Illustrated Physics
physics, drawn
Concept

Magnetic pressure — where it appears

The push a magnetic field exerts on a surface cut across its lines, B²/2μ₀. It adds to a gas pressure in any balance, which is why a magnetised flux tube is lighter than its surroundings and rises, and it is 398 kilopascals at one tesla — the number that sets how hard an electromagnet holds.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Four plasma columns that need exactly the same current. Left: four pressure profiles across a 1 cm column — flat-topped, parabolic, peaked and hollow — each holding the same 10¹⁹ ions per metre at the same 10 keV, so the same total ∫2πr p dr. Right: the current each needs to hold itself together, from solving the force balance J × B = ∇p with Ampère's law, radius by radius. All four come out at 0.800, 0.800, 0.800, 0.800 MA: Bennett's 0.800 MA. The shape of the column decides where the current flows and how strong the field is inside it, and has no effect at all on how much current there must be.

The current that squeezes what carries it

A current flowing along a column of hot gas makes a magnetic field that wraps round the column, and the field pushes the current inward. If the column is a plasma, free to move, it is squeezed. Ampère's law and a balance of forces then fix how much current a column of given temperature needs to hold itself together — and the answer does not depend on how wide the column is or how its plasma is arranged inside. Ten thousand electronvolts in ten million million million ions per metre needs 0.8 million amperes, whatever the shape. Above about 1.4 million amperes the same arithmetic says a hydrogen column radiates faster than its current can heat it, and collapses.

electromagnetism · Ampere law The armature sits where the field has halved. The magnetic field halfway between two parallel rails carrying a current out along one and back along the other, against the distance along the barrel in units of the rails' spacing, as a fraction of the field between infinitely long rails. The rails end at zero, where the armature closes the circuit. A few spacings behind the armature the field is the full value: 98.5 per cent at two spacings back. At the armature it is exactly half, because each rail there is a wire that stops, and a wire that stops gives half the field of one that does not. Ahead of the armature it falls away: 15 per cent half a spacing in front. The current in the armature crosses this field and is pushed forward; the push is the current times the field it sits in, which is half the field behind it, and that is where the half in ½L′I² comes from.

The push at the end of a pair of rails

A railgun is two metal rails, a sliding conductor across them and a very large current, and the push on the slider is half the inductance per metre times the current squared. The half is not a fudge. The slider sits at the end of two rails that stop, and a wire that stops gives exactly half the field of one that does not. Followed through, the same circuit pushes its own rails apart six times harder per metre than it pushes the projectile, stores as much energy in its field as it gives the projectile, and wastes less the faster it shoots.

electromagnetism · Ampere law

Named alongside it

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

Magnetic energyPlasmaFlux freezingMagnetic tensionPlasma betaAlfven waveAmperes lawAnisotropyBiot savart lawBremsstrahlungBuoyancyCoronal heating

All concepts