Illustrated Physics
physics, drawn
Concept

Surface energy — where it appears

The energy cost per unit area of an interface, which makes a surface behave as though it were under tension. The cost is there because a molecule at a surface has fewer neighbours than one inside, so the same number in joules per square metre is a force in newtons per metre.

Named by 10 essays across 5 fields — each of them below, with the objects they name alongside it.

Surface at fixed volume. Four shapes of identical volume with their surface areas evaluated. The sphere's is the smallest at 4.836; the flattest shape drawn carries 1.91 times as much. Surface tension is an energy per unit area, so a free drop of liquid has an incentive to be the first of these and none at all to be any of the others.

The skin that is not a skin

A drop of water behaves as though it were wrapped in a stretched membrane, and there is no membrane. What there is instead is an energy cost per unit of surface, and almost everything the apparent skin does follows from a liquid trying to have less of one.

fluids · Surface tension The small bubble empties into the large one. Two soap bubbles of radius 4 mm and 12 mm joined by an open tube. The excess pressure inside each is 4γ/R — 72.8 Pa and 24.3 Pa — so the 4 mm bubble is at the higher pressure and blows itself into the other. The smaller a bubble gets the harder it pushes, so the process runs away rather than settling: there is no equilibrium anywhere except both bubbles equal.

The small bubble blows up the big one

Connect two soap bubbles of different size and the small one empties into the large one. Everybody expects the opposite, and the reason it happens is one equation with a radius in the denominator — which also means the process runs away rather than settling.

fluids · Surface tension 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.

The size at which a body becomes round

A mountain can be no taller than the height at which the rock beneath it begins to crush, and that height falls as the body gets bigger — so there is a size above which a mountain would have to be taller than the world it stands on. Above it, nothing can be any shape but a sphere.

astrophysics · Self-gravity One volume of liquid, several solids. 4 drops of the same 5 µL of liquid, on 4 solids it meets at 20°, 60°, 90°, 140°. Each is the spherical cap that volume and that angle force, so the footprint radius is computed rather than chosen: 20° gives 2.61 mm, 60° gives 1.71 mm, 90° gives 1.34 mm, 140° gives 0.69 mm. At every contact line the three interfacial tensions are drawn to scale, and only their horizontal components balance — γsv = γsl + γlv cos θ. The vertical pull of the liquid surface is taken up by the solid, which is why the angle belongs to three interfaces at once and to no single liquid: change the solid and nothing about the water has changed.

The angle a liquid makes with what it sits on

Everything capillarity does — climbing, beading, wicking, waterproofing — is the sign and size of one cosine, and that cosine belongs to three interfaces at once rather than to the water. Change the solid and nothing about the water has changed, yet the same five microlitres goes from a footprint 2.61 mm across to one of 0.69 mm.

fluids · Capillarity The curve that makes both fusion and fission release energy. Binding energy per nucleon against mass number: how much energy would have to be supplied, per particle, to take a nucleus apart into free protons and neutrons. The curve is the semi-empirical mass formula, evaluated at whichever proton number binds most tightly for each mass number rather than at a guessed one; the points are measured values. It rises steeply at the light end, peaks at mass number 58, and falls slowly thereafter. Everything about nuclear energy follows from that shape and from nothing else. Two light nuclei joined move up the curve and release the difference; one heavy nucleus split moves up it too, from the other side. Both directions are downhill in energy because the peak is in the middle, and the peak is in the middle because two effects fight — the surface term, which penalises small nuclei for having most of their nucleons on the outside, and the Coulomb term, which penalises large ones because every proton repels every other. The energy released is the height climbed times the number of nucleons carried, and it is a million times a chemical bond for the same reason the vertical axis is in millions of electronvolts rather than in single ones.

The mass that is missing

A helium nucleus weighs less than the two protons and two neutrons it is made of. The shortfall is not an error in the weighing; it is the binding energy, converted at the going rate. One curve of that shortfall against size explains why both fusion and fission release energy.

relativity · Mass-energy The ring picks a whole number and pays for the difference. The kinetic energy of the screening current in a superconducting ring, against the flux applied from outside, in units of the flux quantum. Each parabola belongs to one winding number — the number of times the condensate's phase turns round the ring, which has to be an integer because the wavefunction has to come back to itself. The ring cannot hold an arbitrary flux, so it holds the nearest whole number of quanta and drives a current to make up the difference; the energy of that current goes as the square of the mismatch, which is what each parabola is. The lowest curve at each applied flux is the state the ring is in, and the winding number changes at exactly the half-integers — 0.50 and 1.50 and 2.50 here. So the measurable properties of the ring are periodic in the applied flux with a period of one quantum, which is 2.0678 × 10⁻¹⁵ webers and is about the flux the Earth's field puts through a square micrometre.

Two lengths, and which one is longer

A superconductor has a depth to which a field leaks in and a distance over which superconductivity itself can be built up. Which of the two is larger decides the sign of the energy of a boundary — and therefore whether the material keeps every field out or fills itself with a lattice of holes.

electromagnetism · Superconductivity An angle set by a voltage. The contact angle of water on a fluoropolymer coating with a rest angle of 115°, against the voltage across the coating, for coatings 1 and 3 µm thick with a relative permittivity of 1.93. The Lippmann–Young law cos θ = cos θ₀ + ε₀εᵣV²/2dγ makes the cosine rise as the square of the voltage, so the curve is symmetric about zero volts and steepest where the angle is near 90°. For 1 µm the law reaches complete wetting at 110 V; for 3 µm the law reaches complete wetting at 191 V. It does not get there. In most reported experiments the angle stops falling somewhere in the shaded band and then stays put however high the voltage is raised — contact-angle saturation — and why is still argued: charge trapped in the insulator, ionisation of the air at the sharp edge of the drop, and the thermodynamic stability of the contact line have all been proposed and none accounts for every case.

The angle a voltage can set

A contact angle is treated as a fact about three materials — a solid, a liquid and the air — fixed the moment they are chosen. Put a voltage across a micrometre of insulator under a drop and the angle falls as the square of the voltage, with nothing about the materials changed. Drops can be steered across a chip with no pump, and a lens can focus with no moving part. The law that describes it is exact in its model, and real surfaces stop obeying it at an angle nobody has fully explained.

fluids · Capillarity The column that overshoots its own height. The height of water in a vertical glass tube, as a fraction of Jurin's equilibrium height 2γ/ρgr, against time for tubes of radius 0.2, 0.4, 1, 2 mm, integrated from the capillary pull, the weight and the Poiseuille drag with the inertia of the rising column. The 0.2 mm tube (Jurin height 74.4 mm) creeps up without overshooting; the 0.4 mm tube (Jurin height 37.2 mm) creeps up without overshooting; the 1 mm tube (Jurin height 14.9 mm) overshoots to 1.33 of it; the 2 mm tube (Jurin height 7.4 mm) overshoots to 1.46 of it. The linearised motion about the equilibrium is underdamped when the radius exceeds (32η²γ/ρ³g²)^(1/5) = 0.48 mm for water: wider tubes ring, narrower ones do not.

The pore that lifts highest fills slowest

Jurin's law says how high water climbs in a tube and nothing about how long it takes. The rise has three laws in succession — the column's inertia, then its viscous drag, then its weight — and the same narrowness that lifts water highest throttles it hardest, so for any deadline there is a best pore. A wide enough tube does something Jurin's law cannot: it overshoots and rings. And whatever the route, exactly half the energy the surface releases is lost.

fluids · Capillarity Why silt heaves and sand and clay do not. The relative rate at which water is drawn to a growing ice lens, against grain diameter on a logarithmic scale, for three loads on the soil above it, in a capillary model: the rate is the soil's permeability, which grows as the grain diameter squared, times the suction the lens can sustain less the load. The suction is capped both by the supercooling at the lens, here 2 K, and by the Gibbs–Thomson pressure at which ice would break into the pores. Coarse soil has open pores that ice simply invades, so the suction never exceeds the load: under 10 kPa, half a metre of soil, nothing heaves above 58 μm. Fine clay can sustain the full suction but passes water too slowly to feed it. Between lies a fastest grain size, 29 μm under 10 kPa, 10 μm under 30 kPa and 2.9 μm under 100 kPa: silt, the soil engineers have long called frost-susceptible. Real heave pressures exceed this model's, and the fit is to the shape, not the scale.

The frost that lifts by drinking

Frozen ground rises in winter, sometimes by tens of centimetres, and the usual explanation — water expands by nine per cent when it freezes — is wrong: soil soaked in benzene, which shrinks as it freezes, heaves too. The lift comes from layers of clear ice that grow by pulling liquid water up through the soil from below. Ice and water touching each other need not be at the same pressure, and when the water is free to flow, ice a degree below its melting point can push with 1.12 megapascals while it drinks — twelve times less than freezing water in a sealed pipe, and enough to lift a house at a tenth of a degree.

thermodynamics · Chemical potential Flux leaving a plate in stripes and spreading into a uniform field. A cross-section through a superconducting plate in the intermediate state near its top surface (below the line), with a normal fraction of 0.40: normal laminae (shaded) each carry the critical field straight up, the superconducting laminae between them carry none. Lengths are in units of the period of the pattern. Above the surface the field lines (solid) are computed from the stray field of the stripes: they fan out from each normal lamina and close up over the superconducting ones, and within about a third of a period the field is uniform at 0.40 of the critical field — the applied field, which is all a distant observer measures. The fanning field stores energy, and it stores more the wider the stripes are, because a wider pattern's non-uniformity reaches further out. That is half of what sets the period.

The stripes a superconductor makes of a field

A superconductor expels a magnetic field, but only if the field it pushes aside stays below the critical field everywhere at its surface — and for anything but a long rod held along the field, the field it pushes aside piles up at its edges. A flat plate laid across a field cannot expel any of it. It splits instead into interleaved bands of normal metal carrying exactly the critical field and superconducting metal carrying none, in the proportion flux demands, and the bands have a definite width. Walls between the two cost energy and favour few, broad bands; the field fanning out of the surface favours many narrow ones. The compromise is the geometric mean of the plate's thickness and a microscopic length.

electromagnetism · Superconductivity

Named alongside it

The objects these essays reach for when they reach for this one.

Surface tensionLaplace pressureWettingCapillary lengthContact angleEquilibriumMagnetic fluxScalingSuperconductivityActivation barrierBinding energyCapacitance

All concepts