Illustrated Physics
physics, drawn
Concept

Capacitance — where it appears

The charge a conductor or a pair of them holds per unit of applied voltage, a number fixed by geometry and by the material between. For parallel plates it is the permittivity times the area over the separation, and everything about the charge and the voltage divides out of it.

Named by 10 essays across 4 fields — each of them below, with the objects they name alongside it.

Two plates 0.20 plate-widths apart, with the field traced. The electric field between two oppositely charged plates separated by 0.20 of their own width, traced by following the field of 26 discrete charges on each plate rather than drawn as parallel lines. In the middle the lines are straight and evenly spaced; near the ends they bow outward. The field nine-tenths of the way to the edge is 80 per cent of the field at the centre.

How much charge a shape will hold, before anything is charged

Capacitance is decided by geometry alone. Two pieces of metal have a number attached to them, fixed by their shape and their separation, and it is settled before any charge arrives.

electromagnetism · Conductors The field a polarised sphere makes inside itself. A uniformly polarised sphere, its bound surface charge drawn at the size the cosine gives it, the uniform field that charge makes inside, and the exact dipole field it makes outside. The internal field is the same everywhere and points against the polarisation — that is what makes it a depolarising field — and its size is P/3ε₀, the third being the sphere's share of the one unit the three axes divide between them.

The field the matter takes away

Put a piece of glass in an electric field and the field inside it is smaller. How much smaller is not a property of glass. A needle of it keeps almost the whole field, a sphere keeps three-quarters, a slab across the field keeps a seventh — same material, same applied field, three answers, and the difference is arithmetic about shape.

electromagnetism · Dielectrics The only part of the field that can pull. A dielectric slab part-way into a parallel-plate capacitor. Everywhere except at the slab's edge the field is perpendicular to the plates and therefore perpendicular to the direction the slab can move, so it exerts no force along that direction at all: inside the parallel-plate model, which is uniform between the plates and zero outside them, nothing pulls the slab anywhere. The force lives in the bowed lines drawn at the edge, where the field leaks past the end of the dielectric and acquires a component along the plates. Those lines are what the model throws away as a small correction near the boundary, and they are the entire mechanism. The energy method sidesteps the drawing altogether: differentiate the total energy with respect to the insertion and the answer is 1.328e-3 newtons, inward, without ever asking where on the slab the force is applied.

The force that lives where the model is not

A slab of glass held at the mouth of a charged capacitor is pulled in. Inside the parallel-plate model there is no force at all — the field is perpendicular to the slab's motion everywhere — and the same model's energy nevertheless gives the pull exactly right. The mechanism is entirely in the part of the field the model throws away.

electromagnetism · Dielectrics 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 potential at a point is where its walkers end up. A square divided into a 24-step grid, with its top edge held at a potential of 1 and the other three edges at 0. From the probe point (0.29, 0.71), 4000 random walkers each step to one of their four neighbours with equal chance until they touch an edge; six of them are drawn, each ending with a dot on the edge it reached. The fraction that end on the held edge is 0.405 ± 0.008, one standard error, after an average of 122 steps. Solving Laplace's equation on the same grid by repeatedly replacing every value with the average of its four neighbours gives 0.408, and the series solution for the continuous square gives 0.408. The walkers were never told the equation: a value that is the average of its neighbours and a probability of ending somewhere are the same arithmetic.

The potential is where the wanderers stop

Start a random walker at a point between charged conductors and let it wander until it touches one of them. The average potential of the surfaces the walkers touch is the potential at the starting point — exactly, with no equation solved — and the charge a conductor keeps at each place on its surface is the chance that a walker arriving from far away touches it there first.

electromagnetism · Potential How close a network gets to a load that stores charge. The fraction of a wave's amplitude reflected from a resistance shunted by a capacitance, with RCωc = 2, against frequency in units of the band edge ωc, for the load alone and for networks of inductors and capacitors whose values, with an ideal transformer at the source, were optimised numerically to keep the reflection low across the band. The bare load holds it to 0.707 across the band; the 1-element network holds it to 0.392 across the band; the 2-element network holds it to 0.320 across the band; the 3-element network holds it to 0.289 across the band. The dashed line is Bode and Fano's floor, exp(−π/RCωc) = 0.208, which no network of any size can go beneath over the whole band; each added element brings the design closer to it, and each buys its flatter band with a reflection that climbs to total just beyond the band edge.

The mismatch no network can remove

A quarter-wave layer or a taper can match a resistance to a resistance as well as anyone likes. Put a capacitance across the load and that stops being true for every network that could ever be built from lossless parts: Bode and Fano proved that the total amount of match available is fixed by the load's resistance and capacitance, so a network can only move it about — and a flat match across a band can never be better than e to the minus π over the load's time constant times the band.

waves · Impedance Three ships, three local times, one clock. The fair-weather field at sea level, in the same model, as recorded by ships at longitudes 120° W, 0°, 120° E, each plotting it against its own local solar time. An ordinary daily effect — sunshine, heating, turbulence — would peak at the same local time everywhere. This one peaks at 11.4, 19.4, 3.4 hours local time in turn, because every ship is reading the same global potential, which follows the world's storms and peaks at one moment in Universal Time. Seeing the maxima line up in Greenwich time rather than local time, over clean ocean air far from any storm, was how the global circuit was demonstrated: the field in fair weather is the far end of a circuit whose generators are thunderstorms thousands of kilometres away.

The field that keeps Greenwich time

Over a calm ocean, a thousand kilometres from any storm, the air carries a downward electric field of about a hundred volts per metre, and it rises and falls by a sixth once a day. It peaks at the same instant on every ship on every ocean — near seven in the evening in Greenwich, whatever the local clock says. Nothing local can do that. The field is the far end of a circuit whose upper plate is a single conductor round the whole planet, and a conductor has one potential, so every patch of fair-weather sky is reading the sum of all the world's thunderstorms at once.

electromagnetism · Potential The same rate for every size of molecule. The diffusion-limited rate constant for two equal spherical molecules meeting in water at 25 °C, against their radius from 0.1 to 100 nm: 8kT/3η = 7.43·10⁹ per molar per second, flat. The dashed line is what a larger target would give if its partner kept the mobility of a 0.3 nm molecule — 7.4·10¹¹ at 30 nm — the intuition that a bigger molecule is easier to hit. Stokes and Einstein's law takes it away again: a molecule twice as big is twice as easy to hit and diffuses half as fast, and the two cancel exactly. Every reaction whose partners react at the first touch runs at close to this one number, whether the partners are ions, small organic molecules or proteins.

The speed limit that does not care how big the molecules are

Two molecules in water cannot react faster than they can find each other, and how fast they find each other is set by diffusion. A bigger molecule is an easier target, so it seems a reaction between big molecules should be faster. It is not: a molecule twice as big is twice as easy to hit and wanders half as fast, and the two cancel exactly. Every reaction that happens at the first touch runs at the same rate, about seven billion per molar per second in water, whether the partners are ions or proteins — a speed limit written by the viscosity of water and the temperature, and nothing else.

thermodynamics · Diffusion Water drops that charge two cans to a spark. The voltage between the two cans of a Kelvin water dropper against time, on a logarithmic axis, starting from a stray difference of 1 V, for 10, 20, 40 drops a second in each stream; drops 3 mm across, each with a capacitance of 0.17 pF to its ring, cans of 20 pF. Each drop carries a charge induced by the opposite can's potential into its own can, which raises the opposite can's induced charge on the next drop: the difference grows exponentially, multiplying by e every 6.0 s at 20 drops a second. It passes the 10 kV that will jump a few millimetres of air after 55 s — from one volt to a spark, with nothing but falling water.

The water drops that charge themselves to a spark

Two streams of water drip from one earthed pipe through two metal rings into two metal cans, and each ring is wired to the opposite can. Nothing is charged to begin with, nothing is rubbed, nothing is plugged in. Within a minute a spark cracks between the cans at ten thousand volts. A stray volt between them makes the drops forming in one ring pick up a slight charge, which they carry to the can that makes the other ring's drops pick up the opposite charge — a loop in which every drop strengthens the imbalance that charged it. Kelvin built it in 1867; it is a dynamo made of water and gravity.

electromagnetism · Potential The diamonds where the island's charge is fixed. The stability diagram of a single-electron island with identical junctions: bias voltage (units of e/CΣ) against the charge the gate induces (units of e). Inside each shaded diamond, labelled with the number of extra electrons the island holds, every possible tunnelling step costs energy and no current flows; outside, at least one step pays for itself and electrons pass one at a time. The diamonds repeat every time the gate charge rises by one electron, and touch at gate charges of e/2, 3e/2, … where two charge states cost the same and the island conducts at any bias. Their height is e/CΣ in each direction. Probing the model at three points outside the diamonds gives currents of 0.60, 0.25, 0.53; inside them, nothing.

The island that takes electrons one at a time

A capacitor holds charge in proportion to its voltage, and the proportion is a property of its shape — on any piece of metal large enough to see. Make the metal small enough and cold enough, and joined to the world only by tunnel junctions, and the proportion breaks into steps: one electron's charge costs more energy than the heat or the battery can supply, so no current flows at all until it can, and then the electrons cross one by one. A charge of a single electron becomes something a voltmeter can see.

electromagnetism · Conductors

Named alongside it

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

Boundary conditionsSurface chargeBound chargeConductorDielectricElectric potentialEquipotentialField energyFringing fieldPolarisationRandom walkAnalyticity

All concepts