Potential created by charge distributions

Point charge

The potential created at a point M located at a distance r from a point charge q, taking V = 0 at infinity, is:

V(M) = k * q / r, with k = 1/(4piepsilon0) ~= 9.10^9 (SI units)

Unlike the field E which decreases as 1/r^2, the potential decreases as 1/r: it is "slower" to vanish and often simpler to calculate.

Superposition principle

For a set of point charges qi located at distances ri from point M, the total potential is the algebraic sum (not vector sum) of the individual potentials:

V(M) = sum( k * qi / ri )

This simplicity is a major advantage of the potential: no projections or components to handle, just a sum of scalars (watch out for the signs of the charges).

Example: the electrostatic dipole

Two charges +q and -q separated by a distance a form a dipole of moment p = q*a. At a large distance r (r >> a), the potential on the axis is approximately expressed as:

V(M) ~= k * p * cos(theta) / r^2

Note that the dipole's potential decreases as 1/r^2 (faster than an isolated charge), because the contributions of the two opposite charges partially cancel out.

Case of a continuous distribution

For a volume distribution with charge density rho, the potential becomes an integral:

V(M) = integral( k * rho(P) / PM * dV )

Classic pitfall

Do not confuse algebraic sum (potential) with vector sum (field). Two opposite charges can give a zero potential at a point (on the perpendicular bisector plane of a dipole) while the field E there is nonzero: this is a frequent source of error in exercises.