Illustrated Physics
physics, drawn
How this site is made

The figure library — page 7

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.

gyro-top

2 essays

A 120 g top at 3000 rpm, precessing once every 1.92 s. A disc of radius 30 mm spinning at 3000 revolutions a minute on a shaft 45 mm long, tilted 30° from the vertical. The weight acts at the centre of mass and the pivot holds the bottom, so the torque about the pivot is horizontal and at right angles to the plane containing the axis and the vertical. Angular momentum points along the axis; a torque at right angles to a vector turns it without changing its length, so the axis sweeps round the dashed circle instead of falling. The precession rate is Mgl divided by I₃ω₃ to leading order, which is 3.269 radians a second here, or one turn every 1.92 seconds — slower the faster it spins.

Every branch gyro-top draws, and what checks it

mie-size

2 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

photoelectric

2 essays

Maximum electron energy against the frequency of the light. The greatest kinetic energy a photoelectron leaves with, against the frequency of the light, for caesium (work function 2.14 eV), calcium (work function 2.87 eV), zinc (work function 4.33 eV). The lines are parallel: their common slope is Planck's constant, 4.1357e-15 electronvolt seconds. Each line meets the energy axis at minus its own work function and meets zero at its own threshold frequency, below which no light of any brightness produces an electron.

Every branch photoelectric draws, and what checks it

relativistic-force

2 essays

Pushing one way and going another. The angle between an applied force and the acceleration it produces, against the angle between the force and the body's velocity, at four speeds. At 0° and 90° the two are parallel, because those are the two directions the γ³ and γ divisors do not mix. Everywhere between, they are not: at 0.99c the worst case is 73.9°, reached with the force at 8.0°. A body under a steady sideways-ish push does not travel along it.

Every branch relativistic-force draws, and what checks it

zone-plate

2 essays

Rings whose radii go as the square root of their number. A 20-zone plate for 550 nanometres at 200 millimetres, drawn to scale, beside its zone radii against zone number. The outermost is 1.483 millimetres and the innermost 0.332, and the curve is a square root because the radii come from making each zone's extra path exactly half a wavelength longer than the last. The consequence worth noticing is on the drawing rather than in the formula: every ring has the same area, to 0.003 per cent, so each contributes about equally to what arrives on the axis and the rings get thinner outwards to keep it so. Alternate rings are opaque. Half the light is thrown away and the axis gets brighter, because what is thrown away is the half that would have arrived out of phase with the rest.

Every branch zone-plate draws, and what checks it

kramers-kronig

1 essay

Absorption and refraction, drawn as one function. The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else.

Every branch kramers-kronig draws, and what checks it

noether-charge

1 essay

Three quantities that do not move while everything else does. On the left, an orbit in an inverse-square attraction, integrated from its equation of motion over 2.4 revolutions at an eccentricity of 0.55. On the right, three quantities computed from that same trajectory at every step and plotted against time: the energy, the angular momentum, and the length of the eccentricity vector that points at periapsis. Every one is flat to better than 1.2e-11 in units where the circular speed at r = 1 is 1, and none of them was constrained to be — the integrator was given the force and nothing else. Each is a symmetry seen sideways. The energy is constant because the force law does not mention the time; the angular momentum is constant because it does not mention the direction; and the eccentricity vector is constant because of a symmetry that is not a motion of space at all, which is why the inverse square closes its orbits and its neighbours do not. What the picture cannot show is the direction of the argument: it demonstrates that these three are constant here, and the theorem says something much stronger, that a constant exists for every continuous symmetry whatever the system.

Every branch noether-charge draws, and what checks it