Illustrated Physics
physics, drawn
Concept

Scaling law — where it appears

A relation stating how a quantity changes with size, usually as a power of a length or a mass. Comparing two scaling laws that grow at different rates is often the quickest way to see which effect wins at large sizes and which at small ones.

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

An atom made of powers of n. Five properties of a hydrogen atom in a level of principal quantum number n, each divided by its value in the ground state, against n from 1 to 300 on logarithmic axes. The radius of a circular orbit rises as n², to 0.53 µm at n = 100; the orbital period as n³, to 1.5 × 10⁻¹⁰ s; the radiative lifetime of the circular state, from the Larmor loss on its orbit, as n⁵ — slope 5.000 — to 0.9 s. The binding energy falls as n⁻², to 1.36 meV, and the electric field that pulls the electron away falls as n⁻⁴, to 3.2 V/cm. Across n = 1 to 300 the lifetime changes by 2 × 10¹² and the ionising field by 8 × 10⁹.

The atom the size of a bacterium

Excite hydrogen's electron to the hundredth level and the atom is a micrometre across, bound by a thousandth of an electronvolt, pulled apart by the field of a torch battery, and — in its most circular state — able to live for nearly a second. Every one of those numbers is a power of n, and together they make the most exaggerated atom there is into one of the most useful.

quantum · Atomic spectra How high a jumper of any size can rise. Jump height — the rise of the centre of mass after take-off, in vacuum — against body mass, on logarithmic scales. The flat line is Borelli's limit: if muscle can do a fixed amount of work per kilogram and a fixed fraction of the body is jumping muscle, every animal rises the same height, here 0.61 m. The sloping line is the limit set by power: the legs straighten in a time proportional to their length, and a small animal must deliver its energy in a fraction of a millisecond, faster than muscle can. That limit rises as the 2/9 power of mass and meets Borelli's near 80 kg. Points mark take-off speeds reported for six animals, converted to the height they would give without air: flea 0.18 m, froghopper 1.13 m, locust 0.52 m, frog 0.62 m, galago 2.22 m, human 0.49 m. Every animal well left of the crossing jumps far above its power limit, and every one of them is known to store energy in a spring before it jumps.

The jump that does not get higher with size

A flea, a locust and a person all lift their centres of mass by roughly half a metre when they jump, though their masses differ by a factor of a hundred million. Giovanni Borelli saw why in 1680: the work a muscle can do grows with its mass, and so does the work needed to lift a body, so the height cancels the size. What does not cancel is time. A small animal's legs straighten in a fraction of a millisecond, and no muscle can deliver its energy that fast — a flea would need more than five hundred times the power any muscle can give. Every small jumper reaches Borelli's height anyway, and every one of them does it with a spring.

mechanics · Energy

Named alongside it

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

Air dragBlackbody radiationBohr modelCorrespondence principleElastic energy storageField ionisationKinetic energyLarmor formulaMuscle powerPotential energyPower amplificationResilin

All concepts