The same loop, spanned two ways

At its defaults it draws the same loop, spanned two ways. A 100 cm² capacitor with a 2 mm gap, charged at 1.0 million volts per second. A loop drawn round the wire can be spanned by a flat surface, which the current of 44.271 µA passes through, or by a bag-shaped surface that passes between the plates, which no charge crosses at all. Ampère's law as it stood gave two different answers for one circulation. The rate of change of electric flux between the plates, multiplied by ε₀, is 44.271 µA — the same number to every figure, because C = ε₀A/d is the same ε₀A/d either way. That is the term, and it is not an approximation or a correction: it is what makes the law consistent at all.

em-wave is one function in lib/figures/fields.js — charge, current, flux and the lines drawn between them. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

A 100 cm² capacitor with a 2 mm gap, charged at 1.0 million volts per second. A loop drawn round the wire can be spanned by a flat surface, which the current of 44.271 µA passes through, or by a bag-shaped surface that passes between the plates, which no charge crosses at all. Ampère's law as it stood gave two different answers for one circulation. The rate of change of electric flux between the plates, multiplied by ε₀, is 44.271 µA — the same number to every figure, because C = ε₀A/d is the same ε₀A/d either way. That is the term, and it is not an approximation or a correction: it is what makes the law consistent at all.

The skin depth of copper, stainless steel, seawater, mu-metal against frequency, both axes logarithmic. Every line has the same slope, −½, because the depth goes as the inverse square root of the frequency for all of them; what separates them is conductivity and permeability, which enter the same way. Three points are marked and each is a practical fact. Copper at mains frequency has a skin depth of 9.22 mm, so a busbar of that thickness carries current throughout and a thicker one does not. Copper at a gigahertz has 2.1 µm, so the current in a microwave component runs in a layer thinner than the plating and the surface finish becomes the conductor. Seawater at the frequency navies use to signal submerged submarines has 28.9 m, which is why that link is measured in tens of hertz and carries a few characters a minute. Mu-metal is the outlier and the reason it exists: its conductivity is forty times worse than copper's and its permeability twenty thousand times better, so at low frequency it is the one material on the chart that is thin.

The field inside copper, against depth measured in skin depths. The amplitude falls as exp(−z/δ), reaching 0.368 of its surface value after one skin depth and 1.8e-2 after four, and the field lags the surface by one radian for every skin depth it has travelled — which is why the oscillating curve underneath crosses zero before the envelope has fallen far. Drawn this way the four frequencies give the same curve, and what distinguishes them is the horizontal scale: one skin depth is 9.22 mm at 50 Hz, 2.06 mm at 1.00 kHz, 65.2 µm at 1.00 MHz, 2.1 µm at 1.00 GHz. Six decades of frequency move the scale by a factor of a thousand, because the depth goes as the inverse square root, and that single fact is why a metal box is an excellent radio shield and a poor magnetic one. The picture is of a solution to a diffusion equation rather than a wave equation: inside a good conductor Maxwell's equations lose their second time derivative, and a field entering metal spreads the way heat does rather than the way light does.

The resistance of a copper slab 10.00 mm thick to alternating current, divided by its resistance to direct current, against frequency, both on logarithmic scales. The curve is exact: the current density inside the slab solves the diffusion equation and comes out proportional to cosh(kz) with k = (1 + i)/δ, and the ratio is the real part of ka coth(ka). At low frequency it is one, because the current fills the metal. Once the slab is more than a skin depth thick the ratio becomes the half-thickness divided by the skin depth, drawn as the straight asymptote, and therefore climbs as the square root of the frequency for ever. The resistance has doubled by 741 Hz. Two pieces of engineering follow directly. Thick conductors are wasted metal at radio frequency, which is why waveguides and heavy busbars are hollow; and a bundle of insulated strands, each thinner than a skin depth and woven so that every strand takes every position in the bundle, keeps the low-frequency resistance up to frequencies where a solid conductor of the same copper would be several times worse.

The absorption a 1 mm sheet offers, in decibels, against frequency, for copper, aluminium, mu-metal. One skin depth of any material is 8.69 dB, so the curves are the thickness measured in skin depths and scaled — which means each of them climbs as the square root of the frequency and none of them starts anywhere useful. Copper gives 0.9 dB at mains frequency and 133 dB at a megahertz, and the gap between those two numbers is the whole practical story: the metal box that silences a radio does essentially nothing about the fifty-hertz magnetic field of the cable running past it, and no thickness of copper that anybody would build fixes that. What fixes it is permeability rather than conductivity — mu-metal offers a path the flux prefers rather than absorbing it — which is why magnetically shielded rooms are built of soft iron alloys and not of the best conductor available. The curve is absorption only. A real sheet also reflects, and for an electric field arriving from far away the reflection is usually the larger term; for a magnetic field from a nearby source it is not, which is exactly the case this figure is about.

The three terms of an oscillating dipole's electric field against distance, in units of a reciprocal wavenumber, both logarithmic. They fall as 1/u³, 1/u² and 1/u, so they are all equal at u = 1 — a distance of λ/2π, which is a property of the frequency and of nothing about the antenna. The heavy curve is the field the three actually add up to, and it is not their sum: the static and radiation terms are in antiphase and partly cancel, which is why the total dips below every one of them just inside the crossing before settling onto the 1/u the far field is made of.

physicscheck asserts something about em-wave that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.