Illustrated Physics
physics, drawn
Concept

Plasma — where it appears

A gas ionised enough that its electrical behaviour dominates its mechanics, so that it carries currents and drags magnetic fields about with it. It is the state of almost all visible matter, and its defining approximation is that the field is frozen into the fluid.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

A radar echo past the Sun, delayed by 233 microseconds. The extra time a round-trip radar signal takes when its path passes close to the Sun, against how close, for a reflector 0.723 AU away. Grazing the Sun's limb the delay is 233 microseconds — about 70 kilometres of light travel, on a path of hundreds of millions — and it falls only as the logarithm of the impact parameter, so the effect is still tens of microseconds ten solar radii out. That slow falloff is what makes the measurement possible: the delay can be watched building and fading as the geometry changes, rather than having to be caught at one instant. The dashed curves are the delay the solar corona's plasma adds at three radio frequencies. It is the competing effect, it is larger than the gravitational one close in, and it falls as the square of the frequency while the gravitational delay does not depend on frequency at all — which is how the two are separated, and why the sharpest measurement of this was made with a spacecraft carrying three radio links instead of one.

The delay that is not a bend

The same metric that bends a ray also slows it, and the two are different tests. A radar echo from Venus arrives 233 microseconds late when its path grazes the Sun — and half of that delay is accumulated more than twenty solar radii away, in a field thousands of times weaker.

astrophysics · Light deflection The field near a neutral point. Field lines near a magnetic null, traced by following the local field direction rather than plotted from the closed form. The field is B ∝ (y, k²x) with k = 1, whose lines are the hyperbolae y² − k²x² = constant and whose separatrices are the straight lines y = ±1x. At k = 1 the X is symmetric and the current density is exactly zero: the field is curl-free, and nothing is stored in it beyond the field itself. The four quadrants are four separate flux systems, and which of them a given line belongs to is the quantity a frozen-in field is not allowed to alter.

The knot the field cannot untie

A perfectly conducting fluid cannot change which field line joins which piece of it. So two flux systems pushed together may be squashed indefinitely and can never merge, and the energy of the squashing accumulates with nowhere to go. The release happens only where the perfect conductivity locally fails — in a sheet three metres thick inside a structure ten thousand kilometres across — and the rate that follows is a hundred thousand times too slow for the flares that are observed.

astrophysics · Flux freezing One line, 21 decades of density, and every mirror on it. Plasma frequency against electron density, both logarithmic, with the horizontal rules at frequencies a reader already has a feel for. A wave is reflected by everything to the right of where its own rule meets the line and passes through everything to the left. a fluorescent tube at 1.0e+17 m⁻³ cuts off at 2.84 GHz; ionosphere, D layer at night at 1.0e+8 m⁻³ cuts off at 90 kHz; ionosphere, E layer at 1.0e+11 m⁻³ cuts off at 2.84 MHz; ionosphere, F2 layer at 1.0e+12 m⁻³ cuts off at 8.98 MHz; aluminium's conduction electrons at 1.8e+29 m⁻³ cuts off at 3819.9 THz. So the F2 layer turns back a 1 MHz broadcast and lets a 100 MHz one straight out, which is why one of them is heard across an ocean at night and the other stops at the horizon; and aluminium's cutoff sits in the far ultraviolet, which is why it is a mirror for everything visible and a window above 78 nm. The dependence is a square root, so the line has slope one half: a hundredfold denser plasma reflects only ten times the frequency. Arriving at an angle helps by a factor of sec θ — 1.00 at 0°, 1.15 at 30°, 2.00 at 60° — because only the component of the motion along the density gradient has to be turned round.

The frequency below which nothing gets in

Free charges give a medium a permittivity that is negative, and a negative permittivity is not an absorbing medium — it is one in which no wave exists at all. Below that frequency the reflection is total, exactly rather than nearly, because there is no transmitted wave and nothing to absorb. The same expression puts the number at 9 MHz for the ionosphere and 3.8 PHz for aluminium.

astrophysics · Plasma oscillation The charge a plasma hides. The potential around a charge in a plasma, as a fraction of the bare Coulomb potential at the same distance, against distance measured in screening lengths. The electrons crowd toward the charge and the ions move away until the rearrangement cancels the field, and what is left falls as exp(−r/λ_D) on top of the ordinary 1/r. At one screening length the potential is already down to 37 per cent of the bare value, at three to 5 per cent, and at ten to 4 × 10⁻⁵ — so a charge in a plasma is invisible beyond a few λ_D, and the long range of the Coulomb force, which is what makes electrostatics awkward everywhere else, is simply gone. The screening length at 10¹¹ m⁻³ is 3.8 mm at 300 K, 6.9 mm at 1000 K, 21.8 mm at 10000 K, rising as the square root of the temperature because a hotter electron is harder to hold in place. Drawn this way the three curves coincide exactly: the shape is universal and the only thing a plasma's density and temperature decide is the length written on the axis.

The long-range force that does not reach

The Coulomb force falls off as slowly as gravity does, which is what makes electrostatics awkward — every charge is in principle in contact with every other. Put the same charge into a plasma and it becomes invisible beyond a few millimetres, because the mobile charges around it rearrange until its field is cancelled. What is left is a screened potential with a definite range, and that range is what makes a plasma a plasma.

astrophysics · Plasma oscillation The one place in a plasma where the charges do not balance. The potential and the two densities through a sheath, in Debye lengths from the sheath edge, for a plasma of hydrogen ions. The potential is measured downward in units of the electron temperature and reaches 2.84 at the wall — the value at which the two fluxes balance. The layer is 15.1 Debye lengths thick — for 3 eV electrons at 1e+16 per cubic metre, a Debye length of 129 µm and a sheath of 1.9 mm. Inside it the electron density falls as the Boltzmann factor while the ion density falls only as the ions speed up, so the two separate: at the wall the ion density exceeds the electron density by 85 per cent of itself. That is the whole of the difference between a sheath and the plasma it borders — quasineutrality is not an assumption that can be made here, and Poisson's equation has to be solved instead of replaced.

The wall a plasma builds against itself

A plasma is quasineutral everywhere except where it touches something. Electrons are faster than ions by the square root of the mass ratio, so any surface is struck by far more of them, charges negative, and goes on charging until the two arrivals balance — leaving a layer a few Debye lengths thick in which the charges do not cancel at all.

astrophysics · Plasma oscillation Three charges, three orbits, one drift. Three particles released at rest in crossed fields — 1000 V/m across 0.1 T — with their paths integrated by a scheme that rotates the velocity rather than adding to it, so the magnetic part changes no speeds. The three loops have wildly different sizes and periods: the the electron turns at 2799.25 MHz, the proton turns at 1.52 MHz, the α particle turns at 0.76 MHz. Their guiding centres all creep along the same line at the same rate, measured here from the orbits at -1.000e+4, -1.000e+4 and -1.000e+4 m/s against −E/B = -1.000e+4 m/s, a spread of -0.00 per cent. Neither the charge nor the mass nor the sign appears in the answer. A plasma in crossed fields therefore moves bodily and carries no current from this drift at all, which is the opposite of what an intuition built on ions being heavier than electrons expects.

The drift that does not care what the charge is

A charge in a uniform magnetic field goes round in a circle and arrives nowhere. Add anything at all — an electric field, gravity, a gradient in the magnetic field itself — and the circle's centre creeps sideways, at right angles to both. One of those drifts is the same for every particle regardless of charge, sign or mass; the others are not, and the difference decides what a plasma does.

electromagnetism · Magnetism One magnitude, and a direction that turns the wrong way. The force per unit area a magnetic field of 0.01 tesla exerts on a surface, drawn for surface normals at 0°, 30°, 45°, 60°, 90° to the field. The short grey arrows are the normals; the long coloured ones are the tractions. Every traction has the same length, 40 pascals, because the magnitude of T·n does not depend on the orientation of n at all — and the direction is the normal reflected in the field rather than carried round with it, so as the surface turns one way the force turns the other. A surface cutting across the field is pushed (the magnetic pressure); a surface cutting along it is pulled (the magnetic tension); and at 45° the traction lies in the surface, which is a shear and is neither.

The same force whichever way the surface faces

A magnetic field's stress is usually described as a pressure across the lines with a tension along them, as though it were two effects. It is one. The force per unit area is B²/2μ₀ on every surface however it is oriented, and only the direction changes — the surface normal reflected in the field, so that turning the surface one way turns the force the other. At 45° neither name applies.

astrophysics · Flux freezing Three speeds, drawn against direction. The phase speed of the three magnetohydrodynamic waves against the angle between the wavevector and the field, drawn as a polar diagram with the field horizontal and the sound speed 0.6 times the Alfvén speed. Along the field the two magnetosonic branches take the values of the sound speed and the Alfvén speed themselves and swap which is which as the ratio crosses one; across the field the slow branch vanishes entirely and the fast branch runs at the quadrature sum. The shear branch is the figure-of-eight, v_A cos θ, and it is the only one of the three whose speed contains no thermodynamic quantity — no pressure, no temperature, no sound speed. It does not know what the gas is made of.

The wave that does not know what the gas is made of

Give the magnetic tension of a bent field line an inertia and it becomes a string. The wave that runs along it goes at the same speed at every wavelength, compresses nothing anywhere, and has a speed containing no temperature, no pressure and no sound speed — it is the only wave in classical physics that is indifferent to what the medium is. Its energy travels along the field whatever direction the wave was sent in.

astrophysics · Flux freezing The electrode that walks itself negative. An electrode driven through a blocking capacitor by a symmetric waveform of amplitude 20 kTe, in an argon-like plasma whose electrons are collected 108 times as readily as its ions at saturation, starting uncharged. Potentials are relative to the plasma, in electron temperatures. On the first positive half-cycle the electrode draws a flood of electrons and nothing on the negative half-cycle can return the charge, because the ions arrive at a fixed, small rate; the electrode walks down, and within 6 cycles its mean potential is within half an electron temperature of the self-bias. That bias, found by integrating the charging, is −22.27 kTe, and the charge balance with a Bessel function gives −22.27. The floating potential of the same wall undriven is −4.68. At 3 eV, a 60 V drive makes a 67 V bias with no direct current supplied anywhere.

The bias no battery supplies

Drive an electrode in a plasma through a capacitor with a perfectly symmetric waveform and it charges to a steady negative voltage almost as large as the waveform's own amplitude. No direct current flows anywhere and no battery is connected. The plasma's sheath lets electrons in far more easily than it lets them out, and every semiconductor wafer etched in the last forty years was held at a voltage made that way.

astrophysics · Plasma oscillation The field a pinch gives up relaxing into. The axial and azimuthal field across a cylinder of conducting plasma in the minimum-energy state at fixed helicity, Bz = J₀(λr) and Bθ = J₁(λr), drawn for λa = 1.5 and λa = 3. Solid lines are the axial field and dashed lines the azimuthal field. Both satisfy ∇×B = λB, checked by finite differences at three radii, so the current runs along the field everywhere and the field exerts no force on the plasma. At λa = 3 the axial field passes through zero at r = 0.802a and is reversed outside it. The reversal needs λa above 2.405, the first zero of J₀, and nothing was imposed at the edge to produce it. λa = 1.5: pinch parameter Θ = 0.75, reversal parameter F = 0.688. λa = 3: pinch parameter Θ = 1.50, reversal parameter F = -1.150.

The twist that outlives the turbulence

A plasma pinch driven hard enough goes violently unstable, and then settles into the same quiet state however it was started — with the field at its edge pointing backwards. The explanation is that turbulence destroys almost every constraint a perfect conductor obeys and spares one. The magnetic helicity, a measure of how twisted and linked the field is, decays far more slowly than the energy, and a field that has shed all the energy it can at fixed helicity has only one shape available to it.

astrophysics · Flux freezing Circles that go nowhere, and a current across the line. Gyrating ions in a uniform magnetic field pointing out of the page, with 60 guiding centres drawn from a density that falls by a factor of e every 3 gyroradii to the right, and Maxwellian speeds. Every ion goes round clockwise and none of the circles moves. Of the circles that cross the dashed vertical line, those centred to its left cross it moving down and those centred to its right cross it moving up; in this sample 11 cross moving down and 7 moving up, a count a sample this small could turn either way. Because there are more circles on the left, the ions at the line move downwards on average over every speed and phase, at exactly the thermal speed squared over the gyrofrequency times L, 0.33 thermal speeds, computed by averaging over speeds and phases and checked against that value. It is a current, carried by circles whose centres are still.

The current no particle carries

A magnetised plasma holds its own pressure against the field only if a current flows across the pressure gradient, and the fluid equations say exactly how much. Follow the particles in a uniform field and none of them is going anywhere; every guiding centre is still. The current is real all the same. It is made of circles that are more crowded on one side of a line than the other, and when the field is not uniform, the drifts that do move the guiding centres flow the wrong way.

electromagnetism · Magnetism

Named alongside it

The objects these essays reach for when they reach for this one.

Flux freezingMagnetic energyPlasma betaDebye lengthDispersion relationDissipationDriftEquilibriumFloating potentialGroup velocityGuiding centreLangmuir probe

All concepts