Illustrated Physics
physics, drawn
Electromagnetism

The field before the lines were drawn on it

A field is a vector attached to every point of space. Drawing it as arrows on a grid is honest and ugly; drawing it as lines is beautiful and throws information away.
9 min read 6 figures Fields, not forcesWhere the model stops

Before any lines are drawn, a field is a much plainer object than the pictures suggest. It is a rule that assigns a vector to every point of space — a magnitude and a direction at each location, defined whether or not anything is there to feel it.

That definition is deliberately unglamorous, and it is worth sitting with before reaching for the line picture, because the lines are a compression of it and compression loses things.

The same field, sampled as arrowsThe field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.+−lines and arrows are the same fielddrawn two ways
Fig. 1 The field of two opposite charges, sampled on a regular grid. Each arrow points along the field at its own location and its length is scaled by the local strength, so the picture shows both quantities and shows them nowhere else.

What the arrows mean

Each arrow answers one question: if a small positive test charge were placed exactly here, which way would it be pushed and how hard?

The definition carries an idealisation in the word small. A real test charge has its own field, which pulls on the sources, which moves them, which changes the field being measured. The definition takes a limit — force per unit charge as the charge goes to zero — so that the measurement stops disturbing the thing measured. That limit is fictional, since charge comes in units of ee and cannot be made arbitrarily small, and the field is nevertheless perfectly well defined without any test charge present at all.

That last point is where the field concept earns its keep. Coulomb’s law describes a force between two charges and needs both of them to say anything. The field describes what one charge does to space, in advance, without knowing what will arrive. The two formulations agree on every prediction in electrostatics, and they part company completely the moment anything moves — because a field can carry energy and momentum and take time to propagate, and a force between two objects at a distance cannot.

The same field, sampled as arrowsThe field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.+lines and arrows are the same fielddrawn two ways
Fig. 2 The field of one positive charge. The arrows point radially outward and shorten with distance, and the shortening is computed from the inverse-square law.

Adding fields is why any of it is tractable

Two sources produce a field that is the vector sum of what each would produce alone. Not approximately, and not only when they are far apart — exactly, at every point, always.

Superposition is what makes the subject calculable. Any arrangement of charge, however complicated, can be chopped into pieces small enough to treat as points, and the field of the whole is the sum of the fields of the pieces. A charged sphere, a wire, a capacitor plate, a thundercloud: all of them are integrals of the same one-charge answer.

The same field, sampled as arrowsThe field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.++lines and arrows are the same fielddrawn two ways
Fig. 3 Two like charges. Between them the two contributions oppose, and at the exact midpoint they cancel completely — a point in space where the field is zero although two charges are shouting at it.

The null point between two like charges is the vector sum doing something a scalar sum cannot. It is the same cancellation that produces the quiet directions in an interference pattern, and it has the same character: two influences producing nothing where either alone would produce something.

Superposition is also what makes the field picture linear, and linearity is a stronger claim than it looks. It says the field of a charge is unaffected by the presence of other charges — that two fields pass through each other and add without interacting, exactly as two waves do. Classical electromagnetism is linear in vacuum to every precision yet measured. Quantum electrodynamics predicts a tiny violation, light scattering off light through virtual pairs, and the effect was first seen unambiguously in 2019.

The trade the line picture makes

Field lines are built from the arrow field by a specific procedure: start somewhere, step along the local direction, repeat. Every line is a trajectory of the arrows.

The field of two like chargesField lines from two positive charges. None connects them; between them is a point where the field vanishes entirely.++field is zero here
Fig. 4 The same two like charges, drawn as lines. The lines curve away from each other and none crosses the midpoint — the null the arrow picture showed as a gap now shows as an absence of lines.

Something is gained and something is lost, and the exchange is worth being explicit about.

Gained: continuity. The arrow grid samples. A line follows. The eye reads a curve as a path, and for a field whose whole content is which way from here, a path is the right kind of object. Nothing in the arrow picture makes it obvious that following the arrows from a positive charge always ends on a negative one; in the line picture that is the first thing seen.

Lost: magnitude, as a directly readable quantity. The arrow picture states the strength at each point by length. The line picture encodes it in density — lines per unit area — which is a genuine encoding and a much harder one to read. Judging that one region has twice the line density of another is something the eye does badly, and it is impossible in a two-dimensional drawing of a three-dimensional field, where the density that matters is per unit area and the drawing shows lines per unit length.

Lost: the null points. In the arrow picture, a zero field is an arrow of zero length, drawn at a definite place. In the line picture there is simply nothing there, and nothing there is also what empty space looks like.

Lost: the sampling honesty. A grid of arrows advertises that it is a sample. Lines look like objects. Nothing about the drawing signals that twice as many could have been drawn, all equally valid.

The field of a single chargeField lines radiating from one positive charge, straight and evenly spread.+
Fig. 5 One charge, as lines. The number of lines is a choice; only their radial direction and the way their spacing grows with distance carry any physical content.

The density that does mean something

There is one respect in which the line picture is not a convention at all, and it is the reason the convention survived.

Why the field falls off as the squareThe same number of field lines crossing shells at one, two and three times the distance. The shell's area grows as the square of the radius, so the lines per unit area falls as its inverse.r = 1area × 1strength ÷ 1r = 2area × 4strength ÷ 4r = 3area × 9strength ÷ 9+nothing is lost between the shells —the same lines are simply more spread out
Fig. 6 The same number of lines crossing shells at one, two and three times the distance. The shell’s area grows as the square of the radius, so lines per unit area falls as its inverse — which is the inverse-square law, obtained by counting.

Lines do not begin or end except on charges. So the number crossing any surface enclosing a charge is the same however far out the surface is placed, and if the same count is spread over an area growing as r2r^2, the density must fall as 1/r21/r^2.

This is not an illustration of the inverse-square law — it is a derivation of it, and it is the argument that generalises into Gauss’s law. The same geometry sets the falloff of any wave spreading from a point source, which is why sound and light obey the identical exponent for a reason that mentions neither. The exponent is two because space has three dimensions and the surface of a sphere in three dimensions grows as the square. In a hypothetical four-dimensional space the same argument would give an inverse cube, and stable orbits would not exist.

So the line picture’s density encoding is the one part of it that is forced. Direction is physical, density is physical, and the count itself is arbitrary — a mixed status that no other diagram in physics quite shares.

Where both pictures stop

Four things are true of the electric field and visible in neither drawing.

It is three-dimensional. Both figures are slices. A real field fills space, lines pass in front of and behind the plane, and the two-dimensional density is not the three-dimensional density. Every intuition about spacing built from these pictures is off by a power.

It changes with time. These are static fields, and statics is the special case where the field concept is least necessary — everything in it could be done with Coulomb’s law and more patience. The field becomes indispensable when charges move, because then the field lags: a charge that jumps sends out a kink in its lines that propagates outward at cc, and until the kink arrives, distant space does not know. That kink is light, and it is the strongest argument that the field is a thing rather than a bookkeeping device.

It holds energy. Assembling a charge distribution takes work, and the work is stored in the field itself — in the same sense that a stretched spring stores it, except that the storage medium is space — at a density proportional to E2E^2. Neither picture shows an amount of anything being stored anywhere, and the E2E^2 dependence means the energy is concentrated much more sharply near the charges than the arrow lengths suggest.

It is not the whole field. Electric and magnetic fields are two aspects of one object, and which is which depends on the observer’s motion. A pure electric field for one observer is electric and magnetic for another moving past it. Any picture that draws only the electric part has already chosen a frame, silently — and the frame-dependence is not a subtlety at the edge of the theory but the reason the two fields appear in one set of equations at all.

The ladder from here

Later rungs on this anchor: the potential, a scalar field whose gradient is this vector field, and why one number per point is often enough. Equipotential surfaces, and why they meet field lines at right angles. Gauss’s law as the line-counting argument made exact. Conductors, and why the field inside one is zero. Dielectrics, and a field weakened by matter it polarises. Field energy and its density. The magnetic field, which has no sources for its lines to start on. And the point where the field stops being a convenience and becomes the primary object: radiation, where fields exist with no charges nearby at all.

Faraday drew the lines because he could not do the mathematics, and was patronised for it. Maxwell then wrote the mathematics of exactly what Faraday had drawn, and said so.

Where this goes next

+− Field lines are a choice, not a discovery Nothing in space is arranged in lines. The lines are a drawing convention — and an unusually good one, because three separate facts about the field survive the translation. +every line leaves: net flux counts the chargethe shape of the surface never enters the answer Counting what comes out, and never looking inside Draw any closed surface. The field crossing it depends only on the charge enclosed — not on where that charge sits, not on its shape, not on anything outside. sourcesourcesolid: waves arrive in stepbetween them they arrive opposed When two waves meet, they simply add Waves pass through each other unchanged and their displacements add point by point. From that one impoverished-sounding rule comes interference, beats, and the evidence that light is a wave at all.