Illustrated Physics
physics, drawn
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The figure library — page 6

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.

mean-free-path

6 essays

A path through a crowd. A point crossing a field of 90 scatterers, rebounding off each. The mean length of 4000 such segments is 0.2983 box widths, against the textbook form 1/2nr = 0.3086 — a departure of -3.3 per cent, from a measurement that knows nothing of the formula. It does not agree exactly and should not: the closed form is derived for a vanishingly dilute field and these discs cover 9.2 per cent of the plane. Two finite-density effects pull opposite ways — crowding shortens the path, and discs shadowing one another lengthen it — so which side of the formula a given field lands on is not something the formula can tell.

Every branch mean-free-path draws, and what checks it

normal-modes

6 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

orbit-closure

6 essays

The same start, four force laws. Orbits under 4 different central force laws, every one started at the same radius with the same fraction — 0.72 — of the local circular speed, and every one integrated for 6 radial oscillations. The paths are drawn to different scales because the excursions differ; what is comparable between the panels is whether the curve retraces itself. Under the inverse square it does: the orbit is a closed ellipse and the sixth circuit lies exactly on the first. Under the linear force it does too, and the ellipse is centred on the source rather than focused on it. Under anything in between the path is a rosette that never closes, because the angle between successive closest approaches is not a rational fraction of a full turn. Those angles are 180.0° at n = -2, 145.9° at n = -1.5, 126.4° at n = -1, 90.0° at n = 1. The closure is not a matter of degree — a rosette that nearly closes is not nearly a closed orbit, since after enough circuits it fills the annulus.

Every branch orbit-closure draws, and what checks it

chromatic-focus

5 essays

Two glasses, and how differently they disagree with themselves. The refractive index of N-BK7 and F2 across the visible, each computed from its manufacturer's Sellmeier coefficients. Both curves fall from blue to red, which is why any single lens has a shorter focal length for blue light than for red. What separates the two glasses is not where they sit but how steeply they fall: over the same interval F2 changes index by 17.1 parts in a thousand and N-BK7 by only 8.1. The ratio of a glass's index-minus-one to that difference is its Abbe number, which is 64.2 for the crown and 36.4 for the flint — a single figure of merit saying how much bending is bought per unit of colour spread, and the only property of a glass the achromatic condition uses. The three vertical lines are the Fraunhofer wavelengths the definition is stated at.

Every branch chromatic-focus draws, and what checks it

nonlinear-wave

5 essays

A sine on its way to a vertical face. One period of a sine, followed by the equation whose only nonlinearity is that the local speed depends on the local height. The profiles are at σ = t/t_b of 0, 0.3, 0.6, 0.9, 0.995, each obtained by solving the implicit relation u = u₀(x − (c₀+βu)t) for u at every point by bisection — and checked against the partial differential equation itself, which it satisfies to 2.6e-7. The crest travels faster than the trough, so the descending front leans forward and the ascending one leans back; the wave stays exactly as tall as it started and exactly as long, and only its shape changes. The steepest gradient grows as one over (1 − σ) — measured here as 9.83 times its initial value at σ = 0.9, against ten — so it is infinite at σ = 1 and the curve has a vertical tangent. The picture cannot be drawn past that point, which is not a failure of the drawing: the solution genuinely becomes three-valued, and what actually happens is a jump whose width is set by the dissipation this equation does not contain.

Every branch nonlinear-wave draws, and what checks it

prism-deviation

4 essays

The deviation that has a bottom. The angle by which a 60° and a 90° ice prism bends a ray, against the angle at which the ray arrives, for an index of 1.31. Each curve is traced ray by ray through both faces and stops where it stops: outside the plotted range the ray meets the second face beyond the critical angle and never leaves. Both curves have a minimum, and the minimum is the point of the figure twice over. Its value — 21.84° for the 60° prism and 45.73° for the 90° prism — is where the sky puts a halo. And its flatness is why there is a halo at all: near a minimum the deviation changes only in second order, so a wide band of orientations all deliver light to nearly the same angle, and a cloud of randomly tumbling crystals piles up a bright ring there while sending the rest of the light nowhere in particular. The passage at the minimum comes out symmetric — in at 40.92°, out at 40.92° — which was found by searching the traced curve rather than assumed. The window of incidence that gets through at all is 76.5° wide for the 60° prism against 32.2° for the 90° one, a factor of 2.4, and that is why one of the two halos is common and the other is rare.

Every branch prism-deviation draws, and what checks it

critical-point

3 essays

Two straight lines that are not the same line. Half the difference between the liquid and vapour densities, in units of the critical density, against the distance from the critical temperature — both logarithmic, over 5 decades. The van der Waals curve is solved for the coexisting pair at each temperature and its slope in the last decade is 0.500, which is the mean-field ½. Real fluids give 0.326. The two differ by 11 per cent at t = 0.1 and by a factor of 7.9 at the bottom of the axis, which is why an equation with the wrong exponent in it looked right for eighty years.

Every branch critical-point draws, and what checks it

fresnel-coefficients

3 essays

The two reflectances, and the angle one of them loses. Reflectance against angle of incidence for light going from n = 1 into n = 1.5. The upper curve is light polarised with its electric field along the surface, which reflects more and more strongly until at grazing incidence everything reflects. The lower curve is light polarised in the plane of incidence, and it does something the other cannot: it falls to exactly zero at 56.31°, where tan θ = 1.5000, and then rises again. At normal incidence the two are equal at 4.00% because there is no plane of incidence to tell them apart. The dashed curve is the transmittance, computed from the transmission coefficients and the two media's projected impedances rather than as one minus the reflectance; it agrees with one minus the reflectance to 4.4e-16 across the whole range, which is where the energy accounting can be seen to close.

Every branch fresnel-coefficients draws, and what checks it

gaussian-surface

3 essays

A closed surface with the charge inside. Every field line from the enclosed charge crosses the surface exactly once on its way out, so the net flux counts the charge.

Every branch gaussian-surface draws, and what checks it

mie-size

3 essays

Scattering efficiency, all the way from small to large. How strongly a sphere of index 1.333 scatters, as a multiple of its own geometric cross-section, against the size parameter — the circumference divided by the wavelength. The horizontal axis is logarithmic and covers three and a half decades. On the left the curve is Rayleigh's, rising as the fourth power of size and drawn dashed for comparison; the two are indistinguishable up to a size parameter of about a half and differ by a sixth at one, after which the fourth power runs away and the series does not. The efficiency then climbs to a first and largest maximum of 3.98 at x = 6.49, where the light that went through the sphere emerges 0.69 of a wavelength behind the light that went round it and the two interfere constructively in the forward direction. Past that it oscillates with diminishing amplitude toward two — not one — so a large sphere removes twice as much light from a beam as it geometrically blocks, which is the extinction paradox; the curve here has reached 2.12 by the right-hand edge. Nothing in the figure is a fitted or drawn shape: every point is the Mie series summed at that size.

Every branch mie-size draws, and what checks it

parametric-tongues

3 essays

Where modulating a system sets it going. The regions of the modulation plane in which an oscillator with a damping ratio of 0.02 will not stay still. The horizontal axis is the modulation frequency in units of the oscillator's own; the vertical is how deeply the stiffness is modulated. Inside a shaded wedge the state of rest is unstable and any disturbance grows exponentially; outside it, nothing happens at all. The wedges sit at modulation frequencies of twice, once and two-thirds of the natural frequency, and the first is much the widest — it opens at a depth of 8.0%, against 40.0% for the second. Each boundary is found by integrating one period of the modulation from two independent starts and asking whether the resulting map has a multiplier outside the unit circle, then bisecting on the depth; none of the shape is drawn by hand. Without damping every wedge would come to a point on the axis and there would be no threshold at all — the flat bottom of each is damping, and it is why a swing has to be pumped hard enough before it does anything.

Every branch parametric-tongues draws, and what checks it

rocket-ledger

3 essays

What a mass ratio buys. Change of speed against the ratio of fuelled mass to dry mass, for exhaust speeds of 3.05 km/s, 4.44 km/s, 30 km/s. Each curve is the exhaust speed times the logarithm of the mass ratio, which is a shape with two unforgiving properties. A mass ratio of e buys exactly one exhaust speed and no more, whatever the engine; and doubling the achieved speed requires squaring the mass ratio, not doubling it. Reaching 9.4 km/s needs a mass ratio of 21.8 at 3.05 km/s of exhaust, 8.3 at 4.44 km/s of exhaust, 1.4 at 30 km/s of exhaust. The first of those is why a chemical rocket to orbit is nine parts propellant and one part everything else, and the last is why an ion engine, whose thrust would not lift its own weight, is nevertheless the only way of reaching the outer planets with a useful payload.

Every branch rocket-ledger draws, and what checks it

spreading-loss

3 essays

Three geometries, three exponents. Amplitude against distance for a wave spreading in one, two and three dimensions, on logarithmic axes, over 3 decades. The same power crosses every surface round the source, so the intensity falls as one over the area of that surface and the amplitude as one over its square root. The fitted slopes are 0.000 along a line, -0.500 over a cylinder, -1.000 over a sphere, each fitted by least squares to the drawn curve rather than written on it. A ripple on water is the middle case and a sound in a room is the last, which is why a ripple stays visible so much further than a shout carries.

Every branch spreading-loss draws, and what checks it

stellar-visibility

3 essays

Fringe contrast against baseline, for four stellar diameters. The visibility of the fringes an interferometer would obtain at 550 nm, against the separation of its two apertures, for uniform discs of angular diameter 10, 20, 47, 100 milliarcseconds. Each curve is 2J₁(πθB/λ)/(πθB/λ), the transform of a uniform disc, and each first reaches zero at 13.84 m for 10 mas, 6.92 m for 20 mas, 2.94 m for 47 mas, 1.38 m for 100 mas. Dividing each of those by λ/θ returns the same number, 1.2197, which is the 1.22 in every textbook and is the first zero of J₁ divided by π — recovered here from the four curves rather than written into them. The practical content is that a smaller star needs a longer baseline, in exact inverse proportion, and that the measurement is of a contrast rather than of a picture. Michelson and Pease found the fringes from Betelgeuse vanishing at a 3.07 m separation in 1920 at a wavelength of 575 nm, which by the same arithmetic is a disc 47.1 milliarcseconds across — and no telescope resolved that star for another seventy years.

Every branch stellar-visibility draws, and what checks it

cycloid-path

2 essays

Four beads, four heights, one arrival time. One arch of a cycloid of radius 1, drawn with four beads on it at heights 0.061, 0.235, 0.592, 2.000 — a range of 33.0 to one. Each bead's time to slide, from rest and without friction, to the bottom of the arch is computed as a quadrature of ds/v along the curve as drawn, and the four answers are 1.003205 s, 1.003205 s, 1.003205 s, 1.003205 s: identical to 6.9e-13 of themselves. The closed form for this curve is π√(a/g) = 1.003205 s, which the quadrature reproduces without being told it. A bead let go at the cusp travels 5.7 times as far as the lowest one and arrives with it, because the extra distance is exactly paid for by the extra speed the extra height buys.

Every branch cycloid-path draws, and what checks it

fresnel-edge

2 essays

The Cornu spiral, and the chords that are amplitudes. Fresnel's two integrals plotted against each other, traced from v = -4.2 to 4.2. Arc length along the curve is v — the distance along the wavefront in units of √(λL/2) — and the amplitude arriving from any stretch of that front is the straight chord between the stretch's two endpoints, not the length of curve between them. The two eyes at ±(½, ½) are where the far parts of an unobstructed front pile up, so the whole open front is the chord between them, of length 1.4142. A straight edge blocks half of it and leaves a chord of 0.7071 — exactly half the amplitude, so a quarter of the intensity, at the geometrical shadow's edge. The chord drawn here runs to v = 1.217, is 1.6556 long, and is the longest chord from the eye that exists: its square over the open chord's square is 1.370, which is why the brightest fringe outside a shadow is brighter than no obstacle at all.

Every branch fresnel-edge draws, and what checks it