Illustrated Physics
physics, drawn
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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.

phase-boundary

7 essays

The phase boundary of water, from one equation. Pressure against temperature for water on a logarithmic pressure axis spanning 6.6 decades. The vaporisation curve is integrated from Clausius–Clapeyron between the triple point at 273.16 kelvin and 611.7 Pa and the critical point at 647.096 kelvin, with a single latent heat of 43.32 kilojoules per mole — the value the two published points on the curve imply. The measured latent heats are 45.05 at the triple point and 40.65 at the reference point, and the fitted value sits between them, because a constant latent heat is an average over the interval. The sublimation curve below the triple point is not measured but predicted, from the two latent heats adding where all three boundaries meet: 51.1 kilojoules per mole, which reaches 103.2 Pa at 253.1 kelvin against a measured 253.15. The melting curve is drawn at the slope Clapeyron gives it, -13.5 megapascals per kelvin, which is a volume ratio and nothing else: water's solid is 917 against 1000 kilograms per cubic metre for its liquid, so melting shrinks it and the line leans backwards. Across the whole 6.6 decades of this axis that line moves 5.2 kelvin, and one atmosphere shifts the melting point by 0.0075 kelvin. At 1 atmosphere the boundary is crossed at 373.1 kelvin, where water boils. The one place the curve fails is its top end: a constant latent heat reaches 37.4 MPa at the critical temperature where the measured critical pressure is 22.1 MPa, 70 per cent high, because the latent heat falls to zero at the critical point and this curve does not know that.

Every branch phase-boundary draws, and what checks it

plasma-response

7 essays

Nothing propagates below 8.98 MHz. Frequency against wavenumber for a wave in a plasma of 1.0e+12 electrons per cubic metre, in units of the plasma frequency. The curve is ω² = ωₚ² + c²k², so it starts at ωₚ with zero slope and becomes the light line far above it; the whole band below ωₚ has no real k at all, which is the shaded region. At k = 3.19e-1 m⁻¹ the phase velocity read off the curve is 1.1607c and the group velocity 0.8615c, whose product is c² to a part in 10¹⁰ — the crests outrun light and the signal does not, which is the same arrangement as any other medium with a cutoff.

Every branch plasma-response draws, and what checks it

pressure-depth

7 essays

Three vessels, one pressure. Three vessels filled to the same depth of 3 m. The pressure on each base is 29.4 kPa — identical, because pressure is set by depth — while the weight of water each holds differs by a factor of 4.7. The base of the flaring vessel carries more force than the water standing over it weighs.

Every branch pressure-depth draws, and what checks it

rigid-rotation

7 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

single-slit

7 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

ampere-loop

6 essays

Four paths, one answer. The field of a straight wire carrying 10 A, summed step by step around four closed paths in 4000 pieces each. Three of them enclose the wire and each returns 12.566 µT·m, which is μ₀I; the fourth does not enclose it and returns zero, because the outward stretch of the path and the return stretch cross the same field lines in opposite senses. Nothing about the shape survives into the answer — not the radius, not the centring, not the corners — which is what makes the law usable and also what makes it useless without a symmetry to hand.

Every branch ampere-loop draws, and what checks it

barrier-tunnel

6 essays

A wavefunction crossing a barrier it has not the energy for. An electron of 2 electronvolts meeting a 3 electronvolt barrier 0.3 nanometres wide, with the four matching conditions solved rather than sketched. Left of the barrier the incident and reflected waves add to a standing pattern; inside it the amplitude decays exponentially; to the right a travelling wave continues with amplitude 0.3912 of the incident one, so 15.3 per cent of the electrons get through. Classically none of them do.

Every branch barrier-tunnel draws, and what checks it

bell-correlation

6 essays

The correlation, and the best a shared list of answers can do. The coincidence correlation between two polarisation analysers against the angle between them, over two full turns of the correlation — a polariser turned through 180° is the same polariser, so the picture repeats. The singlet gives −cos 2Δ, drawn through −1.00 at 0°, 1.00 at 90°, −1.00 at 180°, 1.00 at 270°. Beside it is the best correlation any shared list of pre-agreed answers can produce: straight lines between the same four extremes, with corners where the cosine is smooth. The two agree exactly at the multiples of 45° and nowhere else, and they are furthest apart — by 0.2105 — at 19.77° and 70.23°, which is ½ arcsin(2/π) from either end of the quarter turn. The difference is not a matter of degree: it is a curve against a shape with a corner in it, and no list can be bent into the curve.

Every branch bell-correlation draws, and what checks it

buoyancy

6 essays

Where the upward force comes from. A block submerged with its top 1.2 m down. The pressure on the bottom face (19.6 kPa) exceeds that on the top (11.8 kPa) by exactly the weight of a column of water as tall as the block, and the sideways pressures cancel in pairs. Nothing has been added to the physics of pressure to get buoyancy out of it.

Every branch buoyancy draws, and what checks it

concentration-limit

6 essays

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

damped-response

6 essays

One equation, and the number that decides what it does. The same oscillator released from rest at one unit of displacement, with 4 different amounts of velocity-proportional loss. Every curve is a numerical integration of a single equation with no case analysis in it; what changes between them is one dimensionless number, the damping ratio. Below one, the two exponents are a complex conjugate pair and the motion crosses zero over and over inside a decaying envelope. At one they collide into a real double root and the trace reaches the axis and stops. Above one they are two distinct real numbers, the slower of them dominates, and the return is sluggish — an overdamped system takes longer to come back than a critically damped one, which is the thing the word 'over' is doing in its name. The ratios drawn are 0.15, 0.5, 1, 2, and the first zero crossing happens at 1.75 s for ζ = 0.15, 2.42 s for ζ = 0.5, never for ζ = 1, never for ζ = 2.

Every branch damped-response draws, and what checks it

energy-landscape

6 essays

A harmonic well. Potential energy against position, with a horizontal line at the total energy. The motion is confined to where the line lies above the curve, and the turning points are the intersections — computed by solving for them, not marked by hand.

Every branch energy-landscape draws, and what checks it

falloff-compare

6 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

6 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

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

membrane-modes

6 essays

6 modes of a drum, and their frequency ratios. Nodal-line diagrams for 6 modes of a circular membrane, each labelled with its frequency as a multiple of the lowest mode's. A mode (m, n) has m nodal diameters and n − 1 nodal circles, and the circles are drawn at the radii where the computed radial function J_m(j(m,n)·r/R) crosses zero — not at guessed fractions of the radius. The two tints are the two directions the head is moving in at that instant, and the lines between them are the parts of it that never move. The ratios are 1.000, 1.593, 2.136, 2.295, 2.653, 2.917: each one is a quotient of two zeros of Bessel functions, computed here from the power series and checked against their published values to 4.4e-7. Not one is a whole number, which is why a drum has no harmonic series and no pitch in the sense a string has one — and why these same ratios belong to every circular membrane ever stretched, whatever it is made of and however tightly it is pulled.

Every branch membrane-modes draws, and what checks it