Illustrated Physics
physics, drawn
Concept

Spherical harmonics — where it appears

The functions of direction on a sphere that are the natural standing waves there, labelled by a degree l and an order m, with 2l + 1 of them at each degree. They describe atomic orbitals, the shapes of multipole fields and the patterns in the cosmic microwave background.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Three bodies of one mass, one field outside, three fields inside. The gravitational field against distance from the centre, both in units of the body's surface values, for 3 spherical bodies of the same mass and radius: a uniform ball; a dense core under a light mantle; a hollow shell. Outside the surface the three curves are one curve — checked at 1.7 radii by adding up the pull of every mass element in each body, which agrees with the pull of a point of the same mass to better than a part in five hundred. Inside they part: the uniform ball's field falls in a straight line, the layered body's rises to 1.24 times the surface value at the top of its core, and the hollow shell's is zero throughout its cavity. Their moment-of-inertia factors are 0.400 (uniform ball), 0.330 (dense core under a light mantle), 0.551 (hollow shell) — a number that no measurement of the field outside can supply.

The field outside that cannot find the core

Gauss's law gives the field outside a body from what it encloses, and read backwards it is a limit. A uniform ball, a planet with an iron core and a hollow shell of the same mass have identical gravity everywhere outside. The field fixes a list of numbers — the mass, the flattening, higher moments — and leaves free everything else, including the moment of inertia; the Earth's core was weighed by its wobble, not by its pull.

electromagnetism · Gauss's law A linear quadrupole, and the cone on which it vanishes. Equipotentials of a linear quadrupole — charges +q, −2q, +q in a line, a distance d apart — in a plane containing the line, positive in one colour and negative in the other. The total charge is zero and so is the dipole moment, so far away the potential falls as 1/r³ with the angular pattern 3cos²θ − 1: positive along the axis, negative around the waist, and zero on a cone. Found on circles of growing radius, the zero of the exact potential lies at 50.89° at 2d, 53.76° at 4d, 54.49° at 8d, 54.67° at 16d, closing on arccos(1/√3) = 54.74°, the dashed lines.

The shape that takes five numbers

A charge distribution's net charge is one number and its dipole three. The next term, the quadrupole, is where the distribution's shape first reaches the far field, and it is a table of nine numbers of which only five survive: two for the shape and three for which way it points. Its field vanishes on a cone at 54.74° — the same angle at which a spinning sample makes a nuclear magnetic resonance spectrum sharp.

electromagnetism · The field concept What a station hears from one thunderstorm region. The power spectrum of the vertical field from a single region of thunderstorms, received 20° and 90° of arc away, from a sum of the first seven resonant patterns, each with the measured peak frequency and a quality factor of 4 to 6, both taken as inputs. Measured spectra fall off with frequency rather more than this one, because real strokes and the real ionosphere are not as simple. The resonances are so broad that neighbouring peaks run into each other; a quality factor of five means a peak about a fifth as wide as its frequency, and the ringing dies in a few cycles. At 20° every tone is present, the higher ones merging. At 90° the first, third and fifth have vanished, because those patterns have a node a quarter of the world away, and the spectrum is made of the even tones alone. Real stations receive the sum of the world's storms, which move with the time of day, and the heights of the peaks drift with them.

The tone the whole Earth rings at

The ground conducts and so does the ionosphere, eighty kilometres up, and between them lies a waveguide the size of the planet. Two walls, unlike a pipe, carry waves of any length, so the guide takes radio waves forty thousand kilometres long — longer than the world is round — and, closed on itself, it rings. Lightning strikes fifty times a second and keeps it ringing at about 7.8, 14.3 and 20.8 hertz. Perfect walls would put the tones at 10.6, 18.4 and 25.9; the difference is the ionosphere, a ceiling that leaks.

waves · Guided waves

Named alongside it

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

Multipole expansionCutoffElectric potentialGauss's lawGravity fieldGroup velocityGuided wavesIntermolecular forcesInverse problemLaplace equationMagic angleMoment of inertia

All concepts