Definition and properties of potential

Why introduce potential

The electrostatic field E is a vector field, often difficult to handle directly in energy calculations. Since the electrostatic force is conservative, we can associate it with a scalar function, the potential V, such that:

E = -grad(V)

In one-dimensional coordinates: Ex = -dV/dx. The potential is defined up to an additive constant: only the potential difference (pd) between two points has a physically measurable meaning.

Relation with work

The work of the electrostatic force to move a charge q from A to B is written:

W(A->B) = q * (V(A) - V(B)) = -q * (V(B) - V(A))

This relation is fundamental: it links a mechanical quantity (work) to an electrical quantity (pd). The work does not depend on the path taken, only on the starting and ending points, a direct consequence of the conservative nature of the field.

Equipotential surfaces

An equipotential surface groups together all the points where V is constant. The field E is always perpendicular to these surfaces, and directed from high potentials toward low potentials. For a point charge, the equipotentials are spheres centered on the charge.

Quantity	Nature	SI unit
Field E	vector	V/m
Potential V	scalar	V (volt)
Potential difference	scalar	V

Classic pitfall

A common mistake is to believe that V = 0 means there is no field at that point, or conversely. The reference potential (often taken as zero at infinity or at ground) is an arbitrary choice; only the gradient of V matters to obtain E. It is quite possible to have V = 0 at a point where E is intense (the case of a dipole on the perpendicular bisector plane, for example).