What is the electric field?

Coulomb’s force tells us how two charges attract or repel each other, but it always requires thinking of a pair of objects acting directly on one another. The electric field offers a more fruitful perspective: every charge modifies the space around it, creating at every point a vector quantity $\vec{E}$; another charge, placed at that point, feels the force through this field, without needing to refer directly to the distant source. Space stops being a passive container and becomes the true mediator of the interaction.

History — Faraday and the idea of the field

Michael Faraday (1791–1867) was the son of a blacksmith and at thirteen became a bookbinder’s apprentice. It was precisely by reading the article on electricity in the Encyclopædia Britannica, which he bound daily, that he became passionate about science. On Christmas Eve 1812 he received a letter from the great chemist Humphry Davy accepting him as a laboratory assistant at the Royal Institution (Guillen 1995, pp. 132–140) — see Riferimenti bibliografici.

Almost entirely self-taught, Faraday did not master advanced mathematics. It was perhaps precisely for this reason that he arrived at an insight the mathematicians could not have: instead of thinking of the electric force as an “action at a distance” between charges, he imagined space filled with lines of force, a field that permeates all of space. This vision, later translated into equations by Maxwell, became a pillar of modern physics (Simonyi 2012, pp. 329–351).

Principle — Electric field

The electric field generated by a point charge $Q$ at distance $r$ is: $\ev{\vec{E} = k\,\frac{Q}{r^2}\,\uv{r}}$ The field points radially outward if $Q > 0$, inward if $Q < 0$.

The strength of the field decreases with the square of the distance, exactly like Coulomb’s force: it is the same physical law, but reread as a property of space rather than as an interaction between two bodies.

Key formula

$|\vec{E}| = k\,\frac{|Q|}{r^2}$ $\vec{F} = q\,\vec{E}$ The force on a test charge $q$ is simply $q$ multiplied by the field.

The relation $\vec{F} = q\,\vec{E}$ is the bridge between the field and what we can measure: a test charge $q$ immersed in the field feels a force equal to the product of its charge and the local field. This separation brings with it an enormous conceptual advantage, because it splits every electrostatics problem into two independent parts:

1. calculating the field created by the sources, which depends only on the charge distribution that generates the field;
2. calculating the force felt by the test charge, simply by multiplying the field by $q$.

In this way the study of the sources (part 1) is entirely detached from the study of the effect on the test charge (part 2): we can map the field once and for all and then use it for whatever charge we want to place in it.

Note

This way of seeing the field was introduced by Faraday well before Maxwell’s equations formalised it. Today it remains the fundamental visual tool for understanding electromagnetism.

Collegamenti

Argomenti: Campo elettrico e potenziale · Elettrostatica Concetti: Campo elettrico · Carica elettrica · Legge di Coulomb

Esercizi collegati: Field of a point charge at 9 cm · Distance drill for a given field · Ranking the field across various configurations