The orbit that opens when E reaches cB

The drift that does not care what the charge is found one of the most useful results in plasma physics. A charged particle in crossed electric and magnetic fields does not accelerate along the electric field. It gyrates round the magnetic field and, on average, drifts sideways, perpendicular to both fields, at the speed $E/B$ — the same speed for electrons and ions, heavy and light, positive and negative, which is why a plasma in crossed fields moves as a whole.

The speed $E/B$ is a ratio of two fields that can be set independently, and nothing in the formula stops it exceeding the speed of light. The field nobody can transform away gave the reason the formula has to change there: the electric and magnetic fields are two faces of one object, and the combination $E^2 - c^2B^2$ is the same in every frame. When it is negative there is a frame with no electric field, and the drift is the motion of that frame. When it is positive there is no such frame. This essay follows a single charge across that boundary.

The loop in the frame that drifts #

The first figure releases a charge from rest in crossed fields of three strengths and integrates its motion exactly, with the relativistic equation of motion. Each path is a chain of arches: the charge is pulled along the electric field, turned by the magnetic field, brought back to rest at the bottom of each arch, and pulled up again, so that it advances to the right in hops. At low field strength the arches are cycloids, the curve traced by a point on the rim of a rolling wheel, and Newton’s law gives their height as $2mE/qB^2$.

The exact arches are taller, and much taller as the ratio $E/cB$ grows: 10 per cent at 0.3, 56 per cent at 0.6, three and a half times at 0.85. The reason is clearest in the frame that moves with the drift. A boost along the drift direction at speed $v_d = E/B$ turns the fields into

$$E' = \gamma_d(E - v_d B) = 0, \qquad B' = \gamma_d\left(B - \frac{v_d E}{c^2}\right) = \frac{B}{\gamma_d},$$

where $\gamma_d$ is the Lorentz factor of the drift. The electric field is gone, and the magnetic field is weaker by $\gamma_d$. In that frame the charge, which started at rest in the laboratory, is moving backwards at the drift speed, and a charge moving in a pure magnetic field circles at constant speed. The circle’s radius is the momentum over the charge times the field, $\gamma_d m v_d/qB' = \gamma_d^2 m v_d/qB$: one factor of $\gamma_d$ from the charge’s relativistic momentum, one from the weakening of the field. The height of the arch, which is the circle’s diameter measured across the boost and so the same in both frames, is $2\gamma_d^2 mE/qB^2$.

Magnetism is electricity seen sideways found that the magnetic force between currents is the electric force seen from a moving frame. Here it is the other way round: the electric field in the laboratory is part of what the magnetic field looks like from a frame moving at $E/B$, and the charge’s motion, so complicated in the laboratory, is a plain circle in that frame.

The height that runs away #

The second figure follows the height across the whole range. The integrated motion falls on the formula at every ratio checked. At weak electric fields relativity adds a per cent; at half the critical ratio, a third; at 0.99, a factor of fifty. Newton’s arches have a height that never exceeds $2mc/qB$ for any field the drift formula would allow, because Newton puts no limit on the drift speed and none on the charge’s speed at the top of the arch, which is twice the drift speed. Relativity caps the charge’s speed below $c$, so a charge that would have to reach $2E/B > c$ at the top of a Newtonian arch cannot, and it spends longer being turned, rising higher on a slower turn.

Each arch is also slower to complete. In the drift frame the circle takes a time $2\pi\gamma_d^2 m/qB$, and the laboratory, watching that frame move, sees it dilated by another $\gamma_d$: $2\pi\gamma_d^3 m/qB$. Near the critical ratio the charge spends almost all of its time near the top of each arch, moving at nearly the speed of light in the drift direction, and makes progress across the fields only rarely. The arches also change shape. A Newtonian cycloid is π times as wide as it is high, the width of a rolling wheel’s circumference against its diameter. The relativistic arch, drifting at $v_d$ for a time $2\pi\gamma_d^3 m/qB$, is $\pi\gamma_d$ times as wide as it is high: the wheel that rolls in the drift frame is seen from the laboratory with its circumference spread by the time dilation of its motion, and at $E/cB = 0.85$ each arch is nearly six times as wide as it is high.

Bounded below the line, unbounded on it #

The third figure watches the charge’s energy. Below the critical ratio it is bounded: it rises during the climb of each arch and returns to the rest energy at the bottom, and its largest value, $1 + 2\gamma_d^2 E^2/c^2B^2$ in units of $mc^2$, is finite however long the charge is watched. At $E/cB = 0.95$ the charge reaches a Lorentz factor of 19.5 at the top of each arch and is brought back to rest at every bottom. The magnetic field always wins in the end, turning the charge back against the electric push.

At exactly $E = cB$ the bound disappears. The energy grows without limit, and the integration measures its rate: as the time to the power 0.678, within two per cent of two-thirds. Just above the critical ratio the growth is faster, and in the long run it is the steady linear climb of a charge in a pure electric field. A small change in the ratio of the two fields, from 0.95 to 1.05, has turned a charge that returns to rest every few hundred units of time into one with a Lorentz factor of 152 and rising.

The frame that removes the other field #

The fourth figure explains the change in terms of frames. Below the critical ratio there is a frame, moving at $E/B$, in which the electric field vanishes, and the charge’s motion there is a circle; the laboratory’s arches are that circle seen from a moving frame, and they drift with it. Above the critical ratio the drift speed would exceed $c$ and no such frame exists. But a different frame does: one moving along the same direction at $c^2B/E$, slower than light because $cB < E$. In it the magnetic field vanishes and an electric field $E\sqrt{1 - c^2B^2/E^2}$ remains. A charge in a pure electric field is accelerated in a straight line without end — the hyperbolic motion of the wall of silence behind a rocket — and in the laboratory that motion is the run-away of the third figure.

Where the run-away’s energy comes from is answered by the same frames. Below the critical ratio, in the frame with no electric field, the charge’s energy is constant: a magnetic field does no work, and the charge circles for ever at one speed. The energy that rises and falls in the laboratory with each arch is the electric field’s work on the way up, returned on the way down, and the books balance every loop. Above the critical ratio, in the frame with no magnetic field, there is nothing to turn the charge back and the electric field’s work is never returned. The field is uniform and infinite, so it has unlimited energy to give, and in a real arrangement the run-away ends when the charge reaches the edge of the region where the field is held.

So the drift formula does not fail above $E = cB$ by giving a speed faster than light. It stops applying, because the frame it describes has ceased to exist, and a different frame, describing a different kind of motion, takes its place. The two frames’ speeds approach $c$ from opposite sides as the ratio approaches 1, and they meet there.

What decides which frame exists is the quantity $E^2 - c^2B^2$. The one quantity a boost leaves alone and six numbers, one object found that it is invariant: every observer, however they move, measures the same value. Negative, and the field is magnetic at heart, with a frame where it is purely magnetic. Positive, and it is electric at heart. The boundary between a bound charge and a run-away is therefore not a property of the laboratory’s choice of frame, and no observer could see a charge trapped that another sees escape.

Carried off along a semicubical curve #

The critical case itself has an exact solution, and the last figure draws it beside the integration. With $E = cB$ the fields have both invariants zero — the same fields, frozen in time, as a plane light wave travelling along the drift direction — and the motion has a conserved quantity that light imposes: the combination $\gamma - \gamma v_x/c$, the component of the charge’s four-velocity along the direction a light wave would travel, stays at its initial value. Starting from rest that value is 1. In terms of the charge’s own proper time $s$, its momentum along the electric field grows steadily, $\gamma v_y = s$ in these units, and the conserved combination then fixes everything else: $\gamma = 1 + s^2/2$, and the position is $x = s^3/6$, $y = s^2/2$. The path is a semicubical parabola, rising along the electric field as the two-thirds power of its progress along the drift, and it never turns back.

Since the laboratory time is $s + s^3/6$, the energy grows as the two-thirds power of the time, which is the rate the third figure measured. The light that moves an electron and pays it nothing found the same conserved light-front combination governing an electron in a laser pulse, where it guaranteed that the electron is left at rest when the pulse has passed. A pulse ends; the static field here does not, and the same conservation law that returns energy in one case extracts it indefinitely in the other. A field with $E = cB$ is a light wave of zero frequency, and it does to a charge what light does — pushes it forward along the direction of propagation — without ever letting go.

Where crossed fields like these occur #

Crossed fields with $E$ close to $cB$ are not exotic. A magnetron, the microwave source in every microwave oven, works by sending electrons in crossed fields round a cylindrical cavity, and its design depends on keeping the electric field below the value at which electrons cross from the cathode to the anode: the Hull cut-off, the condition that the loops in the first figure are too short to reach the anode. In pulsar magnetospheres, where a rotating magnetised neutron star induces electric fields comparable with $cB$, the regions where $E$ exceeds $cB$ — or where the component of $E$ along $B$ is not screened — are where particles are accelerated to the energies that produce the observed radiation. In the plasma physics of the drift that does not care what the charge is, the crossed-field drift is always non-relativistic because real plasmas screen any electric field larger than $cB$ long before it forms; the critical ratio is the reason they can.

Two other machines use the drift directly, and both show how far from the critical ratio ordinary engineering sits. A Wien filter places crossed electric and magnetic fields across a beam of charged particles, and only those moving at exactly $E/B$ — the drift speed — pass straight through, because for them the electric push and the magnetic push cancel; everything faster or slower is bent aside. It is a velocity selector, and in a mass spectrometer or the electron gun of a high-resolution microscope it sorts particles by speed without ever bringing them to rest. A Hall thruster, the electric engine that keeps many satellites in their orbits, crosses an axial electric field with a radial magnetic field in an annular channel. Electrons there are light enough to be trapped: they drift round and round the annulus at $E/B$, a closed current that never reaches the anode, and the ions they collide with are too heavy for the magnetic field to turn and are accelerated straight out of the channel. Xenon ions falling through 300 volts leave at about 21 km/s.

In both, the ratio $E/cB$ is tiny. A thruster’s field of 30,000 volts per metre across a magnetic field of 15 millitesla gives $E/cB$ of about 0.007, where the relativistic arches of the second figure are taller than Newton’s by five parts in a hundred thousand. The physics of the drift frame is the same at every ratio, and the critical value only matters where the electric field has nearly the strength of the magnetic one in natural units — in the gaps of a pulsar’s magnetosphere, or in the focus of the most intense lasers.

The cut-off a magnetron is built round #

The magnetron’s design question can be answered exactly with a conservation law that the figures have been using implicitly. Take a flat cathode and a flat anode a distance $d$ apart, a voltage $V$ between them and a magnetic field $B$ parallel to both. An electron leaving the cathode at rest climbs towards the anode and is turned back by the magnetic field; if it is turned back before it gets there, no current flows. Two quantities fix how high it gets. Its energy is $mc^2 + eEy$ at height $y$, since the electric field has done that much work. And its momentum along the drift direction is $eBy$, because the magnetic force along the drift, $ev_yB$, adds momentum at a rate proportional to how fast the electron climbs. At the top of its arch the electron is moving only along the drift, so its energy and that momentum are related by $\varepsilon^2 = (pc)^2 + (mc^2)^2$, and the electron just reaches the anode when

$$eBd\,c = \sqrt{(eV)^2 + 2\,eV\,mc^2}.$$

This is Albert Hull’s cut-off condition of 1921, written relativistically. For a slow electron it reduces to Hull’s $B = \sqrt{2mV/e}/d$; at high voltage the field needed to hold the electrons back is larger than Hull’s original formula says, by the extra $(eV)^2$ under the root, because a fast electron carries more inertia and is harder to turn. The figure’s arch heights are this calculation at every height rather than at one; the cut-off is the statement that the arch’s top must lie below the anode.

What the integration takes for granted #

The figures treat the fields as uniform, static and given, with the charge’s own field and radiation ignored. A charge accelerated to Lorentz factors of a hundred radiates, and in the strong fields where the critical ratio matters — near pulsars, or in the focus of an intense laser — the radiation reaction can be large enough to change the motion qualitatively, turning the run-away into a terminal state in which the charge radiates energy as fast as the field supplies it. The fields are also taken to fill all space. A real arrangement has edges, and a charge that is run away with at $E > cB$ eventually leaves the region of field with whatever energy it has collected.

What a single trajectory hides #

Each figure follows one charge from rest. A charge with an initial velocity draws different arches — prolate or curtate cycloids, depending on whether it starts faster or slower than the drift — and its loops have a different height, though the drift speed and the existence of the drift frame do not change. The critical ratio is a property of the fields, not of the charge, and every charge, whatever its starting speed, mass or sign, finds itself bounded below the ratio and unbounded above it. The figures show this for one starting condition only, and the generality comes from the frame argument, not from the drawings.

Still open: how a run-away is stopped in nature #

Fields with $E > cB$ accelerate any charge in them without limit, and in nature something always intervenes: the charges screen the field, radiation reaction caps the energy, or the field’s region ends. In the magnetospheres of pulsars and magnetars, which charges are accelerated where, how the gaps in which $E$ is unscreened open and close, and how the resulting particles produce the observed radio and gamma-ray emission are still worked out with large numerical simulations whose conclusions depend on how the pair production that loads the magnetosphere is modelled. The critical ratio drawn here is the starting point of those calculations; how the plasma arranges itself so that it is almost, but not quite, everywhere obeyed is not settled.

The habit worth carrying away is to ask whether a frame exists before using it. A charge from rest in crossed fields loops in arches $2\gamma_d^2mE/qB^2$ high while it drifts at $E/B$ — 6.13 instead of Newton’s 1.7 at $E/cB = 0.85$ — because in the frame moving at $E/B$ the electric field vanishes and the magnetic field is weakened by $\gamma_d$; at $E = cB$ that frame reaches the speed of light, and beyond it a frame without the magnetic field takes over and the charge runs away, as $E^2 - c^2B^2$, the same for every observer, changes sign. A formula that is the velocity of a frame stops working when the frame would have to outrun light, and what replaces it is a different frame.