Illustrated Physics
physics, drawn
Concept

Circular motion — where it appears

Motion on a circle at constant speed, in which the velocity turns without changing length and the acceleration points always at the centre. The acceleration is the speed squared over the radius even though nothing about the speed is changing, because acceleration is a rate of change of a vector.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

The classical atom, and how long it lasts. An electron in a circular orbit of 5.29·10⁻¹¹ m loses energy at the rate the Larmor formula gives, so its radius obeys r³ = r₀³ − 4k²t/c³ and reaches zero in 1.556·10⁻¹¹ seconds — 16 picoseconds. It completes about 2.04·10⁵ orbits on the way, so the spiral is far too tight to draw. Nothing in this calculation is wrong: the acceleration is right, the radiated power is right, and the conclusion is that matter cannot exist. The curve is the shape of that conclusion.

A charge that turns must glow

An accelerating charge radiates, and a charge going round in a circle is accelerating. Apply that to an electron orbiting a nucleus and classical physics predicts that every atom collapses in sixteen picoseconds — a calculation with nothing wrong in it except its conclusion.

astrophysics · Radiating charge Same field, same charge, three momenta. Electrons entering a 10 mT field at right angles to it, at 1, 4, 9 keV, each drawn for a quarter of its turn. The radius is mv/qB — 10.7 mm, 21.3 mm, 32.0 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 3.57 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment.

The force that does no work

A magnetic field can turn a moving charge through any angle at all and cannot add a joule to it. Everything a magnet is good for follows from that one prohibition — including the fact that a bent track is a reading of momentum, and that a machine built on it stops working at five kilovolts for an electron.

electromagnetism · Magnetism The term that abolishes the inner orbits. The effective potential of the Schwarzschild geometry per unit mass, in units of c², at 4 angular momenta, against radius in Schwarzschild radii. Newton's version has a minimum — a stable circular orbit — at L̃²/GM for every angular momentum there is, however small, and it is the dashed curve at the same L̃, here with its minima at 10.13, 7.61, 6.00, 5.12 rs. General relativity adds one term, −GM L̃²/c²r³, and that minimum stops existing below a definite angular momentum. At L̃ = 4.50 GM/c the barrier and the well are still separate, at 1.83 and 8.29 rs. The two merge at L̃ = √12 GM/c, at 3 rs = 6GM/c² — the innermost stable circular orbit, where the curve drawn here has neither a maximum nor a minimum but a single inflection. Below that momentum — the curve at L̃ = 3.20 GM/c — there is no stationary point anywhere outside the horizon, so no circular orbit exists at all, at any angular momentum whatever. None of that is a property of matter; it is a statement about the geometry, and it fixes the energy of the innermost orbit at √(8/9) = 0.94281 of mc², so 5.719% of the rest mass has been radiated by anything that reached it. Every well drawn here was located on the emitted curve by golden section and checked against the closed form.

The orbit that cannot be made smaller

Newtonian gravity allows a stable circular path at every radius, however tight, and there is no innermost one. General relativity adds a single term to the expression that says so, and below 6GM/c² no stable circular path exists at any angular momentum whatever. Anything arriving there has radiated 5.72% of its rest mass, which is eight times what hydrogen fusion converts.

astrophysics · Orbit stability A governor that hunts. The speed of an engine with a centrifugal governor after its load is suddenly increased, against time in units of √(ℓ/g), for three amounts of friction in the governor's linkage: damping 0, 0.25, 0.6, against Vyshnegradsky's minimum for this governor, 0.397. With no friction the speed oscillates about its new, slightly lower value with a growing swing — the governor hunts, opening and closing the throttle out of step with the engine — until the arms hit their stops and the curve ends. At damping 0.25, below the minimum, it still hunts, more slowly. At damping 0.6, above the minimum, the oscillation dies away and the engine settles at its new speed. The friction that seems to spoil the mechanism is what makes it work.

The governor that needed friction to work

James Watt's steam engines were kept at a steady speed by two heavy balls on hinged arms, spun by the engine itself: run faster and the balls fly out and close the steam valve; run slower and they drop and open it. The balls stand at a height set by the speed alone. As engines grew and governors were made more sensitive, many began to hunt — speeding and slowing in a growing rhythm, the governor always too late. In 1868 James Clerk Maxwell wrote the equations down and showed why: a governor with no friction cannot settle at all, and the more precisely it is meant to hold the speed, the more friction it needs.

mechanics · Circular motion

Named alongside it

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

Angular momentumAtomic structureCentrifugal forceCentripetal forceConical pendulumControl theoryCyclotronDampingDipole radiationEffective potentialEfficiencyEvent horizon

All concepts