Field lines are a choice, not a discovery
Put a positive charge somewhere and a negative one nearby, and the space between them looks exactly as empty as it did before. Nothing is arranged in curves. Nothing is flowing from one to the other. The lines in the picture below are a decision about how to draw, and it is worth being clear about that before being impressed by them.
What is actually being represented
At every point in space there is a vector: the force a small positive test charge would feel if it were placed there, divided by its charge. That is the field. It is defined everywhere, it has a direction and a magnitude at each point, and it exists whether or not anything is there to feel it.
That is an enormous amount of information — a vector at every one of infinitely many points — and no drawing can show it directly. The honest representation is a grid of little arrows:
The arrow picture is more truthful and much harder to read. Arrows near the charges are enormous and everything else is a fuzz; the eye gets no sense of where the field goes.
Field lines solve that by throwing away the sampling grid and following the field instead. Start anywhere, step in the direction the field points, repeat. The resulting curve is a path along which the force always points forwards.
The step is worth taking literally, because it is how the figures on this page are made. Each line is an integrated trajectory: start near a charge, evaluate the field there by summing the inverse-square contributions of every source, move a short distance along it, and repeat a few hundred times. Nothing about the curvature is drawn. The shape of a dipole’s lines is a computed consequence of the exponent in Coulomb’s law, which is why the same procedure with a different exponent would produce visibly different curves.
Three things the convention gets right
The reason field lines survived as a convention, rather than being a pretty but misleading picture, is that three separate properties of the field are preserved by the translation.
Direction is exact. The tangent to a line at any point is the field direction at that point, by construction. Nothing is approximated.
Density encodes strength. This is the non-obvious one and the reason the convention is worth anything. If lines are started evenly around a charge and never created or destroyed in between, then the number crossing a given area falls off exactly as the field does.
That figure is doing more than illustrating. It is a derivation, and the argument in it becomes Gauss’s law when it is made exact. The crowding of the lines is not decoration; it is the field strength, drawn, and the exponent in Coulomb’s law is a statement about the dimensionality of space rather than about electricity.
Lines never cross. Because the field has one direction at each point — the vector sum of every source’s contribution, which is a single arrow however many charges are present. Two lines crossing would mean a test charge at the intersection being pushed two ways at once — and if two lines meet at a point where the field is zero, the meeting is a limit rather than a crossing, since a zero vector has no direction for either line to inherit.
Those three together are what make the picture readable as physics rather than as an illustration.
What it hides
The convention has costs, and they are worth naming because they are invisible in the drawing.
The number of lines is arbitrary. Drawing sixteen lines per charge rather than eight does not represent a stronger field; it represents the same field at a different drawing resolution. Only the relative density carries meaning, so two figures drawn at different line counts cannot be compared by eye.
The picture is two-dimensional and the field is not. Real field lines from a point charge spread over a sphere, and their density falls as . A drawing on a page spreads them over a circle, where density falls as . The picture therefore systematically overstates the field far from the charge, and a reader who counts lines to estimate a ratio will get the wrong exponent.
And the lines suggest motion that is not there. They look like flow, like something travelling from the positive charge to the negative one, and nothing is. The electrostatic field is static. Nothing moves along a field line unless a charge is put there to be pushed. The visual language of flow is borrowed from fluids, where the streamlines really do have something running along them, and the borrowing is why the analogy misleads exactly where it is most persuasive.
That null point between two like charges is a good test of whether the picture is being read as physics or as decoration. It is a place where the drawing has no lines, and the reason is not that the drawing ran out — it is that the field is genuinely zero, and a line through it would have no direction to have.
Comparing the two figures makes the trade concrete. The arrow grid states the magnitude at every sample point and says nothing about where the field goes; the line picture states where it goes and encodes the magnitude in a quantity the eye reads badly. Neither is the field. Both are lossy projections of an object with too much information in it to draw, and knowing which loss each one takes is the difference between using a figure and being led by it.
Superposition is why either picture can be drawn at all
Both conventions rest on a property that is easy to overlook because it is never stated: the field of several charges is the vector sum of their individual fields, exactly.
Without that, there would be no procedure. Tracing a line requires evaluating the field at each step, and evaluating it requires adding up contributions — and if charges affected each other’s contributions, the sum would not be a sum. The whole subject would be as intractable as the three-body problem.
Linearity is what makes it tractable instead. Any distribution of charge, however complicated, is an integral of point-charge fields, and the same superposition that lets two waves pass through each other unchanged lets two fields overlap without interacting. It is one of the deepest facts about classical electromagnetism and one of the least remarked on.
Why the field concept was worth the trouble
Fields were not introduced because they draw nicely. They were introduced because the alternative had become unbearable.
The alternative is action at a distance: charge A pushes charge B directly, across the intervening space, with no intermediary. That works arithmetically and is deeply strange. It requires the force to be transmitted instantaneously, which turned out to be false, and it offers no answer to where the energy is when a charge has been moved but the other has not yet noticed.
Faraday’s move — and it was Faraday’s, arrived at by someone who could not do the mathematics and therefore had to think in pictures — was to say that the charge modifies the space around it, and the second charge responds only to the space it is actually in. Everything becomes local. Nothing acts anywhere except where it is.
That change of view is what makes the rest possible. A field can carry energy and momentum. A field can have a delay built into it, so that a change propagates outward at a finite speed. A field can exist with no charges present at all, which is what light is — and which is why refraction and interference turn out to be phenomena of the same object drawn here. None of that is available in a picture where charges push each other across a gap.
The delay is the decisive argument. Shake a charge and the kink in its field lines travels outward at ; until it arrives, a distant charge behaves as though nothing had happened. During that interval, momentum is missing from both charges and must be somewhere. The field is where. A bookkeeping device does not carry momentum, and a field does — which is what promotes it from a convenience to a constituent of the world.
So the lines are a convention, but the thing they are a convention for is the most consequential idea in classical physics, and it arrived because someone insisted on drawing it.
Where the model stops
Four limits, in increasing order of seriousness.
Statics. Everything drawn here is a field that is not changing. In electrostatics the field picture buys elegance and little else; every result could be had from Coulomb’s law with more patience. It becomes indispensable only once things move.
Vacuum. Inside matter the field polarises the medium, the medium’s own charges produce a field of their own, and the total is weaker than the sources alone would give. The lines are still drawable and their density no longer counts the free charge.
Classical. Charge is quantised, so the small test charge in the definition is a fiction taken to a limit that reality does not offer. Below that scale, the field is a quantum object, and the appropriate picture is not a line. The same fate awaits every classical picture on this site: a definite trajectory, a definite value at a point, a definite direction of push.
The lines can close. With changing fields, an electric field line can form a closed loop with no charge at either end — which is what induction means, and it breaks the intuition that lines start on positive charge and end on negative. Every argument on this page that relies on lines having two ends is a statics argument, and it should be labelled as one.
The ladder from here
Later rungs on this anchor: the vector field before any lines are drawn on it, and what each picture costs. Flux, and Gauss’s law as the statement that the lines leaving a closed surface count the charge inside. Equipotentials, always perpendicular to the lines, and why that is not a coincidence. The energy stored in the field itself, which goes as the square of it. Magnetic field lines, which have no starts or ends at all, and what that absence means. Induction, where a changing field of one kind makes the other. The delay, and the radiation that escapes when a charge is shaken. And the point at which the field picture itself becomes the approximation, and the quantised description takes over.
Faraday drew lines because he could not write the equations, and was condescended to for it. Maxwell then wrote the equations of exactly what Faraday had drawn, and said in his preface that he had found Faraday’s methods to be mathematical ones, merely not written in the conventional symbols.