Electric field

The electric field E at a point is defined operationally: place a small positive test charge q there, measure the force F on it, and divide. E = F/q. The field is whatever the test charge does not care about — its value depends only on the source charges, not on q. Electric field has units of newtons per coulomb, or equivalently volts per metre, and is a vector: it has both magnitude and direction at every point in space.

The field concept replaces "action at a distance" with a local picture. In Newton's theory of gravity, the Sun pulls on the Earth across empty space; nobody liked this, but it worked. Faraday and Maxwell replaced it for electromagnetism with a field: the Sun's charges fill the surrounding space with a pattern, and a distant charge feels only the local value of that pattern. When the source wiggles, the pattern wiggles — not instantaneously, but at the speed of light. This upgrade turned out to be the deeper description; the field carries energy, momentum, and information, and eventually (with Einstein) becomes a dynamical thing of its own.

Every charge distribution produces a field; every field exerts a force on any charge you drop into it. Once you know E(r) everywhere, electrostatics is solved — the force on a charge q at position r is simply qE(r). Most of the subject is about calculating that field for a given source, either directly from Coulomb's law or more cleverly via Gauss's law and symmetry.

Michael Faraday introduced the notion of "lines of force" filling the space around charges and magnets in the 1830s — a pictorial language without mathematics. James Clerk Maxwell converted Faraday's pictures into the vector-field partial differential equations of 1861–65 that now bear his name. Einstein later said Faraday and Maxwell completed what Newton had begun: a description of physics in terms of fields rather than particles-at-a-distance.