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The figure library — page 4

Every picture here is generated from code at build time. This page lists each family of figures in its plainest form, with every essay that draws on it.

osmotic-cell

6 essays

The column a dissolved thing holds up. Two arms of one vessel, joined below by a membrane that passes water and not solute. On the right is 10 mol/m³ of dissolved particles at 25 °C; on the left, pure water. Water crosses into the solution until the extra weight of the right-hand column has raised its pressure by the osmotic pressure — 24.8 kPa, which is 2.53 m of water, drawn here to scale. Nothing is pulling. The solvent is at a lower chemical potential where it is mixed, so it moves that way, and it stops when mechanical pressure has made up the difference. A solute a thousand times more dilute than seawater lifts a column taller than a person.

Every branch osmotic-cell draws, and what checks it

Trajectories at one speed and several angles. Projectile paths launched at the same speed and five different angles. The 45° launch travels furthest, and the 20° and 70° launches land in the same place.

Every branch projectile-angles draws, and what checks it

Which grains the light wins. The radiation force on a spherical grain divided by the gravitational force on it, against the grain's radius, on logarithmic axes. Both forces fall as the inverse square of the distance, so the ratio does not depend on how far away the grain is — only on how big it is. Light acts on the cross-section and gravity on the volume, so the ratio goes as 1/a, and the two are equal at 287 nm for material of density 2000 kg/m³. Anything smaller than that is expelled; anything larger stays.

Every branch radiation-pressure draws, and what checks it

How much each colour is scattered. Scattering strength against wavelength, as the inverse fourth power, normalised to one at 550 nanometres. Light at 450 nanometres is scattered 4.35 times as strongly as light at 650 nanometres — which is the whole reason the sky is the colour it is.

Every branch scattering-spectrum draws, and what checks it

schwarzschild

6 essays

The factor, against distance from the horizon. √(1 − rs/r) against radius in units of the Schwarzschild radius, with its reciprocal beside it. The first is the rate of a clock held still there, as read by somebody far away, and is also the redshift of anything it emits; the second is how much a radial ruler is stretched. At 10 rs the clock runs at 0.949 of its distant rate, which is a large effect for a quantity most of physics is allowed to call one. Both curves are drawn to the horizon, where one reaches zero and the other has no value at all.

Every branch schwarzschild draws, and what checks it

A parcel let go 300 m up comes back, and does not stop there. Height of a parcel released 300 m above where it started, against time, for each environment. Nothing is holding it and nothing is damping it, so a stable column does not return the parcel to where it belongs — it overshoots by as much as it was displaced, every time, which is what an oscillation is. At -5 K/km the integrated period is 4.67 min against 4.67 from 2π/N; At 0 K/km the integrated period is 5.75 min against 5.75 from 2π/N; At 6.5 K/km the integrated period is 9.95 min against 9.95 from 2π/N. The two agree to about a per cent, and the difference is the amplitude: the restoring force is computed from the full temperature difference here, which is not quite proportional to the displacement. Any rate steeper than the adiabat leaves the picture instead of oscillating.

Every branch stratified-parcel draws, and what checks it

vector-field

6 essays

The same field, sampled as arrows. The field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.

Every branch vector-field draws, and what checks it

wire-frames

6 essays

The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

Every branch wire-frames draws, and what checks it

critical-angle

5 essays

The critical angle for n = 1.5 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all.

Every branch critical-angle draws, and what checks it

Wavefronts from a moving source. Circles showing where each crest has reached, centred on where the source was when it emitted them. Ahead of the source the crests are closer together and the frequency heard is higher; behind, they are spread out and it is lower.

Every branch doppler-wavefronts draws, and what checks it

Each stage takes 25.0 per cent of what is left. Entropy against temperature for a spin-½ paramagnet at 0.25 T and 1 T, with the cooling cycle drawn between them: a vertical drop is isothermal magnetisation, a horizontal move is adiabatic demagnetisation. Starting from 1 K the treads are at 1.000 K, 0.250 K, 0.062 K, 0.016 K, 3.91 mK. Each is 0.2500 of the one before — a ratio read back off the drawn treads rather than written into them, and equal to the field ratio 0.25/1 because this refrigerant's entropy depends on the field and the temperature only through their quotient. The steps therefore shrink in proportion to what is left, and no finite number of them arrives.

Every branch entropy-staircase draws, and what checks it

induction-loop

5 essays

A loop leaving the field. A rectangular loop of wire 0.3 metres by 0.2 metres moving at 1.5 metres per second out of a region of magnetic field of 0.6 tesla directed into the page, marked with crosses. 0.08 metres of the loop's width is still inside the field. The induced current runs clockwise, and the force on the side that is in the field opposes the motion.

Every branch induction-loop draws, and what checks it

kam-tori

5 essays

Phase portraits of the standard map at 2 couplings. The standard map p → p + K sin θ, θ → θ + p, iterated 220 times from 12 starting points, at couplings of 0.6 and 1.3. Both coordinates run from 0 to 2π. An orbit that lies on a curve spanning the picture from left to right is an invariant circle, and nothing can cross it; an orbit that fills an area is chaotic; an orbit that circulates round a centre is trapped in a resonance island. At K = 0.6, orbits launched on p = 0 get no further than 1.55 in p, so a spanning curve is still there. At K = 1.3, orbits launched on p = 0 get no further than 10.78 in p, so a spanning curve is gone and transport is global. The point of the pair is that the change between them is not a change of character in any single orbit — chaotic orbits and regular ones coexist on both sides — but the loss of the barriers that kept the chaotic ones local.

Every branch kam-tori draws, and what checks it

larmor-radiation

5 essays

Where the radiation goes. The angular distribution of the power radiated by an accelerating charge. On the left the charge is slow: the pattern is sin²θ about the acceleration, with nothing radiated along it and the maximum at right angles. On the right the same charge is moving at 0.9 of the speed of light, and aberration sweeps the whole pattern forward into a narrow cone — the peak here is at 13.4°, against the 1/2γ = 12.5° the usual estimate gives, inside a cone of half-angle 1/γ = 25.0°. A synchrotron is a searchlight for this reason and no other.

Every branch larmor-radiation draws, and what checks it

normal-modes

5 essays

The two normal modes of a coupled pair at kc/k = 0.1. Two equal masses, each held to a wall by a spring of stiffness k and to each other by a coupling spring of 0.1k. Above: the in-phase mode, in which both masses move the same way by the same distance, the coupling spring never changes length, and the frequency is therefore 1.0000√(k/m) — the coupling does not appear in it at all. Below: the out-of-phase mode, in which the coupling spring changes length by twice the displacement, so each mass feels k + 2kc and the frequency rises to 1.0954√(k/m), a ratio of 1.0954. Both displacement patterns are the eigenvectors of the pair's stiffness matrix, obtained from its trace and determinant and checked against those two square roots. The red arrows are the force each mass is pulled back by, computed as −Kx: 1.00kA in the first mode against 1.20kA in the second, a factor of 1.20, which is the square of the frequency ratio because ω² is a stiffness over a mass. Any motion of the pair whatsoever is a sum of these two and nothing else.

Every branch normal-modes draws, and what checks it

pendulum-period

5 essays

The pendulum's period against its amplitude. The exact period of a simple pendulum divided by the small-angle period, plotted against amplitude. The small-angle formula is the horizontal line at one; the exact curve leaves it slowly and then climbs without limit as the amplitude approaches a half turn.

Every branch pendulum-period draws, and what checks it