- a distinctly more laborious calculation. This is one of the chief practical reasons potential is introduced at all: many electrostatics problems, particularly those involving several charges arranged in some geometric pattern (at the corners of a triangle, square, or other shape), are worked out far more easily via potential first, with the field itself (if it is needed) recovered afterward using the relation of Section 2.3. …