Forces Between Multiple Charges: The Superposition Principle

Coulomb's law, as stated, describes the force between exactly two point charges. Real situations almost always involve more than two -- a collection of charged particles, ions in a crystal, or protons in a nucleus -- and the rule for handling any number of charges is the principle of superposition:

The total force on any one charge, due to a number of other charges, is the vector sum of the individual forces exerted by each of the other charges taken one at a time, exactly as though every other charge acted alone (i.e. the presence of the remaining charges does not, in any way, alter the force any one pair exerts on each other).

For a charge $q_0$ in the presence of charges $q_1, q_2, \ldots, q_n$, the net force is

where $\vec{F}_i$ is the Coulomb force that $q_i$ alone would exert on $q_0$, and $\hat{r}_i$ is the unit vector from $q_i$ toward $q_0$.

The crucial word is vector: each $\vec{F}_i$ has both a magnitude and a direction (fixed by the line joining $q_i$ to $q_0$), and these directions are, in general, all different from one another. It is a common and serious error to add up the magnitudes $F_1, F_2, \ldots$ as though they were plain numbers; the correct procedure is always to resolve every individual force into components along a chosen set of perpendicular axes, add the components separately, and only then combine the resulting components (using the Pythagorean rule for magnitude and inverse trigonometry for direction) to get the net force. …