Electric Potential and Energy

Electric potential (V) — the work done per unit positive charge in bringing a small test charge from infinity to a point:

V = Q / (4 π ε₀ r)

measured in J C⁻¹, which is the volt. It is a scalar, so potentials from several charges add arithmetically.

Sign — unlike gravitational potential, this can be positive or negative. It is positive near a positive charge, since work must be done against repulsion to bring a positive test charge in, and negative near a negative charge. Only the gravitational case is forced to be negative, because gravity has only one sign.

Electric potential energy — for a charge q at potential V:

E = q V = Q q / (4 π ε₀ r)

positive for like charges, which repel and so are stored under tension, and negative for unlike charges, which are bound together.

Potential gradient — the field strength is the negative gradient of the potential:

E = − ΔV / Δr

which is why the uniform-field result E = V/d works: over a constant gradient, the gradient is just the total change divided by the distance.

Equipotentials — surfaces of constant potential, always perpendicular to field lines. Around a point charge they are spheres; between parallel plates they are planes parallel to the plates. No work is done moving a charge along an equipotential.

Example: the surface of any conductor in equilibrium is an equipotential, because if it were not, the potential difference would drive charge until it was. This is also why the field just outside a conductor meets its surface at right angles, and why charge concentrates where a conductor is most sharply curved — the basis of the lightning conductor.

Graph shapes — for a point charge, V varies as 1/r while E varies as 1/r², so the potential curve falls off more slowly. The area under an E–r graph gives the potential difference, and the gradient of a V–r graph gives the field strength.