Physical · the question, drilled to the bottom
What Holds a Magnet Together
Ask how a magnet works and you'll be told opposite poles attract. That's true — and it explains nothing, the way "it falls because of gravity" explains nothing. So drill. Three times the floor gives way, and what holds your fridge magnet together turns out to rest on electricity.
Drill 1 / 3A magnet is a crowd that agreed
Every iron atom is already a tiny magnet — its electrons carry spin, and spin is magnetic. A bar of iron contains some 10²³ of them. Whether the bar as a whole is a magnet comes down to one question: do the little magnets point the same way?
Heat fights order; the neighbours' pull for alignment fights heat. Below a sharp temperature the pull wins and patches of atoms lock into step — domains. Above it, heat wins and the directions scramble. Drag the temperature and watch a magnet switch on and off:
T = 1.70 · ordered (a magnet)
critical point: Tc = 2.2692
This is a real Ising model — the simplest honest cartoon of a ferromagnet. Its switch-over isn't a fudge: in two dimensions Lars Onsager solved it exactly in 1944, and the tipping point sits at Tc = 2/ln(1+√2) = 2.2692…, the critical point marked above. Heat past it and the alignment collapses to zero. This happens to real magnets: every ferromagnet has a Curie temperature where it simply stops being a magnet.
| Metal | Curie temperature |
|---|---|
| Iron (Fe) | 1043 K (770 °C) |
| Cobalt (Co) | 1388 K (1115 °C) |
| Nickel (Ni) | 627 K (354 °C) |
| Gadolinium (Gd) | 292 K (19 °C) |
Representative values, Kittel, Introduction to Solid State Physics. Gadolinium's is near room temperature: a cooled piece sticks to a magnet and lets go once your warm hand takes it past 19 °C.
Drill 2 / 3The force that lines them up isn't magnetic
So neighbouring atomic magnets pull each other into alignment. The natural guess is that they pull magnetically — north-seeking each other, the way two compass needles do. Let's check that guess with a number, because it fails badly.
Two atomic dipoles a lattice-spacing apart in iron interact with an energy of about μ₀/4π · m²/r³. Put in iron's real numbers (r = 0.248 nm, moment 2.22 Bohr magnetons) and that energy is 2.8×10⁻²⁴ J — which, as a temperature, is 0.20 K. If magnetism were the glue, iron would scramble above a fifth of a degree above absolute zero. Instead it holds its alignment all the way to 1043 K — about 5,200× stronger than magnetism can account for.
The check — magnetism is far too weak
U_dipole = (μ₀/4π)·(2.22 μ_B)² / (0.248 nm)³ = 2.77×10⁻²⁴ J = 0.201 K
T_Curie / T_dipole = 1043 K / 0.201 K ≈ 5,197×: computed offline by the published check, every constant from CODATA 2018.
The energy that actually does the aligning is of order k_B·T_Curie ≈ 0.09 eV — an electron-volt-scale energy. That is the energy scale of electrostatics, not magnetism.
The real glue is the exchange interaction, and it is a piece of sleight-of-hand by quantum mechanics. Electrons are identical fermions: the Pauli exclusion principle forbids two with the same spin from being in the same place. Same-spin electrons therefore stay further apart — which lowers their electrostatic (Coulomb) repulsion energy. In iron, aligning the spins is the cheaper arrangement, so the spins align.
Read that again, because it's the punchline of the second drill: the force that makes iron magnetic is the electric repulsion between electrons, refereed by a quantum rule about identity. No magnetic force enters the aligning. Magnetism, at the bottom, is held up by electricity.
Drill 3 / 3And the magnetic force itself is electricity
Fine, but a current still pushes a passing charge sideways. That force is magnetic, surely? Here is the strangest true thing on this page. The magnetic force is what the electric force looks like when you're moving. Einstein's 1905 relativity paper was titled On the Electrodynamics of Moving Bodies for exactly this reason.
Take a wire carrying a current: a fixed lattice of positive ions, and electrons drifting through it. The wire is electrically neutral — equal + and −, no net charge, no electric field. Now send a test charge gliding alongside it. In the lab it feels a sideways magnetic force. But ride along with the charge and switch frames:
Lab frame — wire neutral, force is "magnetic"
In the charge's own frame the two rows move at different speeds, so relativistic length contraction packs them by different amounts. The − charges bunch up relative to the + charges (or the reverse) and the once-neutral wire now carries a net charge. A net charge makes an electric field, which pushes the (now stationary) test charge. Same push, different name. Work both stories out with real numbers and they agree to the last decimal place:
The check — same force, two frames
LAB: F_magnetic = q·u·B = 1.07851×10⁻⁴ N (wire neutral, current 119.9 A)
CHARGE: the wire's net density becomes 1.258×10⁻⁷ C/m → an electric force that, transformed back to the lab, is 1.07851×10⁻⁴ N
Equal to machine precision. The hinge that makes the two stories one is the identity μ₀·ε₀ = 1/c² — the magnetic constant and the electric constant are the same constant, wearing the speed of light. (Speeds idealised to 0.40c and 0.30c so the contraction is visible, and the drawing exaggerates it further; the agreement is exact at every speed.)
The bottomWhere the honest answer stops
So: a magnet is a crowd of spins that agreed; they agreed because of electric repulsion and a quantum rule, not magnetism; and the magnetic force a current puts on a moving charge is electricity seen sideways. (A fridge magnet's own field comes mostly from electron spin, which no change of frame turns into electricity.) Keep drilling and you reach questions no analogy reduces — why does an electron have spin and a magnetic moment at all? Why does nature enforce Pauli exclusion? These bottom out in quantum field theory, in mathematics that doesn't translate into anything you've ever held. Richard Feynman, asked the plain question "why do magnets repel," refused to fake the floor:
"I really can't do a good job, any job, of explaining magnetic force in terms of something else you're more familiar with, because I don't understand it in terms of anything else that you're more familiar with."— Richard Feynman, BBC Fun to Imagine, 1983
That isn't a failure. It's the most truthful thing in the whole chain: an explanation that keeps going until it honestly runs out, and says so, instead of dressing the last gap in a word like "poles" and calling it understood. You now know more about a fridge magnet than almost any account online will tell you — and exactly where the knowing ends.
The apparatus — what's proven, assumed, and idealised
Recomputed offline by research/how-magnets-work/verify.mjs (7/7 pass): Onsager's exact Tc = 2.2692; the dipole energy 0.201 K and the 5,197× shortfall vs iron's 1043 K Curie point; the lab/charge-frame force agreement to machine precision.
Idealised, and named as such: the spin grid is the two-dimensional Ising model — real iron is 3D with a different exact-less Tc; the 2D value is what the live grid actually simulates, and it is exact for that model. The relativity demo uses a single drifting electron line at relativistic speed; real drift is millimetres per second, yet the same effect produces the full magnetic force because the charge density is astronomically large. The exchange-interaction account is the standard qualitative one (Heisenberg 1928); the sign of exchange is what makes iron a ferromagnet rather than an antiferromagnet, and that sign is material-specific.
Corrections.
Corrected 2026-10-18: The “Ride with the charge” readout said the force in the charge’s frame was the same 1.07851×10⁻⁴ N as in the lab. In the charge’s frame it is 1.13058×10⁻⁴ N, which is 1.07851×10⁻⁴ N once transformed back to the lab, as this page’s check prints; found by the assay line, 2026-10-18.