Illustrated Physics
physics, drawn
Concept

Electromagnetic field — where it appears

The electric and magnetic fields taken together as one object, split into its two parts differently by observers in different motion. A field that is purely electric in one frame has a magnetic part in another, while two combinations of the two parts take the same value in every frame.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as field invariants — the same set of essays touches all of them, so they are one junction rather than several.

A charge going past, as the field it delivers. The electric field across the line of motion at a point a distance b from the path of a passing charge, against time in units of b/c, on a logarithmic field axis in units of q/4πε₀b², for Lorentz factors of 1.2, 3, 10, 30. The peak is γ times the Coulomb field at rest and the width at half height is 2.311 b/c at γ = 1.2, 0.542 b/c at γ = 3, 0.154 b/c at γ = 10, 0.0511 b/c at γ = 30: taller by γ and shorter by γ, so the area under each curve — the sideways impulse the passing charge delivers, 2q²/4πε₀bv — is the same for all of them to within the integration's precision. A fast charge's field arrives as a thin, intense flash.

The charge that passes as a flash of light

A charge at rest has a field that no change of frame can turn into light. A charge going past at nearly the speed of light delivers a field that is, for every practical purpose, a pulse of light anyway: thin, perpendicular to the motion, with its electric and magnetic parts equal and crossed. Enrico Fermi saw in 1924 that such a field could be counted as photons, and the Large Hadron Collider now collides lead nuclei's photons with each other.

relativity · Field transformation 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.

Field invariantsThe Lorentz factorThe Lorentz transformationCyclotron motionE cross b driftEquivalent photonsFourier transformThe Lorentz forcePhotonProper timeStopping power

All concepts