The end of the winding is where the field is half
At its defaults it draws the end of the winding is where the field is half. Field on the axis of a solenoid, in units of the infinite-solenoid value μ₀nI, with distance measured in half-lengths so that ±1 is the mouth of the winding whatever its length. At 1:1 the centre reaches 0.7075 of μ₀nI and the end plane 0.4472, a ratio of 0.6322; At 2.5:1 the centre reaches 0.9285 of μ₀nI and the end plane 0.4903, a ratio of 0.5280; At 5:1 the centre reaches 0.9806 of μ₀nI and the end plane 0.4975, a ratio of 0.5074; At 10:1 the centre reaches 0.9950 of μ₀nI and the end plane 0.4994, a ratio of 0.5019. Every point is a sum of Biot–Savart contributions from each of the turns, and the centre value agrees with the closed form for a continuous winding to 0.049 per cent. The half is exact only in the limit: a winding as long as it is wide gives 0.63, and the familiar rule is a statement about a solenoid nobody has.
solenoid-field is one function in lib/figures/fields.js —
charge, current, flux and the lines drawn between them. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
Field on the axis of a solenoid, in units of the infinite-solenoid value μ₀nI, with distance measured in half-lengths so that ±1 is the mouth of the winding whatever its length. At 1:1 the centre reaches 0.7075 of μ₀nI and the end plane 0.4472, a ratio of 0.6322; At 2.5:1 the centre reaches 0.9285 of μ₀nI and the end plane 0.4903, a ratio of 0.5280; At 5:1 the centre reaches 0.9806 of μ₀nI and the end plane 0.4975, a ratio of 0.5074; At 10:1 the centre reaches 0.9950 of μ₀nI and the end plane 0.4994, a ratio of 0.5019. Every point is a sum of Biot–Savart contributions from each of the turns, and the centre value agrees with the closed form for a continuous winding to 0.049 per cent. The half is exact only in the limit: a winding as long as it is wide gives 0.63, and the familiar rule is a statement about a solenoid nobody has.
Same field, same charge, three momenta
The options are the ones Magnetism is electricity seen sideways passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Electrons entering a 20 mT field at right angles to it, at 1, 4 keV, each drawn for a quarter of its turn. The radius is mv/qB — 5.3 mm, 10.7 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 1.79 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment.
The frequency that stops being constant
The options are the ones Magnetism is electricity seen sideways passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
The cyclotron frequency divided by its low-speed value, against kinetic energy measured in units of the particle's own rest energy. Classically the ratio is one everywhere — the period is 2πm/qB and contains no speed — which is the whole basis of a cyclotron. Relativistically it is 1/γ, and γ is one plus that ratio, so the fall is the same curve for every particle. It is 1% low at 0.0101 of the rest energy, which is 5.2 keV for an electron and 9.5 MeV for a proton — a factor of 1836, and the reason cyclotrons accelerate protons while electrons need a machine that changes its own frequency.
Same field, same charge, three momenta
The options are the ones Nothing can be held still by a static field passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Electrons entering a 50 mT field at right angles to it, at 1, 4, 9 keV, each drawn for a quarter of its turn. The radius is mv/qB — 2.1 mm, 4.3 mm, 6.4 mm — so measuring the curvature of a track measures the momentum of whatever made it, which is how every particle detector since the cloud chamber has worked. The time to go once round is 2πm/qB = 0.71 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment.
The field of a solenoid, seen edge on
The options are the ones The coupling that is the same both ways passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Magnetic field lines integrated from the Biot–Savart law for the current shown. Every line closes on itself: there is nowhere for one to start and nowhere for it to end, because no magnetic charge exists to end on.
Circles that go nowhere, and a current across the line
The options are the ones The current no particle carries passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
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.
What checks it
physicscheck asserts something about solenoid-field that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Magnetism is electricity seen sideways
The force on a charge moving beside a current-carrying wire is magnetic in the laboratory and purely electrostatic in the charge's own frame. Both calculations give the same answer, and the drift speed that makes them agree corresponds to a Lorentz factor differing from one in the twenty-sixth decimal place.
ElectromagnetismNothing can be held still by a static field
However many charges are arranged, however cleverly, a charge placed among them has somewhere to fall. The reason is one line of arithmetic — the potential in empty space satisfies Laplace's equation, and a solution of that equation has no interior maximum or minimum — and the consequence is that every real trap for a charged particle works by breaking one of the assumptions rather than by being cleverer.
ElectromagnetismThe 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.
ElectromagnetismThe 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.
ElectromagnetismThe 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.
ElectromagnetismThe 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.
ElectromagnetismThe field that wraps a current
Ampère's law says that going once round a closed path and adding up the field counts the current threaded through it, and nothing else about the path survives into the answer. Four different loops round one wire return the same number to five decimal places — which is exactly why the law is both easy and treacherous.
ElectromagnetismThe field with no ends, and the force that does no work
Magnetic field lines never start and never stop. That single absence is a law, it has survived every attempt to break it, and it makes the magnetic field a different kind of object from the electric one.
ElectromagnetismThe force that does no work
A magnetic field can turn a moving charge through any angle at all and cannot add a joule to it. Everything a magnet is good for follows from that one prohibition — including the fact that a bent track is a reading of momentum, and that a machine built on it stops working at five kilovolts for an electron.
ElectromagnetismThe 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.
QuantumThe 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.