Illustrated Physics
physics, drawn
Concept

Magnetic moment — where it appears

The strength and orientation of a current loop or a spin as a source of magnetic field, and what a field exerts a torque on. Far from the loop its field is a dipole's whatever the loop's shape, so the moment is all that survives of the geometry.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

A torque, and no force at all. A loop of 1 turn enclosing 20 cm² and carrying 10 A has a magnetic moment of 0.02 A m². In a uniform field of 0.05 T the torque on it is m B sin θ, drawn here against the angle between the moment and the field: zero when they are aligned, largest at 0.001 N m across, and zero again when they are opposed. The second curve is the energy, −m·B, whose minimum is the aligned position and whose maximum is the opposed one — which is why a compass needle settles one way round and not the other. The net force is zero at every angle on this axis, exactly and not approximately: the force is I dl × B summed round the loop, the sum of dl round any closed path is zero, and a constant B comes outside the sum. The inset shows the four forces on a rectangular loop; the pair across the axis is the couple, and the pair along it cancels.

The loop that behaves like a needle

Far enough away, a current going round in a circle is indistinguishable from a bar magnet, and one number describes both. That number tells a uniform field how to turn the loop and gives it no way to pull on it at all — which is why two magnets attract by the fourth power of the distance and not the second.

electromagnetism · Magnetism A 120 g top at 3000 rpm, precessing once every 1.92 s. A disc of radius 30 mm spinning at 3000 revolutions a minute on a shaft 45 mm long, tilted 30° from the vertical. The weight acts at the centre of mass and the pivot holds the bottom, so the torque about the pivot is horizontal and at right angles to the plane containing the axis and the vertical. Angular momentum points along the axis; a torque at right angles to a vector turns it without changing its length, so the axis sweeps round the dashed circle instead of falling. The precession rate is Mgl divided by I₃ω₃ to leading order, which is 3.269 radians a second here, or one turn every 1.92 seconds — slower the faster it spins.

The push that comes out sideways

Push down on a spinning wheel's axle and it swings horizontally. Nothing about that is mysterious once angular momentum is a vector — but the steady precession every demonstration shows is a solution nobody's initial conditions select, and a top released from rest does something else first.

mechanics · Rotation How fast a spinning electron would have to turn. The equatorial speed of a uniform sphere with the electron's mass and an angular momentum of ħ/2, against the radius it is given, both logarithmically. The expression is 5ħ/4mr and it passes the speed of light at 4.83e-13 m — half a picometre, which is four hundred times larger than a hydrogen nucleus. Giving it a smaller radius only makes the answer worse: at the experimental upper bound, 10⁻¹⁸ m the equator would move at 4.8e+5 times the speed of light; at the classical electron radius the equator would move at 1.7e+2 times the speed of light; at the reduced Compton wavelength the equator would move at 1.3e+0 times the speed of light. No radius the electron is permitted to have gets anywhere near a legal answer, and the experimental bound is off the scale by eight orders of magnitude. So the angular momentum is not the angular momentum of anything going round. It is a property the particle has, in the same way a charge is, and the only thing it shares with a spinning top is the algebra it obeys — which is, admittedly, the whole of what angular momentum means in physics.

The angular momentum that is not a rotation

An electron has angular momentum, and it is not going round anything. A sphere of its mass carrying that much angular momentum would need its equator moving at half a million times the speed of light at any size the electron is allowed to have. What survives of the analogy is the algebra — and the algebra turns out to require that turning the thing all the way round leaves it changed.

quantum · Spin One level, split by the speed of the electron in it. Hydrogen's n = 2 level on the left at the scale of the whole spectrum, and the same level on the right at a scale ten thousand times finer. The gross structure puts it 3.4014 electronvolts below the ionisation limit; the relativistic and spin–orbit corrections then split it into a lower pair, j = ½, and an upper, j = 3/2, separated by 45.3 microelectronvolts. That gap is 1.331e-5 of the level's own binding energy, which is exactly α²/4 — so the size of the splitting is a measurement of how fast the electron is going, α being its speed in units of c on the innermost orbit. The same physics in sodium is larger by four orders of magnitude: its two D lines at 588.995 and 589.5924 nanometres differ by 2.133 millielectronvolts, which is 1.01e-3 of the transition energy, because the outer electron of sodium penetrates to where the nuclear charge is far from screened and the correction goes as the fourth power of the charge it sees.

The line that is really two

Sodium's yellow line is two lines six-tenths of a nanometre apart, and the gap is not a property of the light. It is a splitting of the atom's own level, caused by the electron's magnetic moment sitting in the field it sees because it is moving — and its size, relative to the level it splits, is exactly α²/4.

quantum · Atomic spectra Every line that goes in has to come out. Field lines of a 5:1 solenoid in the plane through its axis, each traced by stepping along the local direction of a field summed turn by turn from Biot–Savart. Inside the winding they are parallel and evenly spaced, which is the picture the textbook argument is about. Outside they are not absent: they are spread over the whole of the rest of space, which is why the field there is small — 1.60e-2 of the centre value at 2 radii off the axis — and why it cannot be zero. A line has no end, so every one of the lines through the bore returns outside, and a field with no outside would be a field whose lines stop.

The field outside the solenoid, which is not zero

Ampère's law says the field outside a solenoid vanishes, and every step of that argument is exact — for a winding of infinite length. A real one is a bar magnet seen from outside, its external field falls as the inverse square of its length rather than to nothing, and the "exactly zero" that makes the derivation so satisfying is the one part of it a laboratory cannot have.

electromagnetism · Ampere law A loop at rest carrying 2.2e-12 kg m/s. A square loop of area 100 cm² carrying 20 amps, sitting still in a uniform electric field of 1.00 megavolts a metre. The carriers going up the field on one side cross 100 kilovolts on the way, so the ones in the top wire are less energetic than those in the bottom by that much per unit charge. The current is the same all the way round, so the same number of carriers pass per second in each wire — but they carry different energy, and momentum is energy times velocity over c². The two wires therefore contribute unequally, and the difference is 2.23e-12 kilogram metres a second, pointing across both the field and the dipole. Nothing in the picture is moving as a whole. The electromagnetic field round the loop carries exactly that momentum the other way.

The momentum of something that is not moving

A current loop sitting still in an electric field has momentum in the space around it. Nothing is moving, so something must be carrying an equal and opposite amount — and it is the loop, whose carriers on the high-potential side are more energetic than those on the low.

electromagnetism · Field energy 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 The levels that refuse to cross. The six energies of sodium 3p against an applied field from zero to 4 crossover fields, 147 T, from diagonalising spin–orbit coupling and the field together. At zero field there are two levels, 515.5 GHz apart. The two states with the largest |mⱼ| are straight lines at every field, because nothing else shares their mⱼ. The other four bend. The two states with mⱼ = −½ approach within 486.0 GHz near 12.3 T and then separate again — √2 ζ exactly, since the coupling between them sets how close they may come — while the mⱼ = −3/2 state falls straight through the lower mⱼ = +½ state at 16.6 T as if it were not there. States that the field cannot mix cross freely; states it can mix exchange character instead, and that exchange is the passage from one pattern to the other.

The field an atom calls strong

A magnetic field splits sodium's two yellow lines into ten, at spacings set by Landé's factors — until the field grows past the one the electron already feels from its own motion, when the ten reorganise into the three a theory without spin predicts. Nothing about the magnet decides which pattern appears. The atom does: the same 45 tesla is a strong field for hydrogen, a middling one for sodium and a weak one for caesium.

quantum · Atomic spectra No beam is narrow enough and wide enough at once. Three lengths at the far end of a Stern-Gerlach magnet 10 centimetres long with a gradient of 1000 tesla a metre, against the width of the beam entering it, for an electron at 100 electronvolts. The spin splitting is a horizontal line at 2.9e-6 metres: it does not depend on the beam's width. The Lorentz blurring rises in proportion to the width, because the field a particle sees depends on where in the beam it is, and the divergence-free condition ties a gradient in one component to a gradient in another. It overtakes the splitting at 1.0e-9 metres. Narrower than that and diffraction has already spread the beam by 3.9e-3 metres, which is larger still. There is no width at which the splitting is the largest of the three, and the magnet's length and gradient cancel out of the comparison entirely — so no magnet helps.

The experiment that defines spin and cannot be done on it

A Stern–Gerlach magnet separates magnetic moments and is how spin was discovered. It cannot be made to work on a free electron, and the obstruction is not the apparatus: the field gradient that splits the beam also deflects the charge by an amount that varies across it, and the ratio of the splitting to that blurring comes out as the de Broglie wavelength over the beam width — with the magnet's length and gradient cancelling exactly.

quantum · Spin Circles that go nowhere, and a current across the line. Gyrating ions in a uniform magnetic field pointing out of the page, with 60 guiding centres drawn from a density that falls by a factor of e every 3 gyroradii to the right, and Maxwellian speeds. Every ion goes round clockwise and none of the circles moves. Of the circles that cross the dashed vertical line, those centred to its left cross it moving down and those centred to its right cross it moving up; in this sample 11 cross moving down and 7 moving up, a count a sample this small could turn either way. Because there are more circles on the left, the ions at the line move downwards on average over every speed and phase, at exactly the thermal speed squared over the gyrofrequency times L, 0.33 thermal speeds, computed by averaging over speeds and phases and checked against that value. It is a current, carried by circles whose centres are still.

The current no particle carries

A magnetised plasma holds its own pressure against the field only if a current flows across the pressure gradient, and the fluid equations say exactly how much. Follow the particles in a uniform field and none of them is going anywhere; every guiding centre is still. The current is real all the same. It is made of circles that are more crowded on one side of a line than the other, and when the field is not uniform, the drifts that do move the guiding centres flow the wrong way.

electromagnetism · Magnetism

Named alongside it

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

Angular momentumSpinThe Lorentz forceMagnetic fieldSymmetryAtomic spectraConservation lawsDegeneracyDipole radiationDriftElectric fieldEnergy levels

All concepts