Illustrated Physics
physics, drawn
Concept

Averaging — where it appears

Replacing a rapidly varying quantity by its mean over the fast motion, so that a slower motion can be followed without it. The technique is exact only in a limit, and its value is bounded by the ratio of the two timescales — which has to be computed rather than assumed, because everything the method concludes is wrong when that ratio is not small.

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

The potential a shaken pivot creates. The effective potential of a pendulum whose pivot is shaken vertically, against the angle from hanging, for shaking rates of 8, 14, 20, 30 times the pendulum's own frequency at an amplitude of 0.12 of its length. The shaking averages to no force at all — it is up as often as down — and yet it adds a term to the potential, because the pendulum's position is correlated with the phase of the shake rather than independent of it. The added term is proportional to sin²θ, so it is largest sideways and zero at both the hanging and the inverted positions, and it turns the maximum at 180° into a minimum once 14× and 20× and 30× the natural frequency is reached. The criterion is (aΩ)² > 2gL: the shake speed must beat the speed a fall through the pendulum's own length would give. Upside down then becomes a stable equilibrium, with a restoring force and a period of its own.

Held up by a force that averages to nothing

Shake a pendulum's pivot up and down fast enough and the pendulum will stand upside down, balanced, and push back if it is nudged. The shaking supplies no average force at all — it is up as often as it is down — and the reason it nevertheless holds is that the pendulum's position and the phase of the shake are not independent.

mechanics · Pendulum Three charges, three orbits, one drift. Three particles released at rest in crossed fields — 1000 V/m across 0.1 T — with their paths integrated by a scheme that rotates the velocity rather than adding to it, so the magnetic part changes no speeds. The three loops have wildly different sizes and periods: the the electron turns at 2799.25 MHz, the proton turns at 1.52 MHz, the α particle turns at 0.76 MHz. Their guiding centres all creep along the same line at the same rate, measured here from the orbits at -1.000e+4, -1.000e+4 and -1.000e+4 m/s against −E/B = -1.000e+4 m/s, a spread of -0.00 per cent. Neither the charge nor the mass nor the sign appears in the answer. A plasma in crossed fields therefore moves bodily and carries no current from this drift at all, which is the opposite of what an intuition built on ions being heavier than electrons expects.

The drift that does not care what the charge is

A charge in a uniform magnetic field goes round in a circle and arrives nowhere. Add anything at all — an electric field, gravity, a gradient in the magnetic field itself — and the circle's centre creeps sideways, at right angles to both. One of those drifts is the same for every particle regardless of charge, sign or mass; the others are not, and the difference decides what a plasma does.

electromagnetism · Magnetism

Named alongside it

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

Adiabatic invariantConfinementDriftEffective potentialElectric fieldEquilibriumGuiding centreLarmor radiusThe Lorentz forceMagnetic fieldMagnetic momentMathieu equation

All concepts