How can one piece of metal grab another — without touching it?
Hold two magnets close. Feel them snap together — or push apart — before they even touch. Something is reaching across empty space. That something is the magnetic field.
Humans have used lodestones (naturally magnetic rocks) for navigation for over 2,500 years. But we only understood what magnetism actually is in the 20th century, when quantum mechanics revealed it emerges from the spin of electrons deep inside atoms.
Magnetic field lines around a bar magnet. They emerge from N (red) and re-enter at S (green) — always forming closed loops.
Those curved lines aren't drawn on anything. They're a map of direction: at every point in space, the line shows where a compass needle would point. Pack the lines closer together, and the field is stronger there.
Iron, nickel, cobalt — and their alloys like steel — are ferromagnetic. A plastic spoon won't stick to your fridge; an iron nail will. The difference is buried in atomic structure. We'll get there in Section 3.
Iron, Nickel, Cobalt, Gadolinium. Atomic magnetic moments cooperate, lock into "domains," and stay aligned permanently.
Aluminium, platinum, liquid oxygen. Weakly attracted while in an external field — but lose all magnetism the moment you remove it.
Bismuth, graphite, water, frogs. Weakly repelled by strong fields. A live frog was levitated in 1997 using 16 T in a Bitter magnet.
Manganese oxide, chromium. Adjacent atoms anti-align — every atom is magnetic, but neighbors cancel each other exactly.
Drag the poles. See the topology of the field.
Michael Faraday invented field lines in the 1840s — not because he had equations, but because he needed a picture. He had no formal mathematics training. Yet his insight — that electricity and magnetism could be understood as fields pervading space — became the conceptual foundation for Maxwell's equations and then Einstein's general relativity.
Drag N (red) and S (green) poles independently. Try placing them on top of each other — or switching to two N poles.
Field lines are mathematical constructs that encode real physics:
This stands in stark contrast to electric fields, where ∇·E = ρ/ε₀: field lines start and end on charges. But magnetic field lines have no source point. They loop forever. Despite decades of searching in cosmic ray detectors, ancient rocks, and particle accelerators, no magnetic monopole has ever been detected.
A bar magnet is a magnetic dipole. Far from it, its field drops much faster than gravity (1/r²) or a point charge (1/r²). This is why a magnet that lifts a paper clip from 1 cm barely affects it from 5 cm away.
Magnetism lives inside the electron — in two distinct ways.
Zoom into any iron atom. Inside every atom, electrons do two things that each generate magnetism: they orbit the nucleus (a tiny current loop), and they possess an intrinsic "spin" — a quantum property with no classical analog, but one that acts exactly like a tiny bar magnet.
Electron orbiting the nucleus (blue arrow = orbital moment μ_L) and carrying spin (red arrow = spin moment μ_S). Both contribute to the atom's total magnetic moment.
A moving charge is a current. An orbiting electron is a tiny current loop. Any current loop creates a magnetic dipole moment proportional to its angular momentum:
Spin is not the electron physically rotating — that picture breaks down (an electron would have to spin faster than light to produce its observed magnetic moment). It's an intrinsically quantum mechanical property. Yet it couples to magnetic fields exactly as a classical bar magnet would:
In an unmagnetized iron bar, atoms are grouped into domains — regions where spins are aligned. But different domains point in different directions. They cancel out. Net magnetism: zero.
Apply an external field: domains aligned with the field grow at the expense of others. Eventually most moments point the same way. The bar is magnetized.
Magnetic domains. Each arrow = net spin direction of a region. "Apply Field" watches them align. "Heat" randomizes them above the Curie temperature.
Pauli's exclusion principle forces electrons into pairs with opposite spins in each orbital. Paired electrons cancel. An atom is only magnetic if it has unpaired electrons — found in partially filled d or f shells.
Iron (Fe, Z=26) has the electron configuration [Ar] 3d⁶ 4s². By Hund's rules, the six 3d electrons distribute as 5 spin-up + 1 spin-down, giving 4 unpaired electrons and a large magnetic moment of ~2.2 Bohr magnetons per atom. This is why iron, not copper or silver, is the archetypal magnet.
Moving charges create magnetic fields. Changing fields create currents. They are inseparable.
Before the 1820s, electricity and magnetism were thought to be completely separate. Then Hans Christian Ørsted noticed something odd: when he ran current through a wire near a compass during a lecture, the needle swung. He had discovered that every moving charge creates a magnetic field.
Solenoid cross-section. Each circle = one loop of wire, with current into (X) or out of (dot) the page. Their combined field is nearly uniform inside and weak outside.
For any current-carrying wire, the field contribution from an infinitesimal segment is:
Change the magnetic flux through a loop, and a voltage appears. Every generator, transformer, wireless charger, and guitar pickup runs on this:
A charge moving through a magnetic field feels a force perpendicular to its velocity — so it curves but doesn't speed up. In a uniform field, it moves in a perfect helix. This is why:
Maxwell, Einstein, Dirac — three revolutions that unified it all.
In 1865, James Clerk Maxwell combined Faraday's insights with Ampere's law and added one critical term (the displacement current ε₀ ∂E/∂t). The result was four equations that completely describe classical electromagnetism — and predicted the existence of electromagnetic waves traveling at speed c, which Hertz confirmed experimentally 22 years later.
Take the curl of III, substitute IV, and the wave equation drops out:
This may be the most surprising fact about magnetism: it is not a separate force from electricity. It is what electricity looks like to a moving observer.
Consider a wire with current (electrons drifting left). A positive charge at rest nearby feels no force — the wire is electrically neutral. Now set that charge in motion to the right. Length contraction (special relativity) causes the negative charges in the wire to appear Lorentz-contracted — denser — so the wire looks negatively charged. The charge feels attraction. That "electric force" in the moving frame is what we call "magnetic force" in the lab frame.
In relativity, E and B are not separate. They are components of a single antisymmetric rank-2 tensor, the electromagnetic field tensor:
In 1928, Paul Dirac sought a relativistic quantum equation for the electron. He didn't put spin in by hand. It emerged automatically from the mathematics of combining quantum mechanics and special relativity. And so did the electron's magnetic moment, with g = 2 exactly.
Dirac's prediction g = 2.000... was later refined by quantum electrodynamics. Julian Schwinger computed the first correction in 1948:
In condensed matter, magnetic fields and quantum mechanics together produce phases of matter characterized not by symmetry but by topology — mathematical properties that survive continuous deformation:
2D electrons in strong B form Landau levels. Hall conductance is quantized at σ = νe²/h — a topological invariant (the Chern number). Won the 1985 Nobel Prize.
Topological "knots" in spin textures. Like a knot in a rope, they can't be smoothly untied. Proposed as ultra-dense, low-power magnetic memory bits.
Materials with band crossings that act like magnetic monopoles in momentum space — not real space. The chiral magnetic anomaly is macroscopically observable.
A quantum phase acquired when a spin traces a loop in parameter space. Equivalent to a magnetic field in momentum space (the Berry curvature). Foundation for modern topological band theory.