The path a charge takes is not its field line

Field lines are a choice built every line on the page by the same procedure: start near a charge, find the direction of the field, take a small step that way, repeat. The field before the lines called the result, accurately, a trajectory of the arrows. The phrase invites a misreading that almost everyone makes on first seeing field lines and that some textbooks reinforce: that the line is the trajectory of a charge, the route a small positive charge would follow if it were let go there.

The misreading is easy to test, because the motion of a charge in a known field is as easy to compute as the field line. The two calculations differ in one word. A field line is the curve whose direction, at every point, is the direction of the force. A charge’s path is the curve whose acceleration, at every point, is the force divided by the mass. Direction and acceleration are different things, and the difference is inertia.

Released at rest #

The first figure lets four charges go. Each is placed close to the positive charge of a dipole, at rest, and its motion is integrated from Newton’s law with the dipole’s field. Beside each path is the field line through the same starting point, drawn the way a field line is always drawn.

At the first instant the two agree. A charge at rest has no velocity, so the only directional thing about it is its acceleration, and the acceleration points along the field. Each path leaves its starting point exactly tangent to its field line, which is the precise sense in which the naive reading is right.

Then the path and the line separate, and always in the same way. The field line curves round towards the negative charge. The charge, which has been accelerating outward from the positive charge and is now moving quickly, curves too, since the field is still pulling it round — but a moving body resists having its direction changed, and the sideways pull that would make it follow the line’s curvature is not there. The curvature a path can have is the sideways force divided by the mass and by the square of the speed, and that combination has nothing to do with how sharply the field line bends. The charge goes wide.

A landscape makes the same point without any electricity. On a hillside the field lines of gravity’s pull along the slope are the lines of steepest descent, the routes that cross every contour at right angles, and a drop of syrup let go on the hillside runs down one of them. A ball let go at the same place does not. It starts down the steepest line, gathers speed, and when the slope curves the ball carries straight on across the contours and up the far side of the valley, the way the hill that gives it back traced energy moving between height and speed. The syrup and the ball feel the same slope at every point. The syrup is held to the steepest line by its viscosity; the ball is free of it by its inertia, and its path records the history of the slopes it has crossed rather than the slope where it is.

In this figure it goes very wide. Each of the four charges, released close to the positive charge, gains more kinetic energy falling away from it than the negative charge’s attraction can take back, and all four leave the frame for good. Every field line through their starting points ends on the negative charge. A field line from a positive to a negative charge always runs downhill in potential energy for a positive charge, so a charge that somehow followed one would arrive with more energy than it started with; a charge that does not follow one is free to arrive nowhere.

What friction does #

Some charged things do follow field lines, and the second figure shows what they have that the charge in a vacuum does not. The same release is repeated with a drag force proportional to the velocity, the force a viscous fluid exerts on a small particle moving through it, and the drag is increased step by step. With a little drag the charge still escapes. With more it is captured, and its path, which still sets off along the line, now ends at the negative charge after a detour. With more still the detour shrinks until the path cannot be told from the line.

The reason is that drag takes away the charge’s freedom to have a velocity of its own. When the drag is strong, the charge reaches the speed at which drag balances the electric force almost at once, and thereafter it moves at that speed:

$$\mathbf{v} \approx \frac{q\mathbf{E}}{m\beta}.$$

Its velocity is then always parallel to the field, which is exactly the defining property of a field line. Inertia, which carried the undamped charge round the outside of every bend, has been made irrelevant: there is no stored velocity to carry round anything, only a velocity set afresh at each point by the local field.

The third figure measures how fast the convergence is. The largest distance between path and line falls, for strong drag, as the inverse square of the drag coefficient. One factor of the drag comes from how far the charge’s velocity lags behind the value $q\mathbf E/m\beta$ — the lag is the time $1/\beta$ the velocity takes to relax, multiplied by how fast the target velocity is changing — and a second from how far the lagging velocity has to carry the charge before it has caught up. At a drag of a hundred in these units the departure is three ten-thousandths of the distance between the charges.

That regime covers most of the charged particles anyone sees moving. An ion drifting through a gas, its velocity limited by collisions with molecules, moves along field lines at a speed proportional to the field, its mobility times $E$; this is how ion-mobility spectrometers, which identify molecules at airport security by how fast their ions drift down a tube, work. A charged speck of smoke in an electrostatic precipitator, or a charged droplet of paint between a spray gun and a car body, is swept along field lines to the collecting surface, which is why electrostatic painting coats the back of a panel that the gun cannot see. The drift that does not care what the charge is found a similar slaving of velocity to field in a plasma, where crossed fields move every particle at $E/B$. In each case the particle can be said to follow the field because something has removed its inertia from the problem.

The fluid Maxwell imagined #

The overdamped limit has a history that explains why the misreading is so persistent. When James Clerk Maxwell first tried to make Michael Faraday’s lines of force mathematical, in 1856, he did not picture them as paths of charges. He pictured an imaginary incompressible fluid flowing through a medium that resisted it with a force proportional to its velocity — a fluid with no inertia at all, pushed by pressure and held back by drag, whose velocity at each point was therefore proportional to the force there. The streamlines of that fluid were Faraday’s lines; the pressure was the potential; and because the fluid was incompressible, the density of streamlines told the strength of the flow, which is why the density of field lines tells the strength of the field.

Maxwell was explicit that the fluid was a device for reasoning and not a claim about the world, and the device works for exactly the reason the second figure shows. In a fluid without inertia, the path of every element of fluid is a streamline, because each element moves at the velocity the local force dictates and carries no memory of where it has been. The streamlines of an ordinary fluid are paths of its particles only when the flow is steady, and the field lines of an electric field are paths of charges only when those charges are as inertia-free as Maxwell’s fluid. The picture that made field lines into a theory was the picture of an overdamped flow, and the reading of field lines as routes is that picture remembered without its drag.

An electron in a vacuum tube is the other case. It has no drag at all, and the electron optics that the voltage that bends an electron like glass described is the optics of particles that do not follow field lines: an electron lens focuses because electrons cross the field lines at angles determined by their velocity, exactly as light rays cross the surfaces of glass.

A path that crosses every line #

The fourth figure removes the last ambiguity, which is that the charges in the first figure started from rest and so began on their lines. Give a charge a sideways velocity and it need never touch its field line at all. Round a single fixed charge of the opposite sign every field line is a straight radial spoke, and a charge launched sideways moves on an ellipse with the fixed charge at one focus — the inverse-square attraction is the same law as gravity, and the orbits are Kepler’s, closing on themselves as the orbit that does not come back to itself showed only an inverse-square or a linear force allows. The ellipse crosses every spoke once per lap. At no point is the path along a field line, and at no point is it trying to be.

In the orbit the field says something definite and limited: at each instant, which way the charge is being pushed. The push is towards the centre, perpendicular to the path at the ends of the ellipse and oblique elsewhere, and the path bends in response. A planet’s orbit and the Sun’s gravitational field lines are related the same way, and nobody is tempted to say that a planet follows a field line of the Sun, because the lines are plainly radial and the orbit is plainly not. The temptation arises only with curved field lines, where a curved path seems as though it might be the same curve. What the field fixes in an orbit is not the path but the rate at which the velocity changes, and the velocity’s own path is the cleaner object: the velocity that always lies on a circle found that for an inverse-square force the tips of the velocity vectors, drawn from one point, lie exactly on a circle. The field determines that circle; the orbit in space is what the circle integrates to.

The field that charges do follow #

There is one field whose lines charged particles do follow without any friction, and it is the magnetic field. The last figure launches a charge in the field of a magnetic dipole — the Earth’s, roughly — and integrates its motion in three dimensions. Drawn in the plane through the dipole’s axis, its path is a thickened copy of a field line. It spirals round the line, sliding along it towards the pole, and turns back where the field has become strong enough to reflect it, at a latitude of 33 degrees for a launch at 30 degrees to the field; then it returns along the same line to the opposite hemisphere and back again.

The magnetic force behaves differently because it is always perpendicular to the velocity, so it changes the charge’s direction and never its speed, as the force that does no work established. The part of the velocity across the field is turned round and round into a small circle, the gyration; the part along the field is untouched. A charge in a strong magnetic field therefore cannot move across the field lines except by a gyroradius, but moves freely along them, and the field line is its track in the most literal sense. The reflection near the poles comes from a quantity the gyration conserves, the magnetic moment, which forces the speed along the line to fall as the field strengthens, until it reaches zero and reverses. The Earth’s radiation belts are populations of charges held this way, bouncing between the hemispheres on field lines for days or years, and the aurora is where charges whose turning point lies inside the atmosphere end their bounce.

The two cases are opposite in mechanism and alike in effect. The electric field line is followed only when friction destroys the charge’s velocity; the magnetic field line is followed because the magnetic force preserves the speed while locking the direction to the line. In both, what makes the field line a path is that the charge’s motion across it is suppressed — by drag in one case and by gyration in the other — and in neither does Newton’s law say anything as simple as “go where the line goes”.

The model under the figures, and where it stops #

The figures treat the test charge as a point that does not disturb the charges making the field, has no radiation of its own and moves slowly compared with light. All three are good approximations for the drawn cases, and each has limits. A test charge accelerating as violently as the undamped charges in the first figure radiates, losing energy, and an electron released beside a real nucleus would radiate its energy away in a fraction of a nanosecond if classical physics held, which is one of the failures that led to quantum mechanics. The drag in the second figure is linear in velocity, which is right for small particles in viscous fluids and for ions in gases at modest fields, and wrong at high speeds, where drag grows faster than the velocity. The magnetic case assumes that the field changes little over one gyration; when it does not, as for the most energetic particles in the Earth’s field, the magnetic moment is no longer conserved, the bounce is irregular, and particles leak out of the belts.

What a field-line drawing cannot say #

The drawings of field lines this essay set out to correct do not contain any information about time, and no drawing of them can. A field line records the direction of the field at each point and, through its density, the strength; it does not record what a charge does there, because that depends on what the charge is already doing, which the field knows nothing about. Two charges released from the same point, one at rest and one moving, follow different paths in the same field, and both paths are compatible with the same lines. The figures here add the missing ingredient — the initial velocity, the mass, the drag — and the result is a family of paths for each field rather than one. One number for every point found that a potential, a single number per point, contains everything a field contains; the paths show what neither contains, which is a history.

Still open: what a field line means when the field changes #

Everything above concerns static fields. When fields change in time the question of what a field line is becomes harder, and for magnetic fields in moving conductors it is a question with consequences. In a perfect conductor the magnetic field lines move with the material, so it makes sense to say that a line is the same line from one moment to the next; in a real plasma, lines can break and reconnect, and whether a given line at one instant is “the same” as a line at another has no unique answer. Defining a velocity for magnetic field lines in general, so that the motion of charges along them can be described when they are moving too, is possible in several inequivalent ways, and which one is physically meaningful in turbulent plasmas — in the solar corona, in fusion devices — is still argued.

The habit worth carrying away is to ask what a curve is the curve of. A field line is the curve whose direction is the force; a charge’s path is the curve whose bending is the force divided by its mass and speed squared — so a charge released at rest sets off along its line and swings outside every bend, and only drag strong enough to remove its velocity (the departure falling as one over the drag squared) or a magnetic field that turns it without slowing it keeps it on the line. Any time a curve is drawn through a field, it is worth asking whether it records where something is pushed or where it goes.