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The figure library — page 3

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.

polariser-stack

13 essays

Malus's law. The fraction of polarised light passing a filter, against the angle between the light's own direction of shaking and the filter's axis. It is the cosine squared: half at 45 degrees, nothing at 90.

Every branch polariser-stack draws, and what checks it

rigid-rotation

13 essays

Which axis is unstable, and by how much. The three linear exponents for a torque-free rigid body, against the value of its intermediate principal moment, with the outer two held at 3.068e-3 and 9.750e-3 kg m². Two of the curves are oscillation frequencies: a perturbation of a spin about the least or the greatest moment goes round and comes back. The third is a growth rate, and it belongs to the intermediate axis alone. It reaches zero at both ends of the range and is positive everywhere between, so the instability is a property of the ordering rather than of any shape — it disappears exactly when two moments become equal. Its largest value is at 6.409e-3 kg m², the midpoint of the outer two; this body sits at 6.817e-3, giving a growth rate of 0.606 per turn of the spin, or 2.425 s⁻¹ at 4 rad/s — an e-folding time of 0.41 s.

Every branch rigid-rotation draws, and what checks it

Two sources 3 wavelengths apart. Circular wavefronts from two sources, with the lines along which they arrive in step drawn through the pattern. Those lines are where the path difference is a whole number of wavelengths.

Every branch two-source-interference draws, and what checks it

larmor-radiation

12 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

single-slit

12 essays

Single-slit diffraction at three slit widths. Intensity against angle behind a single slit, evaluated from the integral across the aperture, for slits two, six and twenty wavelengths wide. A wide slit throws a nearly sharp shadow; a narrow one spreads light through a wide angle.

Every branch single-slit draws, and what checks it

stratified-parcel

12 essays

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

capillary-rise

11 essays

Four tubes, four heights. Water in tubes of radius 0.2, 0.4, 0.8, 1.6 mm, with each meniscus drawn as the spherical cap a 20° contact angle forces and each height computed from it. The narrowest rises 70 mm and the widest 9 mm — in inverse proportion to the radius, with nothing about the glass or the volume of water entering it.

Every branch capillary-rise draws, and what checks it

heat-capacity

11 essays

The staircase equipartition cannot climb. The heat capacity of hydrogen at constant volume, in units of R, against temperature on a logarithmic axis. The three translational directions contribute 3/2 at every temperature. The two rotations switch on near 85.4 K — the temperature at which kT matches the first rotational step — taking the total to 5/2, computed here by summing the rigid rotor's partition function over two hundred levels rather than by assuming the plateau. The vibration switches on near 6332 K, taking it to 7/2. At 50 K the value is 2.469; at 300 K the value is 2.502; at 5000 K the value is 3.376. Classical equipartition predicts 7/2 at every temperature, including at four kelvin, and the size of the steps is set by ħ — which is how a heat capacity measures a quantum constant.

Every branch heat-capacity draws, and what checks it

horizon-thermo

11 essays

Hotter as it shrinks. The temperature of a horizon and its evaporation time, against mass, on logarithmic axes. A solar-mass horizon is at 6.17·10⁻⁸ K — far colder than the coldest thing anybody has made — and takes 2.1·10⁶⁷ years to evaporate. Both lines are straight, with slopes of exactly −1 and +3, and the first slope is the whole difficulty: losing mass makes the object hotter, so the process accelerates and ends in a burst rather than fading out.

Every branch horizon-thermo draws, and what checks it

hysteresis-loop

11 essays

A material that remembers, and the curve it can never return to. The major loop of a Preisach material — 9216 elementary switches with a spread of coercivities and interaction fields, each carrying its own sign — together with two minor loops driven inside it and the initial curve rising from a demagnetised state. Coercivity 0.63 and remanence 0.77 are read off the drawing. The initial curve is inside the loop everywhere and is the one part of this figure that cannot be revisited: reaching it again means demagnetising the sample.

Every branch hysteresis-loop draws, and what checks it

pendulum-period

11 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

simultaneity

11 essays

Simultaneity at β = 0.5. Two events on the same horizontal line happen at the same time for the stationary observer. The moving observer slices spacetime along the tilted line, and for them one event happens before the other.

Every branch simultaneity draws, and what checks it

superposition

11 essays

Two waves 180° out of step, and their sum. Two sine waves differing in phase by 180 degrees, drawn faintly, with their sum drawn solid. The sum is computed point by point.

Every branch superposition draws, and what checks it

falloff-compare

10 essays

The same law, three shapes of source. Field strength against distance on logarithmic axes, for a point, a long line and a wide plane carrying charge. The exponent is the slope, and it is set by how the area of the enclosing surface grows rather than by anything about the force law.

Every branch falloff-compare draws, and what checks it

fermat-path

10 essays

Snell's law, found by searching. Paths from a point in a medium of index 1 to a point in one of index 1.5, and the optical path length of each against where it crosses the boundary. The curve is that length; the marked point is its minimum, located by golden-section search and not by any use of a law of optics. The angles there are 55.80° and 33.46°, which satisfy n₁sin θ₁ = n₂sin θ₂ to 1.0e-8. Every other drawn path is longer, and the flatness of the curve near the bottom is why light is not fussy: a path a tenth of the way off costs almost nothing.

Every branch fermat-path draws, and what checks it

field-energy

10 essays

The energy of a capacitor, booked as a density. The energy stored by a parallel-plate capacitor of 100 square centimetres — 0.0100 square metres — against the separation of its plates, drawn twice. Held at 10 nC the energy rises in proportion to the separation; held at 113 V it falls as the inverse. Both curves are obtained by integrating the energy density ½ε₀E² over the volume between the plates, and each agrees with ½QV to better than a part in 10¹². The two describe the same capacitor at 1.00 mm, where they cross at 565 nJ, and there their slopes are equal and opposite: the attraction between the plates is 565 µN, or 5.647·10⁻⁴ N, whichever quantity is held fixed. That force is Q²/2ε₀A — a property of the field in the gap and of the area it crosses, with no reference to the plates at all.

Every branch field-energy draws, and what checks it