Illustrated Physics
physics, drawn
Concept

Magnetic flux — where it appears

The amount of magnetic field crossing a surface, whose rate of change is what drives an electromotive force around the surface's boundary. Because the field has no sources, the flux through any surface depends only on its boundary — so the surface can be chosen freely and the answer will not move.

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

What a collapse does to a field. A body of radius 700,000 km carrying a field of 0.01 T, collapsing to 10 km. The flux through every comoving loop is fixed, so B goes as 1/R² and the field reaches 4.9·10⁷ T — a compression of 7·10⁴ in radius bought a factor of 4.9·10⁹ in field. The line has slope exactly −2 and that is the only claim being made: what a real object ends up with also depends on how well the flux was held, and on what generated it.

The field that cannot get out

Squeeze a lump of conducting fluid and its magnetic field comes with it, because the flux through any loop that moves with the material cannot change. Halve the radius and the field goes up fourfold; collapse by a factor of seventy thousand and it goes up by five thousand million.

astrophysics · Flux freezing A metre of tube, with and without the metal. The magnet's position against time over a 1.00 m drop, integrated from Newton's second law with the eddy-current drag included, and the same fall with the drag switched off. The braked descent takes 2.09 s against 0.45 s in free fall. After a transient of about 50 milliseconds — the mass divided by the drag coefficient, and the only timescale in the problem — the trace is a straight line, which is to say the magnet is falling at a constant 49.0 cm/s. Every joule of the 118 mJ of gravitational energy released has gone into resistive heating of the wall, and none into the magnet's kinetic energy, since that is the same at the bottom as it was a moment after the start.

The magnet that falls slowly

Drop a magnet down a copper pipe and it takes several seconds to fall a metre, drifting rather than falling. Nothing touches it, the copper is not magnetic, and there is no circuit anywhere. The pipe arrives warm.

electromagnetism · Induction A disc that generates 0.785 V with no changing flux anywhere. The force per unit charge on a carrier in a conducting disc of radius 100 mm spinning at 3000 revolutions a minute in an axial field of 0.5 tesla, against how far out the carrier is. Each carrier is moving through the field at ωr, so v × B pushes it radially with a force per unit charge of ωrB — zero at the axle and largest at the rim. The area under the line is the potential difference between the axle and the rim, which is ½BωR² = 0.7854 volts. Nothing in this calculation mentions a loop, a flux or a rate of change, and the flux through the circuit formed by the disc, the contacts and the external wire does not change at all while the machine runs.

The rule that is two laws wearing one coat

The flux rule folds two entirely different pieces of physics into one number, and they agree exactly, which is why nobody notices there are two. A disc spinning in a steady field generates a voltage with no changing flux anywhere, and the folding comes apart.

electromagnetism · Induction 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 Two loops, two calculations, one number. The coupling between a circular loop of radius 10 cm and a rectangle of 5 by 3 cm tilted at 35° and offset sideways, against how far apart they are along the axis. The two loops differ in area by a factor of 21 and in shape entirely. Two quantities are plotted and they lie on top of one another. One is the flux through the rectangle when a current runs in the circle, obtained by integrating the circle's Biot–Savart field over the rectangle's tilted surface. The other is the flux through the circle's disc when the same current runs in the rectangle, integrated over a disc a hundred times the area. Nothing is shared between the two calculations except the positions of the wires, and they agree to 0.075 per cent — a difference which is the quadrature's, not the physics's. This is reciprocity, and it is not obvious: there is no reason from the geometry why a small loop should catch as much of a big one's field as the big one catches of the small one's. It follows from the double line integral the two quantities can both be reduced to, which is symmetric in the two loops — and that reduction is the argument, not this figure, which is the check that the argument is true of the actual fields.

The coupling that is the same both ways

A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand.

electromagnetism · Induction How much of μ₀N²A/ℓ a real coil actually has. The inductance of a 140-turn coil of radius 10 mm, divided by the long-solenoid formula μ₀N²A/ℓ, against how long the coil is compared with its diameter. The inductance is computed as a sum of Maxwell's mutual inductances between every pair of turns plus each turn's own, so the long-coil formula appears nowhere in it. At a length equal to the diameter the real coil has 68% of what the formula promises, and at a quarter of the diameter about a third. The reason is that the formula assumes every turn is threaded by the full interior field, and near the ends of a short coil the field has already begun to spread. The ratio is Nagaoka's coefficient, tabulated since 1909, and the computed points agree with the table.

The circuit that fights its own change

Every circuit is threaded by the field its own current makes, so every circuit resists having that current altered. It is the reason a coil takes time to start and the reason breaking one makes a voltage a thousand times the supply's.

electromagnetism · Induction The rotation that mixes electricity into magnetism. The two Lorentz invariants of a field — E² − c²B² across and 2E·cB up — as the field is rotated by the duality transformation that takes E into cB and cB into −E. Every configuration moves on a circle, so the combination of the two invariants is preserved while neither is. A light wave sits at the origin and stays there, which is why a wave cannot be turned into anything else by this rotation; a static charge starts on the positive axis and is carried round to a pure magnetic field a quarter turn later. The source-free equations are unchanged by the whole family, so a universe with no charges in it has no way to say which field is which.

The symmetry one missing charge would complete

Maxwell's equations with no sources are unchanged by rotating the electric field into the magnetic one. With sources they are not, and the only thing missing is magnetic charge — which, if one existed anywhere, would force every electric charge in the universe to be a multiple of a fixed unit.

electromagnetism · Maxwell equations The ring picks a whole number and pays for the difference. The kinetic energy of the screening current in a superconducting ring, against the flux applied from outside, in units of the flux quantum. Each parabola belongs to one winding number — the number of times the condensate's phase turns round the ring, which has to be an integer because the wavefunction has to come back to itself. The ring cannot hold an arbitrary flux, so it holds the nearest whole number of quanta and drives a current to make up the difference; the energy of that current goes as the square of the mismatch, which is what each parabola is. The lowest curve at each applied flux is the state the ring is in, and the winding number changes at exactly the half-integers — 0.50 and 1.50 and 2.50 here. So the measurable properties of the ring are periodic in the applied flux with a period of one quantum, which is 2.0678 × 10⁻¹⁵ webers and is about the flux the Earth's field puts through a square micrometre.

Two lengths, and which one is longer

A superconductor has a depth to which a field leaks in and a distance over which superconductivity itself can be built up. Which of the two is larger decides the sign of the energy of a boundary — and therefore whether the material keeps every field out or fills itself with a lattice of holes.

electromagnetism · Superconductivity Outside the winding, where nothing is supposed to be. The mid-plane of a solenoid 20 radii long, wound at 8 turns per radius, with the field summed turn by turn rather than assumed. The upper curve is the axial field against distance from the axis: it holds up across the winding and collapses outside, reaching 0.47 per cent of its central value at 2 radii. The lower curve is the flux enclosed by a circle of that radius, which is what the vector potential integrates to. The two behave completely differently, and that difference is the whole subject: the field an electron outside can feel has effectively gone, and the flux it encircles has not. The enclosed flux does fall, by 3.29 per cent out to four radii, because the lines that leave the ends come back through the plane outside the coil — that return flux is the leak a real experiment has to defeat, and the reason the definitive versions used a closed toroidal magnet with no ends at all.

The phase a magnet leaves on a path it never touched

An electron beam split around a solenoid comes back with its fringes displaced, although neither path ever entered a magnetic field. What the electron responds to is the flux it went round, and the only local quantity that knows about that flux is the potential.

quantum · Matter waves The field a pinch gives up relaxing into. The axial and azimuthal field across a cylinder of conducting plasma in the minimum-energy state at fixed helicity, Bz = J₀(λr) and Bθ = J₁(λr), drawn for λa = 1.5 and λa = 3. Solid lines are the axial field and dashed lines the azimuthal field. Both satisfy ∇×B = λB, checked by finite differences at three radii, so the current runs along the field everywhere and the field exerts no force on the plasma. At λa = 3 the axial field passes through zero at r = 0.802a and is reversed outside it. The reversal needs λa above 2.405, the first zero of J₀, and nothing was imposed at the edge to produce it. λa = 1.5: pinch parameter Θ = 0.75, reversal parameter F = 0.688. λa = 3: pinch parameter Θ = 1.50, reversal parameter F = -1.150.

The twist that outlives the turbulence

A plasma pinch driven hard enough goes violently unstable, and then settles into the same quiet state however it was started — with the field at its edge pointing backwards. The explanation is that turbulence destroys almost every constraint a perfect conductor obeys and spares one. The magnetic helicity, a measure of how twisted and linked the field is, decays far more slowly than the energy, and a field that has shed all the energy it can at fixed helicity has only one shape available to it.

astrophysics · Flux freezing

Named alongside it

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

Faraday's lawFlux freezingLenz's lawElectromagnetic inductionVector potentialDissipationEmfField linesMaxwell equationsQuantisationReciprocitySolenoid

All concepts