Illustrated Physics
physics, drawn
Concept

Skin depth — where it appears

The distance a changing magnetic field penetrates into a conductor before falling by a factor of e. It is the square root of two over the permeability, conductivity and angular frequency — nine millimetres in copper at fifty hertz, two micrometres at a gigahertz, and infinite for a steady field.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Skin depth against frequency, over eleven decades. 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.

How far a field gets into metal

The first three rungs of this ladder all say the field inside a conductor is zero, and all three assume the electrons have had time to move. Give them less time and the field gets in — 9.2 millimetres into copper at mains frequency, 2.1 microns at a gigahertz — and the metal box that silences a radio does almost nothing about the cable running past it.

electromagnetism · Conductors The screening that survives to zero frequency. How far a magnetic field gets into copper against how fast it is changing, drawn beside the London depth of a superconductor with a carrier density of 6.0e+28 per cubic metre. The metal's curve falls as one over the square root of the frequency — measured on the drawn curve as -0.5000 against −½ — and it is a straight line on these axes with no bottom: at a hundred hertz the field reaches nine millimetres in, and at zero frequency it reaches all the way through, because a normal metal screens by dissipating and a steady field dissipates nothing. The superconductor's line is flat at 21.7 nanometres. The frequency does not appear in the expression for it, so there is nothing for it to depend on, and the screening is as complete at zero frequency as at any other. The two lines cross at 9.0e+12 hertz, in the far infrared, and above that the ordinary metal is actually the better screen — which is a useful corrective, because a superconductor's advantage is not that it screens harder but that it does not need the field to be changing. What the picture cannot show is where the flat line stops: above the energy gap the pairs break, the superconductor becomes an ordinary metal, and the flat line turns into a sloping one.

The field that is pushed out

A perfect conductor keeps whatever field was inside it when its resistance vanished. A superconductor expels the field either way, and the difference between remembering and expelling is the experiment that showed superconductivity is a state of matter rather than a very good conductor.

electromagnetism · Superconductivity The year, arriving underground at four different times. Temperature against depth in soil of diffusivity 0.5 mm²/s, at four times of year, measured from the annual mean. The surface swings by ±12 K; the swing underground is smaller and later, and both are governed by one length, δ = √(2D/ω) = 2.24 m. The amplitude falls as e^(−z/δ) — the dashed envelope — and the phase lags by z/δ radians, so the curves lean over as they go down and cross the axis at different depths. At 7.0 m the lag is half a year: the ground there is at its coldest in August and its warmest in February, in antiphase with the sky, with a swing of 0.52 K left. That is a cellar, and it is also why a water main below about a metre and a half does not know that it froze last week.

The summer that reaches the cellar in December

Drive the diffusion equation at its boundary instead of releasing something into it and the solution is a decaying, lagging oscillation with a single length in it. That length governs both the shrinking and the delay, which is why the depth at which the ground is coldest in August is fixed by the same number as the depth at which the seasons stop being felt at all.

thermodynamics · Diffusion 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 A wave with two directions in it. Light at 60° entering silver, 550 nm, whose index is 0.055 + 3.32i. Phase matching along the boundary fixes the transmitted wave's tangential wavenumber and leaves the normal one to the medium, which supplies a complex answer. The real part decides where the phase goes and the imaginary part where the amplitude does, and they are not the same direction: the surfaces of constant amplitude are parallel to the interface, because the decay is entirely into the metal, while the surfaces of constant phase are tilted by 86.5° from the normal. There is therefore no single refracted angle to quote. The familiar picture, in which one set of parallel planes carries both, requires the absorption to be exactly zero.

The angle that is two angles

Snell's law survives a complex index by giving a complex answer, and a complex angle is not an angle. What the phase-matching argument actually fixes is the tangential wavenumber, and when the medium absorbs, the surfaces of constant phase and the surfaces of constant amplitude stop being parallel. In silver at 550 nanometres the phase fronts run within four degrees of the surface while the amplitude decays straight into it.

optics · Refraction

Named alongside it

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

Boundary conditionComplex wavenumberPhase transitionSuperconductivityAbsorptionAttenuationCirculationCondensateConductivityConductorsCritical fieldDiffusion

All concepts