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.

The counting also gives the inverse square, which is worth noticing because it is usually assumed. Fix the number of lines leaving a charge and ask how densely they are spread at a distance: the area they cross grows as the square of the radius, so their density falls as its inverse. The law is not put into the convention — it comes out of it, provided lines are neither created nor destroyed on the way, and that proviso is the whole physical content.

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 $1/r^2$. A drawing on a page spreads them over a circle, where density falls as $1/r$. 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.

What it costs to draw one honestly #

Every line in these figures is the output of an integration, and integrations have error. Being explicit about that is worth a section, because the difference between a computed figure and a plausible one is the whole justification for drawing it.

The procedure is: evaluate the field at the current point by summing the contributions of every charge, normalise it to get a direction, take a step of fixed length, repeat. The figures here take steps of about three units and up to nine hundred of them. Each step commits a small error, because the field direction changes across the step and only the value at the start was used, and the errors accumulate along the line.

The error is worst exactly where the picture is most interesting. Near a charge the field direction changes rapidly over short distances, so a fixed step is too coarse; near a null point the field is small and its direction is ill-conditioned, so a small numerical error rotates the step by a large angle. Both are places where a naively integrated line can wander somewhere the real field would never take it.

Two things keep the figures honest. The first is a cutoff: the integrator refuses to evaluate the field inside a small radius of any charge, where the inverse-square law diverges and no step size is small enough. The second is the no-crossing property, which is a genuine test rather than a description. Lines of a real static field cannot cross, so two lines crossing in a drawing is a bug — either the step is too large or the field is being evaluated wrongly. A property that is guaranteed by the physics and not guaranteed by the code is exactly the kind of property worth checking, because it fails loudly when the code is wrong and costs nothing when it is right.

That is the general shape of the discipline this site runs on. A figure computed from the physics can be given a test it would fail if the computation were wrong, and a figure drawn to look right cannot be given any test at all.

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 $c$; 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.

The convention that turned out to compute something #

The title of this essay says the lines are a choice, and that is true of the drawing. It is less true than it looks of the picture, and the reason is the strangest thing on this page.

Faraday did not regard his lines as bookkeeping. He thought space near a charge was under a physical strain, and he described the strain in mechanical terms: the lines are in tension along their length, pulling their ends together, and they repel each other sideways, so a bundle of them pushes outward.

Read the figures with those two rules in mind and the forces come out correctly. In the dipole, lines run from the positive charge to the negative one, and lines under tension pull the two together — which is the attraction. Between two like charges no line connects them, and the lines from each are crowded against each other in the gap, and lines pushing sideways force the charges apart — which is the repulsion. Two entirely different-looking figures, one mechanical rule, both signs right.

That is not a coincidence or a mnemonic. Maxwell made it exact: the electromagnetic field carries a stress with a tension of $\varepsilon_0 E^2/2$ along the field direction and an equal pressure across it, and integrating that stress over any surface enclosing a charge gives precisely the Coulomb force. The calculation is standard, it is used daily in machine design to compute the pull of a magnet or the force on a capacitor plate, and it never mentions the other charge at all — only the field on a surface drawn around the one whose force is wanted.

So the convention is a choice, and the choice landed on something with more content than a choice ought to have. The lines were drawn because they made a picture legible, and it turned out that treating the picture as a description of stress in space reproduces the dynamics. That is a considerable amount of luck, or else a sign that Faraday’s eye was doing something more than drawing.

Iron filings are not the lines #

The commonest argument that field lines are real objects rather than a drawing convention is a physical demonstration: sprinkle iron filings around a magnet and the lines appear, unbidden, in the pattern every textbook draws. It is a good demonstration and it does not show what it is usually taken to show.

Each filing is a small piece of soft iron, so the field magnetises it and it becomes a tiny magnet with a north end and a south end. Its neighbours do the same, and a north end beside a south end attracts, so the filings link up nose to tail into chains. What the pattern displays is filings aligning with the local field direction and then attracting each other into strings — which is the same direction information the drawn lines carry, arrived at by a mechanism the drawing does not have.

Three of the things this page has said about the convention survive the demonstration intact. The number of chains is not a measurement: it depends on how much iron was sprinkled and how coarsely it was ground, in exactly the way the number of drawn lines depends on the draughtsman. The chains carry no sense of direction, only an axis — nothing in the pattern distinguishes north from south, which is why every photograph of filings is published with arrows added. And the filings change the field they are displaying, because a chain of magnetised iron is itself a source; the pattern is of a magnet plus its filings, not of the magnet.

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. Conductors, where the lines are forced to meet the surface at a right angle. 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.