Illustrated Physics
physics, drawn
Series

Flux freezing — the series

8 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. What a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.

    The field that cannot get out

    Squeeze a lump of conducting fluid and its magnetic field comes with it, because the flux through any loop that moves with the material cannot change. Halve the radius and the field goes up fourfold; collapse by a factor of seventy thousand and it goes up by five thousand million.

    part 1 · astrophysics
  2. 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.

    part 2 · astrophysics
  3. 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.

    part 3 · astrophysics
  4. 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.

    part 4 · astrophysics
  5. 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.

    part 5 · astrophysics
  6. The ring where an axisymmetric field vanishes. Field lines of the slowest-decaying dipole field of a conducting sphere, drawn in a plane through its axis (the axis is the left edge, the sphere's surface the arc). Each line is a contour of the flux function, which is zero on the axis and at infinity and positive between, so it must have a maximum. Here the maximum is on the equator at 0.873 of the radius: there the lines shrink to a point, which in three dimensions is a ring round the axis on which the meridional field is exactly zero. 7 of the 11 drawn lines leave the sphere and close outside; the other 4 close inside, round the ring. The toroidal current that makes the field is not zero on the ring: it is 89 per cent of its largest value.

    The field no symmetric flow can keep

    The Earth's core would forget its magnetic field in about forty thousand years, and the field is at least three and a half thousand million years old, so something must be making it again all the time. The obvious candidate is the flow of liquid iron. In 1934 Thomas Cowling proved that the simplest version cannot work: no motion of a conductor, however fast or however arranged, can keep up a field that is symmetric about an axis. The reason is a single ring inside the field where the current it needs cannot be driven.

    part 6 · astrophysics
  7. A disc that makes its own field once it turns fast enough. A self-exciting disc dynamo started from rest, its motor's torque set so that, with no current, the disc would run at 0.8, 1.5, 2.5 times the critical speed R/M. Dashed: the disc's speed, in units of the critical speed. Solid: the current in its coil, in units of √(G/M), starting from a stray seed of a millionth, of either sign. Time is in units of the circuit's own L/R. At 0.8 the disc settles at its free-running speed, below critical, and the seed dies. Above it the disc spins up past the critical speed and nothing happens for a while — the current is growing exponentially from a millionth — until it reaches a tenth, at t ≈ 29 for 1.5 and 11 for 2.5 and becomes large enough to brake the disc. Speed and current then ring and settle at exactly the critical speed and a current of ±√(g − 1): 0.71 and 1.22, of the sign the seed happened to have.

    The dynamo that reverses with nothing pushing it

    A copper disc spinning in a magnetic field drives a current from its axle to its rim. Wire that current through a coil wound round the disc and the coil supplies the field the disc needs — and above one definite speed, a resistance divided by an inductance, the arrangement makes its own field out of any stray one, with nothing else magnetic anywhere. It cannot choose which way the field points; the stray field chooses. Couple two such discs, each feeding the other's coil, and with the motors' torque held perfectly steady the field flips direction again and again at intervals that follow no pattern. The simplest dynamo that works is also the simplest one that reverses.

    part 7 · astrophysics
  8. A drift that twist turns into a circle. An ion's guiding centre at 10 keV in a torus the size of ITER, seen in the plane across the bore, with a vertical drift of 609 m/s from the field's curvature and gradient. With no twist it drifts straight up and reaches the wall in about 2.8 ms, ions one way and electrons the other. With the field lines twisted so that they circle the bore once for every three trips round the torus, the same drift is carried round by the motion along the line, up on the outside and back down on the inside, and the integrated path closes into a circle offset by only 1.3 cm. The offset is the drift speed over the rate of circulation. The twist does not stop the drift; it makes the drift average to nothing, which is all a bottle needs.

    The twist a bottle of plasma has to have

    A straight magnetic bottle leaks out of its ends, so the obvious repair is to bend it round into a ring. A ring of field with no twist in it loses its plasma in about three millionths of a second. Adding a twist repairs that by making a drift average to nothing, and adding too much makes the whole column kink like an overwound rubber band. A tokamak is built in the narrow band between those two failures.

    part 8 · astrophysics

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