Illustrated Physics
physics, drawn
Concept

Multipole expansion — where it appears

Writing the field of a charge or mass distribution, seen from outside it, as a sum of terms that fall off ever faster with distance. The terms are the monopole, dipole, quadrupole and on; far away only the first non-zero one matters, which is why a neutral molecule's field is that of its dipole.

Named by 4 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 Each unit of angular momentum costs a factor of tens of thousands. Weisskopf's estimates of the half-life of a gamma-ray transition in a nucleus of mass number 100, against the photon's energy, for electric multipoles of order 1 to 5 — the photon carrying away 1 to 5 units of angular momentum. At 1 MeV the estimates are 10⁻¹⁵ s for E1, 10⁻¹¹ s for E2, 2.0 µs for E3, 292.5 ms for E4, 17.3 h for E5. Each extra unit multiplies the half-life at 1 MeV by between 60,000 and 210,000: about a thousand of that is (λ/2πR)², the square of the photon's reduced wavelength over the nucleus's radius, which grows as the energy falls, and the rest is a numerical factor that grows with the order. A transition that must carry away five units at 100 keV has an estimated half-life longer than the age of the universe; the same energy carried off with one unit takes less than a nanosecond.

The spin a photon has to carry away

An excited nucleus sheds its energy as a gamma ray in a millionth of a millionth of a nanosecond — or in minutes, or centuries, or never within the age of the universe. The energy available hardly matters. What matters is how much angular momentum the photon must carry off, because a source much smaller than its wavelength is very bad at launching a wave that twists, and each extra unit of twist costs a factor of tens of thousands in time. Selection rules are usually stated as what cannot happen; they are really a price list, and it spans forty decades.

quantum · Decay The field along the axis of two coils. The magnetic field along the common axis of two identical coaxial loops carrying the same current, against distance from the midpoint in units of the loops' radius, each normalised to its value at the midpoint, for separations of 0.6, 1, 1.4 radii. Closer than one radius the field peaks at the centre; further apart it dips there between two peaks; at exactly one radius — Helmholtz's spacing — the curvature at the centre vanishes and the field is flat, staying within a part in a thousand of its central value out to 0.17 radii either side.

The two coils one radius apart

A single loop of wire makes a field that is strongest at its centre and falls away in every direction. Two loops on the same axis, spaced exactly one radius apart, make a field that is flat at the midpoint to a part in a thousand over a region a third of the coils' size — and flat sideways as well as along the axis, although only the axis was designed. The sideways uniformity is free because a magnetic field in empty space obeys Laplace's equation, which ties its variation in one direction to its variation in the others, term by term.

electromagnetism · Magnetism

Named alongside it

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

Laplace equationSpherical harmonicsAngular momentumBiot savart lawElectric potentialForbidden transitionGamma-rayGauss's lawGravity fieldHalf-lifeHarmonic functionHelmholtz coil

All concepts