Illustrated Physics
physics, drawn
Concept

Penetration depth — where it appears

The distance a field or wave reaches into a region where it cannot propagate before its intensity falls by a factor of e. It measures how far an evanescent field, a magnetic field in a superconductor or light in a metal gets beyond a boundary.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

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 A sheet of light a tenth of a micrometre thick. The intensity of the evanescent field in water beyond a glass coverslip (index 1.518 against 1.33), against distance from the glass, for 488 nm light totally reflected at 62°, 65°, 70°, 75°, s-polarised, relative to the light arriving in the glass. The critical angle is 61.2°. At 62° the field at the surface is 3.79 times the incident intensity and falls by e in 234 nm; at 65° the field at the surface is 3.07 times the incident intensity and falls by e in 110 nm; at 70° the field at the surface is 2.01 times the incident intensity and falls by e in 75 nm; at 75° the field at the surface is 1.15 times the incident intensity and falls by e in 63 nm. Ordinary illumination (dashed) lights the whole depth of a cell equally; totally reflected light lights only the layer against the glass, which is where a cell's membrane and the proteins in it are.

The mirror that lights a tenth of a micrometre

Total internal reflection sends back every photon, and it leaves behind a field in the medium beyond that dies away within a fraction of a wavelength. Put a living cell on a coverslip and reflect a laser inside the glass, and that field lights only the hundred nanometres of the cell pressed against the glass: the membrane, and the proteins moving in it. Nothing else glows. The same field works in reverse. A molecule close to the glass sends part of its own light into angles no ray from the water could reach, and how much it sends there says how close it is.

optics · Total internal reflection Rotation is a magnetic field, and a superconductor cancels it. Magnetic fields along the spin axis inside a spinning metal, in units of 2mω/e, as seen by its electrons in the frame turning with it. In that frame the Coriolis force on an electron is exactly the force a magnetic field of 2mω/e would exert, pointing against the spin for a negative charge (Larmor's theorem). A normal metal spun in no field has no real field inside, and its electrons feel the rotation's equivalent field in full. A superconductor will not let its carriers feel any field in their interior — that is the Meissner effect — so it generates a real field of exactly the same size pointing the other way, and the sum vanishes. The real field is the London moment.

The field a spinning superconductor makes of itself

Spin a lump of superconductor with no magnet anywhere near it, and a magnetic field appears inside it, pointing along the axis of spin: about eleven trillionths of a tesla for every radian per second, the same in lead, niobium or a copper-oxide ceramic. The field is not left over from anything. In a turning frame the Coriolis force on a moving charge is indistinguishable from a magnetic force, a superconductor will not let its carriers feel any field inside it, and so it makes a real field to cancel the one rotation imitates. The size depends on nothing but the carriers' mass and charge, which makes the field a balance for weighing them — and the one time it was read to a part in a hundred thousand, it gave an answer theory has not matched.

electromagnetism · Superconductivity Two ways to be wrong, and the apparatus that sees each. The fractional sag of the potential at the centre of a charged spherical shell below the shell's own, on a logarithmic axis, against the shell's radius in metres, also logarithmic, for two different departures from Coulomb's law. If the force fell as 1/r^(2+ε) with ε = 10⁻¹⁶, the sag is 3.07·10⁻¹⁷ whatever the shell's size (dashed): the law has no length in it, so neither does its failure. If the photon had a mass of 10⁻¹⁴ eV, the sag is (μR)²/6 (solid): 4.3·10⁻²⁰ for a centimetre shell and 4.3·10⁻¹⁶ for a metre. A mass introduces a length, and the signal grows as the square of the apparatus compared with it — so the strongest bounds on a photon mass come from the largest fields, not the most sensitive laboratories.

The mass that would make Gauss's law leak

Gauss's law says the flux out of any closed surface counts the charge inside, whatever the surface's size. If the photon had even a tiny mass, that would stop being true: the field of a charge would die off exponentially beyond a length ħ/mc, the flux through a large sphere would fall short of the charge, and the inside of a charged shell would no longer be field-free. There are two different ways Coulomb's law could fail, and they are caught by opposite strategies. A wrong exponent has no length in it and shows equally in any apparatus; a mass has a length, and shows as the square of the apparatus's size. That is why the tightest limit on the photon's mass comes not from a laboratory but from the magnetic field the solar wind carries past Pluto.

electromagnetism · Gauss's law A current limited by the wire's circumference. The largest supercurrent a wire can carry before the field of its own current destroys superconductivity at its surface, Ic = 2πaHc, against the wire's diameter, for lead, mercury and tin at 1.5 K, on logarithmic axes; dashed, a niobium–titanium wire carrying 3,000 A/mm² in a field of 5 tesla, which grows with the cross-section rather than the circumference. A lead wire 1 mm across carries 192 A — 244 A/mm² — and one 1 cm across only ten times as much, at a tenth of the density. Below 81 micrometres lead carries more than niobium–titanium; above it niobium–titanium wins by a factor that grows with the wire, and it does so in a field of 5 tesla, where lead cannot superconduct at all.

The current a superconductor stops with its own field

A superconducting wire has no resistance, so it seems it should carry any current at all. It carries a definite amount and no more, and the amount is set not by the metal's ability to conduct but by the field the current itself makes: when that field at the wire's surface reaches the critical field, superconductivity fails there. The rule makes the largest current proportional to the wire's radius rather than its area, makes a thick wire worse per square millimetre than a thin one, and explains why the first superconducting magnet, wound in 1913, did not work.

electromagnetism · Superconductivity Where the heat would land. The average field strength squared on a horizontal plane half-way up the oven from 3 of the in-band modes taken together, 2,3,2; 3,2,2; 2,4,1, darker meaning stronger. The bright patches are spaced by about half a wavelength of the mode along each wall, some 7 to 15 cm for these indices, and between them are nodes where the heating falls to almost nothing. A plate of cheese on a stopped turntable shows this as melted spots, and a measurement of their spacing against the frequency gives the speed of light to within several per cent, a popular school experiment. Since many modes can be involved and the load disturbs them, the method is not as clean as it looks, and here the load's own effect is not drawn.

The oven that cooks in spots

A microwave oven is a metal box, and a metal box has only so many ways of ringing at a given frequency. At 2.45 GHz a kitchen oven allows a handful, and the field inside is the sum of those few standing patterns, with hot patches and nodes several centimetres apart. Food on a stopped plate shows them as a pattern of scorch and frost. A turntable can smear this only into rings, and frozen food has a further trap, because ice absorbs thousands of times less than water.

waves · Standing waves

Named alongside it

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

Meissner effectSuperconductivityAmpere lawAttenuationBoundary conditionCavity modeComplex wavenumberCooper-pairCoriolis forceCritical angleCritical currentCritical field

All concepts