The drift that does not care what the charge is

A magnetic field alone cannot make a charge go anywhere. The force that does no work is the reason: the force is always across the motion, so it turns the velocity and never lengthens it, and what results is a helix that advances along the field and circles endlessly across it. The field itself is whatever the currents elsewhere have made, as in the field that wraps a current, and for the whole of this rung it is simply given. A particle in a uniform field is confined in two directions and free in the third, which is a good start and nothing more.

Everything interesting happens when something else is added. And what is added need not be large: a force a millionth of the magnetic one still produces a steady sideways creep, because a small force acting on a circle does not merely bend it — it displaces its centre, once per turn, in the same direction every time.

The circle that does not close #

The mechanism is worth having in words before any formula.

Take a positive charge circling in a field pointing out of the page, and add an electric field pointing right. On the half of the circle where the particle moves with the electric force, it gains speed; on the other half it loses it. A faster particle turns on a wider radius and a slower one on a tighter radius, so the circle is not a circle: it is wide on one side and tight on the other, and it fails to close by a small amount every turn.

The failure is always in the same direction, and that direction is perpendicular to both the field and the added force. Over many turns the small failures add up to a steady velocity, and the fast circling can be averaged away entirely — leaving a fictitious point, the guiding centre, that moves smoothly. It is fictitious in the same sense that a centrifugal force is — a device for describing motion in terms simpler than the motion, as in the forces that are not there — and like that one it is exactly as trustworthy as the frame it is defined in. Its velocity is

$$\mathbf{v}_d = \frac{\mathbf{F}\times\mathbf{B}}{qB^2}$$

This is the same manoeuvre as held up by a force that averages to nothing, where a fast oscillation is averaged out and leaves behind an effective force that was invisible in the original equation. Averaging over a fast motion is one of the most productive things in physics and one of the easiest to do wrongly, and both essays are examples of doing it carefully enough to say when it fails.

The charge in the denominator is what makes the electric case special. For an electric force $\mathbf{F} = q\mathbf{E}$, the charge cancels, and every particle drifts at $\mathbf{E}\times\mathbf{B}/B^2$ — the same speed, the same direction, for an electron and for a uranium ion. Nothing about the particle survives into the answer.

Why that is more surprising than it looks #

An intuition trained on unmagnetised plasma expects the opposite. In an electric field with no magnetic field, electrons accelerate away from ions immediately, and charge separation is the first thing that happens; that is what makes the long-range force that does not reach a subject at all, because a plasma spends its time cancelling out its own electric fields.

Magnetise it, and the electric field stops separating anything. The whole plasma slides bodily across the field lines at $E/B$, ions and electrons together, carrying no current at all from this motion. A perpendicular electric field in a magnetised plasma is not a driver of current; it is a statement about the velocity of the frame the plasma prefers to sit in.

Any force that is not proportional to charge does separate the species, and gravity is the clean example. The drift velocities are minute — a proton in a tenth of a tesla creeps at less than a millimetre a second under Earth’s gravity — but the sign is opposite for the two species, so a current flows. That current is not a curiosity. It is the seed of the instability that makes a heavy plasma supported by a magnetic field fall through it: the separated charge makes an electric field, the electric field drives an $\mathbf{E}\times\mathbf{B}$ drift of everything together, and that drift feeds the disturbance that separated the charge in the first place.

Where the drift is the machine #

Two devices exist mainly to exploit the fact that the drift does not care about the particle.

A magnetron — the thing in a microwave oven — is a cylinder of cathode inside an anode, with an axial magnetic field and a radial electric field. The electrons emitted from the cathode do not cross to the anode: they drift around the annulus at $E/B$, forming a rotating space-charge cloud whose bunching drives the resonant cavities cut into the anode block. At a fifth of a tesla and four kilovolts across a couple of millimetres, the drift is some thousands of kilometres a second — enough to carry the cloud once round an anode bore of a centimetre in a few nanoseconds, which is where the gigahertz comes from. The device is a drift given something to talk to.

A Hall thruster does the reverse: it uses the drift to hold electrons where they are wanted. A radial magnetic field of a few hundred gauss across an axial electric field of tens of kilovolts per metre makes the electrons circulate azimuthally rather than reaching the anode, so they linger long enough to ionise the propellant — while the ions, whose Larmor radius is far larger than the channel, are unmagnetised and simply accelerate straight out. The same field magnetises one species and not the other, because the Larmor radius carries the mass, and that asymmetry is the whole design.

Neither device would work if the drift depended on the particle. In both, the useful behaviour is a population moving together at a speed that has nothing to do with what it is made of.

A gradient in the field itself #

The magnetic field’s own non-uniformity does the same job. If the field is stronger on one side of the orbit than on the other, the radius is smaller there — since the radius goes as $1/B$ — and the loop again fails to close.

This drift keeps the charge in it. Electrons and ions go opposite ways, so a plasma confined by a field with a gradient — which is every plasma confined by any real magnet, since a field that is strong somewhere and weak elsewhere is the only kind that confines — separates charge along its own field lines and drives a current. What to do about that current is a large part of what distinguishes one confinement scheme from another.

Two drifts, and only one of them cares about the species. That is the practical summary of the last three figures, and it is the reason plasma physics keeps two lists rather than one: drifts that move the fluid, and drifts that move current through it.

The quantity that survives #

The picture so far treats the field as though its strength were a fixed backdrop. Follow a particle into a region where the field really strengthens and something better appears.

The invariant quantity is $\mu = mv_\perp^2/2B$, and it is exactly the magnetic moment of the tiny current loop that a gyrating charge amounts to — the same object as the loop that behaves like a needle, except that here the loop is the particle’s own path.

It is not conserved. Nothing forbids it changing, and there is no symmetry behind it. What can be proved is stranger and, for practical purposes, better: its change is smaller than any power of the slowness of the variation. A field changing over a thousand gyrations does not disturb $\mu$ by a part in a thousand, or a part in a million, but by an amount that falls off exponentially. That is what an adiabatic invariant is, and it is why a quantity with no conservation law behind it can be relied on more heavily than most conserved ones.

The payoff is a reduction of the problem. A three-dimensional orbit becomes a particle sliding along a field line in a potential $\mu B$, with $\mu$ fixed. Everything about trapped particles follows from that one-dimensional picture.

The bottle, and the hole in it #

If a field is weak in the middle and strong at both ends, the potential $\mu B$ has a well in it, and a particle sliding along the line runs uphill at each end. Its perpendicular energy has to grow as it does, and the growth is paid for out of its motion along the field. When that motion is used up, the particle turns round.

The condition depends on the direction of the velocity and not on its size, which is the first surprise: a fast particle and a slow one moving at the same angle to the field are trapped or lost together. The second surprise is the shape of the curve. Making the end fields ten times stronger shrinks the escaping cone by a factor of about three, and there is no ratio at which the cone closes.

There is a third timescale hiding in that description, and it completes the hierarchy. A trapped particle gyrates fastest, bounces between the mirrors more slowly, and drifts around the machine slowest of all — and because the bounce is itself a periodic motion, it has its own adiabatic invariant, the action taken around one bounce. A particle whose bottle is slowly squeezed keeps that second invariant too, and gains energy as the mirrors approach: an argument first made about cosmic rays being accelerated by moving magnetic clouds, and the same argument that says a ball bouncing between a wall and a slowly advancing bat speeds up.

A magnetic bottle therefore leaks by construction. Worse, collisions continually scatter particles into the cone, so the loss is not a one-off emptying of an unlucky population but a steady drain fed by the plasma’s own thermal collisions. That is the reason mirror machines were pursued hard and then abandoned in favour of closed field lines, where a particle following its line never arrives anywhere it can leave from.

The same physics is what makes the Earth’s field trap charged particles at all, and there the leak is the point: the particles that fall inside the loss cone are the ones that reach the atmosphere.

What the invariant is worth as an instrument #

An invariant that holds to a part in a thousand is a measuring device, and the standard use is to read a particle’s history off its present pitch angle.

Because $\mu$ is fixed, the ratio $v_\perp^2/B$ is the same everywhere along a particle’s path. Measure the angle between its velocity and the field at one place, and the field strength at that place, and the field strength at the place it turned round is fixed: $B_{\text{turn}} = B/\sin^2\alpha$. Nothing about the intervening journey is needed — not the path, not the time, not the shape of the field between the two. A single-point measurement of a distribution of pitch angles therefore reports on a region nobody has been to.

That is how a spacecraft with one particle detector maps a magnetic geometry it is not inside, and it is the same logic as the loss cone: a hole in the pitch-angle distribution at a known angle says where the field stops rising. A distribution with no hole in it says the particles have been scattered since they last mirrored, which is itself a measurement — of collisions, or of waves at the gyrofrequency, which are the two things that break the invariant.

The general habit is worth naming. An adiabatic invariant converts a quantity that is hard to observe, the history, into one that is easy, a present angle — and it does so exactly to the extent that the assumption behind it holds, so an invariant that is measured to fail is reporting on the thing that broke it.

The gap the whole picture lives in #

Everything above is an approximation, and it is worth being explicit about what it is an approximation in.

The expansion parameter is the Larmor radius divided by the distance over which the field changes, or equivalently the gyroperiod divided by the time a drift takes to matter. When it is small the fast motion can be averaged away and the guiding centre is a good description of where the particle is. When it is not, there is no guiding centre — not a poor one, none — because the orbit does not close well enough for a centre to be defined.

The places where the ratio fails are exactly the interesting ones. At a magnetic null the field vanishes and the Larmor radius is unbounded, so nothing here applies within a region around it; in the knot the field cannot untie that region is precisely where the field changes its connectivity. A collisionless shock has a field profile with a width comparable to an ion orbit, so ions cross it in one turn while electrons remain perfectly guided, and the two species have to be described differently in the same place.

That pattern — a description that works everywhere except in the thin layers where the physics happens — is common enough to be a warning rather than a curiosity.

Where the model stops #

The drifts here are first order in the small parameter. There are more of them at higher order — polarisation drift when the electric field changes with time, the curvature drift that goes with a bent field line, corrections from a finite orbit sampling a curved field — and a serious calculation carries the whole list. What does not change is the structure: each drift is a force crossed into the field and divided by $qB^2$.

Relativistic particles are not covered. The gyrofrequency becomes $qB/\gamma m$, so the mass in every expression above is the relativistic one, and the mirror condition and the invariant have to be rewritten in terms of momentum. Nothing qualitative moves, but no number here survives.

Collisions are absent throughout. Every claim about invariance assumes the particle is only pushed by the fields. A collision randomises the pitch angle in one event and destroys $\mu$; the whole subject of how a magnetically confined plasma leaks is the competition between the adiabatic picture and the collisional one.

And the fields are prescribed rather than solved for. The particle moves in a given field and does not contribute to it. That is the wrong assumption for anything dense enough to be called a plasma, where the currents the drifts carry make fields of their own — which is where magnetism is electricity seen sideways stops being a change of viewpoint and starts being a coupled problem.

What the pictures cannot show #

The orbits are drawn projected onto a plane, and every one of them is a helix with motion along the field that the projection removes. The particle in the mirror figure is travelling along the axis the whole time; the plot follows quantities rather than positions, precisely because a picture of the path would show a spring being wound tighter and would not show why.

The scale figure draws two timescales as bars and gives no sense of the third, which is the time between collisions. In a laboratory plasma that time often falls between the gyration and the drift, which does not invalidate the picture but changes which of its conclusions survive — and there is no honest way to put a bar for it on a plot without specifying a density and a temperature the rest of the figure does not have.

Where the ladder goes next #

The magnetism ladder began with the field with no ends and the force that does no work, and went on to the loop that behaves like a needle, where a circulating current becomes a magnetic moment and therefore an object a field can push on. This rung takes the same loop, notices that a real particle is one, and follows what happens when the field it sits in is not the same everywhere.

The rung after it is the one where the particles make the field they are moving in, and the drifts and the currents have to be solved for together. The habit to carry there is the one this rung is built on: separate the fast motion from the slow one, average over the fast, and then check the ratio that made the separation legitimate — because the answer is only as good as that ratio, and the ratio is a number that can be computed.