For a fixed electrostatic source configuration, work done by the field is W=q∫[A→B]vec(E)· dvec(l). Conservative-force accounting gives ΔU=-W. Divide by the charge:

A volt is a joule per coulomb. The zero of potential can be chosen conveniently; potential differences have physical consequences. Using zero at infinity for an isolated point charge gives

This follows by integrating the inverse-square field, not by confusing potential with field strength. Potentials of several point charges add as signed scalars: V=Σᵢ kQᵢ/rᵢ. Field vectors still require directional addition.

Slope, not height, determines force

In one dimension,

Positive charges accelerate toward lower potential; negative charges accelerate toward higher potential. Both move toward lower potential energy when released from rest under the electric force. Their initial velocity can of course carry them uphill temporarily.

Between ideal large plates separated by 5.0 mm with a 200 V difference, the approximately uniform field has magnitude E=200/0.0050=4.0×10⁴ V/m. It points from the positive, higher-potential plate toward the negative plate. An electron traveling the other way gains 200 eV=3.20×10⁻¹⁷ J of kinetic energy. Starting nearly at rest, its nonrelativistic speed is

This is about 2.8% of light speed, so the nonrelativistic approximation is adequate here. The calculation assumes negligible collisions, radiation, and gravity and ignores edge fields.

Zero potential need not mean zero field

Return to equal opposite charges symmetrically placed about an origin. At the midpoint their potentials cancel because distances are equal. Their fields add because “away from positive” and “toward negative” point the same way there. A charge at zero potential can therefore feel a force. Conversely, the interior of a charged conducting sphere has constant, generally nonzero potential and zero field.

An equipotential surface connects points of equal potential. Movement along it costs no electrostatic work, since ΔV=0. The field is perpendicular to it: any tangential component would make potential change along the surface. Closely spaced equal-voltage contours indicate a steep potential gradient and stronger field, much as close elevation contours indicate a steep hillside.

The limit that matters for circuits

In static electrostatics, returning to the starting point gives ∮vec(E)· dvec(l)=0; a unique potential difference does not depend on the route. This underlies the ordinary voltage bookkeeping in DC circuits.

When magnetic flux changes, an induced electric field can have a nonzero circulation. Then a single electrostatic potential cannot describe the entire field, and a loop can have an induced emf. Induction does not violate energy conservation; it identifies a regime in which “just subtract the two scalar potentials” is incomplete.