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The figure library — page 5

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.
The bargain no optical system gets out of. A beam entering an aperture 208 units across, filling a half-angle of 12°, and leaving one 3 times smaller. The angle it fills on the way out is not 12° but 38.6°, because the product of the aperture and the sine of the angle is the same at both ends — that product is the étendue, and no lossless system can reduce it. Measured off the drawing, 43.246 at the entrance against 43.246 at the exit. The consequence is the one that matters: flux divided by étendue is the radiance, so a smaller spot is a larger angle and never a brighter image. No lens, no mirror and no arrangement of either has ever made anything brighter than the thing it was looking at.

Every branch concentration-limit draws, and what checks it

One resonance, and the two halves of a permittivity. The real and imaginary parts of the permittivity of a single Lorentz oscillator, against frequency in units of its own resonance. Away from the resonance ε′ rises slowly with frequency and ε″ is negligible: that is ordinary dispersion. Between 0.938 and 1.058 of the resonant frequency, ε′ falls — the anomalous band, shaded — and it falls precisely where the absorption is large, which is not a coincidence but the content of the dispersion relation. ε″ peaks at 7.500 at the resonance itself. The two curves are not two properties of the material. They are one analytic function evaluated on the real axis, and either determines the other everywhere.

Every branch dielectric-response 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

schwarzschild

8 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

Where a body can no longer be any shape it likes. The tallest mountain a body can carry, against the body's radius, on logarithmic axes, beside the line on which a mountain would be as tall as the body. The first falls as 1/R and the second rises as R, so they cross exactly once — here at 282 km, for rock of 200 MPa strength and density 3000 kg/m³. Below that radius a body's own gravity cannot enforce anything and it stays whatever shape it was made; above it, the shape is decided by gravity and the answer is a sphere. The crossing moves as the square root of the strength, so it is an order of magnitude and not a boundary.

Every branch self-gravity-limit draws, and what checks it

Molecular speeds at 4 temperatures. The distribution of molecular speeds in a gas, with each curve enclosing the same area. Raising the temperature moves the peak right and lowers it: the same molecules, spread over a wider range of speeds.

Every branch speed-distribution draws, and what checks it

surface-wave

8 essays

A speed that hardly knows what the material is. The Rayleigh speed against Poisson's ratio, in units of the shear speed, found by bisection on the free-surface residual at each point rather than from a printed formula. Across the whole range a solid can have, the answer moves from 0.8837 to 0.9541 — a span of 0.0704 — while the pressure speed over the same range goes from 1.45 to 7.1 shear speeds and is unbounded at an incompressible limit. So a surface wave's speed is a shear speed to within a few per cent, whatever the rock, and the dashed curve is the customary rational approximation, which is within 0.0019 of the computed root everywhere. It is always slower than the shear wave, and it must be: a surface disturbance travelling faster than the bulk shear wave would radiate into the bulk and stop being confined.

Every branch surface-wave draws, and what checks it

survival-decay

8 essays

Flat, then exponential, then a power law. Survival probability against time in lifetimes, both logarithmic, for a resonance 20 linewidths above the bottom of its band and 400 below the top. The straight dashed line is exp(−Γt), and the computed curve sits on it through the middle — a fitted rate of 0.9998 per lifetime between one and eight — and leaves it at both ends. Below 1.57e-2 lifetimes the curve is flat, falling as (t/τ_z)² with τ_z = 0.1253 lifetimes; beyond 22.6 lifetimes it is an inverse square, which any exponential eventually loses to. Both departures are forced: the head by the state being normalisable and the tail by the band having a bottom.

Every branch survival-decay draws, and what checks it

vortex-lattice

8 essays

Circulation against how fast the bucket turns. The circulation round the rim of a bucket of radius 1 mm, against the angular velocity it is spun at. An ordinary liquid ends up rotating with the bucket, and its circulation is 2Ω times the area — the straight dashed line, continuous in Ω and with no special value anywhere on it. A superfluid's velocity is the gradient of a phase, so it can carry circulation only in whole units of h/m = 9.969e-8 m²/s. Below 0.256 radians per second it carries none at all: the bucket turns and the liquid does not, which is what Hess and Fairbank measured. Above it the circulation is a staircase of 13 steps, each exactly one quantum high and each 1.59e-2 radians per second wide. The staircase runs below the classical line by the ln(R/a) quanta the threshold costs, a fixed lag: at 102 radians per second the two agree to 0.24 per cent, which is why a rotating superfluid looks like a rotating liquid at any speed a bucket is normally spun at.

Every branch vortex-lattice draws, and what checks it

conductor-field

7 essays

A conductor in a field, with the surface charge solved for. Field lines approaching an isolated conducting cylinder. The surface charge was found by requiring the conductor to be an equipotential, and the lines then end on that charge, meeting the surface at right angles and leaving the interior empty.

Every branch conductor-field 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

fermi-sea

7 essays

How much of copper's electron sea a temperature can reach. The probability that a state of a given energy is occupied, in copper, at 4 temperatures, with energy measured in units of the Fermi energy — 7.04 eV here. At absolute zero the curve is a step: every state below the ceiling is full and every state above it is empty. Raising the temperature rounds the step, and rounds it over a range of about kT, which is the whole point — at room temperature kT is 0.0259 eV against a ceiling of 7.04 eV, so the rounding is 1.6 per cent of the way down the sea and everything deeper is untouched. An electron in the deep is not held there by a force; it simply has nowhere to go, because every state it could be promoted to is occupied. at 0 K the step is spread over 0.00 per cent of E_F, at 300 K the step is spread over 1.61 per cent of E_F, at 3000 K the step is spread over 16.13 per cent of E_F, at 20000 K the step is spread over 107.52 per cent of E_F.

Every branch fermi-sea draws, and what checks it

harmonic-compare

7 essays

One parabola, several wells. Unlike potential wells, each divided by its own curvature at the bottom, against the single parabola ½x² drawn through all of them. They agree near the minimum because a function with a minimum has no linear term there, so the quadratic term is the first thing it has. The labels give where each well departs from the parabola by more than 1% of the parabola's own value there: a pendulum at 0.35, a chemical bond at 0.01, a pair of atoms at 0.0015. A symmetric well has no cubic term and stays close for a long way; a well that is steeper on one side than the other has one, and leaves the parabola almost at once — which is why those numbers differ by factors of hundreds and not by a few per cent.

Every branch harmonic-compare draws, and what checks it

Four supports, and a whole line of answers. The same top on four supports at the corners of a square, with the load in the same place. Five sets of reactions are drawn and every one of them satisfies all three equilibrium equations exactly — worst residual 1.1e-16 across the whole family — so statics does not prefer any of them. They differ by a multiple of the pattern plus, minus, plus, minus around the square: pressing one diagonal pair harder and the other pair less adds no net force and no moment about either axis, which is exactly what it means for the problem to have a fourth unknown and only three equations. The rigid-body idealisation has not been applied carelessly here; it has been applied correctly, and the answer it gives is that there is no answer. Every member drawn keeps all four reactions positive, so the requirement that a leg can only push narrows the family without closing it. What decides is left out of the model entirely — how much each leg gives under load.

Every branch redundant-supports draws, and what checks it

action-paths

6 essays

The action along a family of paths. On the left, seven paths between the same two events: the true trajectory of a projectile and six deformations of it, each fixed at both ends and differing by one arch of a sine. On the right, the action of each — the time integral of kinetic minus potential energy — against how much it has been deformed. The true path has the least action, 0.45833 in these units, and every neighbour has more. The curve on the right is a parabola about that minimum with curvature 4.935, so the excess action grows as the square of the deformation and its slope at the true path is zero. Nothing here was minimised: the true path was obtained by solving the equation of motion, and every action on the chart including its own is the same quadrature along a stated curve. What the figure establishes is that the two ways of specifying a trajectory — obey a differential equation at every instant, or make one integral over the whole path stationary — pick out the same curve.

Every branch action-paths draws, and what checks it

cross-section

6 essays

The same law, across twenty-eight decades. The mean free path 1/nσ against cross-section, for a target density of 6.83·10³⁰ targets per cubic metre — solid lead. It is a straight line of slope minus one, because there is only one thing in the law. At 10⁻²⁸ m² the path is 1.46 mm; at 10⁻⁴⁷ m² the path is 1.55 light-years. Nothing about the physics changes between those ends. Only the area does.

Every branch cross-section draws, and what checks it