Illustrated Physics
physics, drawn
Concept

The Lorentz force — where it appears

The force q(E + v×B) on a charge, whose magnetic part is perpendicular to the velocity and can therefore change direction and never energy. That is why a magnetic field bends a particle track without accelerating it, and why the radius of the track measures the momentum directly.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

The field of a current loop, seen edge on. 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.

The 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.

electromagnetism · Magnetism Same field, same charge, three momenta. Electrons entering a 10 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 — 10.7 mm, 21.3 mm, 32.0 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 = 3.57 ns for all three: the faster particle travels a proportionally longer way round and arrives at the same moment.

The 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.

electromagnetism · Magnetism The same wire, seen twice at 0.6c. Above: the wire in the laboratory. The lattice is at rest and the electrons drift, so the electrons are the contracted ones — and the wire is neutral, which means their contracted spacing is what the manufacture of a neutral wire produced. Below: the same wire seen by something moving with the electrons at 0.6c. Now the electrons are at rest and the spacing between them stretches by γ = 1.250, while the lattice moves and its spacing contracts by the same factor. The two densities no longer cancel and the wire is charged. Nothing was done to the wire; the only thing that changed is who is looking, and the magnetic force in the first frame is the electric force in the second.

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.

relativity · Field transformation 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 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 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 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 An orbit that flattens without being told to. The path of a classical electron relative to its nucleus, in the laboratory, before and after the nucleus has been accelerated gently — over 200 orbits — from rest to 0.6 of the speed of light along the horizontal, computed from Maxwell's field of a moving charge and the relativistic law of motion alone. Before, the orbit is a circle of radius 1. After, it is an ellipse 0.810 wide along the motion and 1.012 across it, a ratio of 0.8003 against 1/γ = 0.8000, and one turn takes 1.2504 times as long as before, against γ = 1.2500. Nothing in the calculation assumed a contraction or a slowing clock: the forces between the moving charges produced both.

The contraction the forces work out for themselves

Relativity usually introduces length contraction as a consequence of how moving observers disagree about simultaneity, not as something that happens to matter. There is a second way to reach the same number, and it is worth knowing because it is the way the idea was first found. Take an atom, accelerate its nucleus gently, and compute what the electron does using only Maxwell's field of a moving charge and the law of motion. Nobody tells the orbit to contract, and it does, by exactly 1/γ, while each turn takes exactly γ times as long. The forces holding matter together work out the contraction for themselves.

relativity · Length contraction The armature sits where the field has halved. The magnetic field halfway between two parallel rails carrying a current out along one and back along the other, against the distance along the barrel in units of the rails' spacing, as a fraction of the field between infinitely long rails. The rails end at zero, where the armature closes the circuit. A few spacings behind the armature the field is the full value: 98.5 per cent at two spacings back. At the armature it is exactly half, because each rail there is a wire that stops, and a wire that stops gives half the field of one that does not. Ahead of the armature it falls away: 15 per cent half a spacing in front. The current in the armature crosses this field and is pushed forward; the push is the current times the field it sits in, which is half the field behind it, and that is where the half in ½L′I² comes from.

The push at the end of a pair of rails

A railgun is two metal rails, a sliding conductor across them and a very large current, and the push on the slider is half the inductance per metre times the current squared. The half is not a fudge. The slider sits at the end of two rails that stop, and a wire that stops gives exactly half the field of one that does not. Followed through, the same circuit pushes its own rails apart six times harder per metre than it pushes the projectile, stores as much energy in its field as it gives the projectile, and wastes less the faster it shoots.

electromagnetism · Ampere law Electrons that reach the anode, and electrons that turn back. Paths of electrons leaving a cathode 4 mm across at rest, in a cylindrical diode with an anode 8 mm across held at 4000 V, with no space charge, for axial magnetic fields of 0.80, 0.98, 1.02 and 1.30 times Hull's cut-off field of 142.2 millitesla; computed by integrating the Lorentz force. Below the cut-off the field only bends the path, and the electron still reaches the anode — at 0.98 of the cut-off it arrives almost tangentially. Just above it the electron turns back just short of the anode and returns to the cathode, then repeats the arc further round; at 1.30 times the cut-off it turns back well inside. Nothing reaches the anode, and the current stops completely within a few per cent of the field.

The field at which no electron reaches the anode

Put a hot wire inside a metal tube, make the tube positive, and electrons stream across from wire to tube. Add a magnetic field along the axis and they curve on the way, and still arrive — until, at one definite field, every one of them turns back a hair's breadth short of the tube and the current stops dead. The field at which it happens can be found without knowing anything about how the voltage is distributed between the wire and the tube, because one quantity the electron carries is fixed by the magnetic field alone. The microwave oven runs just inside that line.

electromagnetism · Magnetism Thrown across a uniform field, a fast charge does not draw a parabola. The paths of a charge launched across a uniform electric field (which points up the page) at 0.30, 0.60, 0.90, 0.99 of the speed of light, from the exact relativistic motion (solid), beside the parabolas Newton's law gives for the same launches (dashed); lengths are in units of mc²/qE, the distance over which the field does work equal to the charge's rest energy. Near its launch each exact path is a parabola, but one with the inertia γ₀m in place of m, so it is deflected less than Newton says — 1.05, 1.25, 2.29, 7.09 times less for the four. Further out each turns steeply along the field: one unit across, the 0.3c charge is 11.6 units along where Newton says 5.6, because its sideways speed has fallen while the field kept pushing. Each exact path is a hyperbolic cosine. The field adds to the momentum along it while the sideways momentum stays fixed, and the charge's growing energy is a growing inertia that slows its sideways progress.

The throw a uniform field bends into a chain

Throw a stone sideways across the Earth's gravity and it draws a parabola. Throw a charge sideways across a uniform electric field and, at low speed, it does the same. At high speed it draws a different curve, a hyperbolic cosine — the shape of a chain hanging between two posts, stretched — and the reason is that nothing pushes it sideways yet its sideways speed falls. Its sideways momentum is fixed, the field keeps adding energy, and energy is inertia. The charge gets only logarithmically far across the field however long it flies, and its sideways distance turns out to be a reading of its own clock.

relativity · Relativistic dynamics Loops that swell as the electric field approaches cB. A charge starting from rest in crossed uniform fields — the electric field up the page, the magnetic field out of it — for E/cB = 0.3, 0.6, 0.85, integrated exactly; lengths in units of mc/qB. Each traces a chain of loops drifting to the right at E/B. Newton's law gives loops of height 2mE/qB² — 0.6, 1.2, 1.7 — whatever the speed. The exact loops are 0.66, 1.88, 6.13 high, larger by γ² of the drift speed, 1.10, 1.56, 3.60: in the frame moving at the drift the electric field vanishes and the magnetic field is weakened to B/γ, and the charge circles at the drift speed on a larger orbit than a weaker field would explain alone. As E approaches cB the drift approaches c and the loop grows without limit.

The orbit that opens when E reaches cB

A charge in crossed electric and magnetic fields drifts sideways at E/B, looping as it goes, and the textbook leaves it there. The textbook result has a speed in it that can be made larger than light, and something has to give. What gives is the loop. As the electric field approaches c times the magnetic one, the loops swell without limit, and at E = cB the charge stops looping and is carried off for ever. Above that ratio no frame can make the electric field vanish; one can make the magnetic field vanish instead, and in that frame the charge simply runs away. The transition from a bound orbit to a run-away is decided by a single quantity every observer agrees on, E² − c²B².

relativity · Field transformation

Named alongside it

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

Magnetic fieldMagnetic momentCyclotron motionE cross b driftElectric fieldField linesLength contractionThe Lorentz factorMagnetic monopoleProper timeReference framesAdiabatic invariant

All concepts