Illustrated Physics
physics, drawn
Concept

Scale height — where it appears

The distance over which an atmosphere's density falls by a factor of e, being the ratio of thermal energy to gravitational energy per molecule. It is about 8.5 kilometres for the Earth's air, so the pressure halves every 5.5 kilometres and the same exponential describes a colloid settling in a beaker.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The isothermal atmosphere against the real one. Pressure as a fraction of its sea-level value, against altitude. The curves are the isothermal barometric formula at 220, 288, 400 kelvin, whose scale heights are 6.4, 8.4, 11.7 kilometres. The points are the measured standard atmosphere. At 20 kilometres the 288 kelvin model is 71 per cent out, because the air up there is not at 288 kelvin.

Why the air thins with height, and why that is the same law as the speeds

The pressure of the atmosphere falls exponentially with altitude, and the distribution of molecular speeds falls exponentially with energy. These are not two results that happen to look alike. They are one statement read on two axes.

thermodynamics · Kinetic theory Rays that turn round without meeting anything. Rays leaving an eye 1.5 m above a road, at 0.18°, 0.28°, 0.36°, 0.42°, 0.52° below the horizontal, in air whose refractive index is reduced by 3.0e-5 at the hot surface and recovers over 5 cm. Each is traced by integrating the ray equation, with the conserved quantity n cos θ fixed by where and how the ray set out. The shallow ones come back up without touching anything — the lowest gets to 0.6 cm above the surface and turns — and a ray that returns to eye level from below is seen as sky lying on the road. The steep ones run out of gradient first and hit it. The dividing angle is 0.444°, which is what the whole effect is made of: an unremarkable temperature difference and an angle a fiftieth the width of the Moon. Heights are exaggerated 128× against distances; at true scale every ray here would be indistinguishable from the axis. The conserved n cos θ holds to 4.3e-14 over every trace, which is what says the turns are the physics and not the integrator.

The ray that bends without a surface

Snell's law is about a boundary, and light bends in air where there is no boundary anywhere. Let the index vary continuously and the law of angles becomes a differential equation — one that carries a conserved quantity, forbids the ray from reaching certain heights, and turns a hot road into a mirror a hundred metres long.

optics · Fermat

Named alongside it

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

The Boltzmann factorConservation lawsDeflection angleFermat's principleFocal lengthGradientHydrostatic equilibriumMaxwell–Boltzmann distributionMean free pathOptical path lengthPotential energyRefractive index

All concepts