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 2 essays across 2 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

Named alongside it

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

AttenuationBoundary conditionComplex wavenumberCritical angleDiffusion equationEvanescent waveFluorescence microscopyHeat conductionNear fieldPeriodic forcingPhase lagReciprocity

All concepts