The field before the lines were drawn on it

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.

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 $e$ 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.

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 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.

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 number of lines drawn is a choice, and nothing physical depends on it. Double them and the field is unchanged; only their direction and the way their spacing grows carry content — so a reader who counts lines is reading the draughtsman. Worse, the convention has no way to draw a null: at a point where the field vanishes there is simply no line, and an absence looks the same as somewhere the draughtsman happened not to start one.

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.

The density that does mean something is the one the lines were drawn to represent. The same number of lines crossing shells at one, two and three times the distance gives a density falling as the inverse square, because the shell’s area grows as the square — so the spacing of lines carries the field’s magnitude, and the individual line carries nothing at all. That is the trade in one sentence: a picture that shows the direction everywhere and the magnitude only by comparison.

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 $r^2$, the density must fall as $1/r^2$.

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.

What the field costs #

Two charges in a room can be specified with eight numbers: two charges, and two positions of three coordinates each. The field describes the identical situation with a vector at every point of space — three numbers at each of infinitely many places, almost all of them somewhere nothing will ever go.

That is an enormous inflation of the bookkeeping, and it is worth calling it a trade rather than an upgrade. What the extra numbers buy is locality: every statement the theory makes is about what is happening at one point and its immediate neighbourhood. A charge does not feel a distant charge; it feels the field where it is. The distant charge’s influence had to travel, and while it was travelling it was somewhere, holding energy and momentum, in a form that can be written down. Action at a distance offers a shorter description at the price of a physics in which quantities disappear from one place and reappear in another with nothing in between.

The bill for that is not only philosophical. Field energy is stored at a density proportional to $E^2$, and for a point charge $E$ grows as $1/r^2$, so the energy inside a sphere of radius $r$ around it goes as $\int r^{-4} \, r^2 \, dr$ — which diverges as $r \to 0$. A point charge carries infinite field energy. The classical dodge is to ask at what radius the stored energy equals $mc^2$, which for an electron gives about $2.8 \times 10^{-15}$ m, and then to declare that classical electromagnetism is not to be trusted below it. The number is a real prediction of the theory and the theory does not survive it: the infinity is the field concept’s own accounting saying that a point source is not something it can describe.

There is a computational cost too, and it is the one that decides how the subject is actually practised. Solving for the field of an arbitrary arrangement means carrying values on a grid — $N^3$ of them for a cube $N$ cells on a side — where the action-at-a-distance formulation carries only the sources. For a handful of charges the sum over pairs wins easily. The field representation overtakes it when the sources are continuous, when matter is polarisable, or when anything moves, and those cover nearly everything of interest, which is why the trade is nearly always taken.

The measurement the picture claims #

An arrow of a stated length at a stated point is a claim about an experiment, and it is worth knowing which one, because a figure that is persuasive and untested is the failure mode this whole site is arranged against.

The definitional measurement is the one in the arrow’s own terms: place a charge of known size, measure the force, divide. In practice the charge that is easy to place is an electron or an ion in a beam, and what is measured is a deflection — the field is inferred from a curved trajectory, which means the arrow at a point is really a statement about a whole path through the region. Fields inside matter are measured differently again, by the voltage between two places, which measures a potential difference and returns the field only after a derivative.

Every one of those routes measures something integrated, and the arrow is drawn as though the value at a point were directly available. It is not: there is no instrument that reports the electric field at a mathematical point, and there could not be, since any probe occupies a volume and averages over it. The arrow grid is a picture of a limit, in exactly the way that the small test charge is.

That has a consequence the figures above quietly rely on. Both pictures are drawn from the field of idealised point charges, computed by superposition, and no measurement has ever confirmed the inverse-square law at short range to anything like the precision the drawings imply. What has been tested extremely well is the exponent: if the law were $1/r^{2+\epsilon}$, a charged conducting shell would have a field inside it, and the absence of that field bounds $|\epsilon|$ below $10^{-16}$. The strongest evidence for the shape of these pictures comes from an experiment that consists of finding nothing at all — the same null result that Cavendish ran with a pair of concentric spheres before Maxwell repeated it.

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 $c$, 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 $E^2$. Neither picture shows an amount of anything being stored anywhere, and the $E^2$ 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 convention hiding inside the honest picture #

The arrow grid was described above as the honest drawing, and it is, but it has a convention of its own that is easy to miss because it does not have a name.

The trouble is range. Near a charge the field goes as $1/r^2$, and a grid drawn around a dipole samples points whose distances from the nearest charge differ by a factor of ten or more — so the strengths differ by a factor of a hundred, and for the dipole’s far field, which falls as $1/r^3$, by considerably more. Draw the arrows true to scale and one of them fills the frame while the rest are shorter than the arrowheads. The picture becomes a few black blobs surrounded by dots.

Every vector-field plot ever published therefore does one of three things, and none of them is the stated encoding. It clips, capping the longest arrows and silently reporting several different strengths as the same. It normalises, drawing every arrow the same length and moving the magnitude into colour — which abandons the property that made the picture honest. Or it compresses, scaling the length by a square root or a logarithm of the strength, so that lengths remain comparable and their ratios no longer mean what they appear to.

That is the same species of defect the page charges against the line picture: a quantity that looks directly readable and is not. It is worth stating plainly rather than leaving as a footnote, because the arrow picture is normally introduced as the one without a convention in it. There is no such picture. A field has too much dynamic range for a drawing, and something has to give in every rendering of it.

What else turned out to be a field #

The word arrived in electromagnetism and did not stay there, and the reason is that “a quantity defined at every point of space” turned out to be the shape of nearly everything physics went on to describe.

The temperature in a room is a scalar field: one number per point, obeying an equation that says how it diffuses. The flow of a fluid is a vector field, with the same type as this page’s — and a great deal of fluid mechanics is done by drawing its lines, called streamlines, and reading them by exactly the rules above, including the trap that their spacing means speed and the eye judges spacing badly.

The stress inside a loaded beam is neither. It takes a direction in and returns a direction out, because the force transmitted across an internal surface depends on how that surface is oriented — six independent numbers at every point rather than three. That is a tensor field, and the geometry of spacetime is another one: the metric is ten numbers per point, and general relativity is the equation that says what sources them.

So the ladder of field types runs by how much structure sits at each point, and the electric field is the second rung of it. What generalised was not the electricity. It was the decision to describe a situation by what is true at every location, rather than by a list of objects and the forces between them — and by the 1930s the same move had been made on matter itself, which is what a quantum field is.

Why the picture won the argument #

Faraday drew the lines because he could not do the mathematics, and was patronised for it. He had no university training, spent the 1830s filling notebooks with sketches of what he called lines of force, and treated them as physically real — as though space near a magnet were under a strain that the iron filings were revealing rather than creating.

The continental physicists had a complete and successful alternative. Ampère and Weber wrote forces acting instantaneously between charges at a distance, with no intervening medium and no picture, and it reproduced every electrostatic and magnetostatic result then known. On the evidence available in 1850 there was nothing to choose between the two, and the fashionable choice was the one with the equations.

Maxwell’s first electromagnetic paper, in 1855, is called On Faraday’s Lines of Force, and it does what the title says: it writes the mathematics of exactly the thing Faraday had been drawing, and says so in the opening pages. What settled the matter twenty years later was neither elegance nor preference for pictures. It was that the field formulation predicted something the action-at-a-distance formulation could not contain — a disturbance that leaves its source and travels at a finite speed, carrying energy through space with no charge anywhere near it. Hertz produced and detected one in 1887.

At that moment the field stopped being a way of drawing forces and became the thing the theory is about. The lines in these figures are older than the mathematics that justifies them, which is unusual, and worth remembering whenever a diagram is dismissed as merely an illustration.

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.

The rung immediately after this one is the potential, which replaces the vector at every point with a single number and loses nothing — the most economical trade in the subject, and the one that makes everything after it calculable.