Illustrated Physics
physics, drawn
Concept

Anisotropy — where it appears

The dependence of a material's response on the direction it is asked in, which turns a constant into a tensor or a preferred axis. In a magnet it is the energy difference between magnetising along the easy axis and across it, and it is what decides how wide a hysteresis loop is.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

The index that depends on which way the light is going. The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves.

The crystal that answers twice

Lay a piece of calcite on a printed page and the print appears twice. One image sits still when the crystal is turned and the other goes round it. Nothing has been done to the light except pass it through a material whose response to a field is not a number.

optics · Polarisation Where a magnet actually sits on its own curve. The second quadrant of a magnet's B–H curve, for a material with a remanence of 1.28 T, and the load lines four shapes of it impose. A magnet's own poles put it in a reverse field of N·M, so the working point is where the curve meets the line B = −μ₀(1−N)/N·H. A long thin magnet with N = 0.02 keeps 98 per cent of its remanence and a squat one with N = 0.7 keeps 30 per cent — the same material, cut differently.

The magnet that has to fight its own field

A bar magnet's own poles put it in a reverse field, so the same material cut short and fat is weak and cut long and thin is strong. And a magnetised material does not have a magnetisation — it has a magnetisation and a history, which is why the word for what it does is the Greek for coming late.

electromagnetism · Magnetisation The cross a shaken cylinder leaves in a stratified fluid. The beams radiated by a small body oscillating in a fluid of buoyancy frequency 1.053e-2 per second — a period of 9.9 minutes — at 0.3, 0.6, 0.9 times that frequency. The disturbance does not spread in circles. It leaves along four rays, and the angle of those rays to the horizontal is fixed entirely by the ratio of the driving frequency to the buoyancy frequency: 17.5° at 0.3N, 36.9° at 0.6N, 64.2° at 0.9N. Nothing about the size of the body, the amplitude of the shaking or the wavelength enters. Drive it faster and the beams stand up; drive it slower and they lie down; drive it above N and there are no beams at all, because the dispersion relation ω = N cos φ has no solution. The short arrows across each beam are the wavevector, which is perpendicular to the beam — the dot products drawn here are 6e-17 — so the crests travel sideways across the ray while the energy travels along it, and a fluid doing this looks, in a photograph, as though its waves are moving at right angles to where they are going.

The wave that picks an angle

Shake a rod slowly in a tank of salty water layered by density and the disturbance leaves along four straight beams, at an angle fixed entirely by how fast the shaking is. Change the wavelength and the angle does not move. Change the frequency and it does. Above the buoyancy frequency there is no wave at all.

fluids · Stratification 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 pattern that stands still while the air goes through it. Streamlines of a steady 20 metre-per-second wind over a bell-shaped ridge 800 metres high and 6.0 kilometres wide, in air whose buoyancy frequency is 1.05e-2 per second. Nothing in the picture is moving: the air crosses it from left to right at 20 metres a second and the waves stay where they are, because standing still is what selects them. The vertical wavelength is 2πU/N = 11.9 kilometres, and the crests lean upstream as they rise — checked here by finding the maximum displacement a quarter of a wavelength up, which sits well upstream of the ridge. That tilt is the signature of energy travelling upward, and it is the feature every hand-drawn version of this picture gets backwards. Cloud forms at the crest of each wave where the air is highest and coldest, which is why the lens-shaped clouds sit stationary in a moving airstream.

The wave that is required to stand still

A stratified fluid supports internal waves at every wavelength there is. Put a steady wind over a ridge and the requirement that the pattern stay put picks exactly one of them — 2πU/N, and nothing about the mountain appears in it. The clouds that mark the crests sit still while the air goes through them at twenty metres a second, and the momentum the wave carries away is delivered thirty kilometres up.

fluids · Stratification A reflection that keeps the angle to gravity and not to the wall. An internal wave beam of frequency 0.5N reflecting from a slope of 12°, with the wavelength ratio for slopes of 12°, 20°, 28° computed beside it. The frequency fixes the angle the energy makes with the horizontal — 30.0° here — because the restoring force is gravity and gravity is vertical, so the reflected beam must leave at that same angle whatever the wall is doing. Incident and reflected rays are therefore not mirror images, and the wavelength changes on reflection by sin(θ+α)/sin(θ−α). A flat floor gives one, checked exactly. A slope approaching the ray's own angle gives infinity, checked as the limit, and that is where a basin's internal tide is compressed until it breaks.

The reflection that changes the wavelength

An internal wave's frequency fixes the angle its energy makes with gravity, so a sloping wall cannot send it back the way a mirror would. The reflected beam leaves at the same angle to the vertical rather than the same angle to the wall, its wavelength changes by a factor that diverges when the slope matches the ray, and in a closed basin the changes accumulate until every ray in the fluid lies on one line.

fluids · Stratification A blackbody in every direction, at a different temperature in each. The spectrum of a blackbody at 100 kelvin in its own frame, seen by an observer it is moving past at 0.5 of the speed of light, in 5 directions. Each curve is a Planck spectrum exactly — the Planck form survives a Doppler shift, with the temperature multiplied by the shift — and the temperatures run from 57.74 kelvin looking one way to 173.21 looking the other. So the body is a perfect blackbody in each direction and has no single temperature. A thermometer placed in the radiation reads something between, and what it reads depends on where it is put and on how much of the sky it sees — which is the reason a transformation law for temperature was argued about for sixty years without being found.

The body that has no temperature when it moves

Energy, momentum, length, duration and field strength all change when the observer moves. Temperature was argued about for sixty years, with three transformation laws proposed and each defended by people making no mistake. The resolution is that a moving blackbody is a perfect blackbody in every direction at a different temperature in each — so a thermometer's reading depends on where it is put, and the quantity the law was for is not there.

relativity · Relativistic thermodynamics

Named alongside it

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

Dispersion relationGroup velocityBuoyancy frequencyInternal wavesStratificationBoundary conditionDissipationEvanescenceAlfven waveBirefringenceBlackbodyCircular polarisation

All concepts