Electric Field: Formula, Definition, and Electric Potential

Around any electric charge there’s a region where another charge feels a push or a pull: that’s the electric field. HyperPhysics defines it as the electric force per unit charge, written E = F/q.

The field isn’t just a size, though: it also has a direction, the one in which a positive test charge placed at that point would be pushed. That’s why physicists call it a vector quantity rather than a plain number — a single value can’t capture it fully.

For a point charge, the field comes straight out of Coulomb’s law: HyperPhysics describes it as a vector equation, with the force acting along the line that joins the two charges, growing with the charges involved and shrinking with the square of the distance between them. In a vacuum, the formula reads E = kQ/r², where Q is the charge producing the field, r the distance to the point in question, and k Coulomb’s constant, a fixed number built into the equation. Same-sign charges repel, opposite-sign charges attract, and that rule governs both the force and the field it produces. The pull or push weakens quickly with distance, but according to HyperPhysics it never truly disappears — at least in theory, some trace of it remains at any distance.

Real-world example: rub two balloons against your hair and they pick up the same kind of charge; bring them close together and, instead of sticking, they push apart. It’s the same rule Coulomb’s law lays out for a point charge’s field: same-sign charges repel.

A fairly common misconception is that the electric field only comes into being once a test charge arrives to measure it. That’s backwards: the field is already there, independent of anyone measuring it, and the test charge’s job is simply to reveal it.

Treccani’s encyclopedia entry on electrostatics describes the electric field as a conservative vector quantity, defined as the ratio between the force on a test charge and the value of that charge, provided the test charge is small enough not to disturb the field it’s probing. The field can also be obtained from the potential V, by taking its gradient and flipping the sign, E = −grad V. The potential V, though, isn’t a vector at all: it’s a scalar, a single number standing for the potential energy per unit charge at that point.

OpenStax defines potential V at a point as U/q, the electric energy a charge would hold there, divided by that charge. From this comes another widely used quantity, the potential difference between two points A and B: it shows up whenever a charge travels from one to the other, and it equals the change in that charge’s energy along the way, divided by the charge’s own value. The unit for potential difference is the joule per coulomb, known as the volt in honor of Alessandro Volta. In everyday speech, potential difference goes by another name, voltage — the same concept behind electric current: without it, charges in a circuit would have no reason to move at all.

OpenStax also notes that the electric field always points toward lower potential: the direction the field vector takes and the direction in which potential drops are one and the same. It’s a handy rule for sketching a field around a charge, and it also helps make sense of very different devices. A panel in a solar panel system, for instance, develops a potential difference across its terminals — the voltage that then powers whatever is plugged in. Inside a microwave oven, the electric field instead acts on the charges within food molecules, in a geometry quite different from a single point charge but built on the same basic definition: force per unit charge.

Field and potential stay connected but distinct: one describes a force with a direction, the other the energy available at a point. And it’s the potential difference between two points, voltage, that explains why charges move at all.