Magnetism Explained: From Spinning Electrons to Field Lines

Atomic origins: spin and orbital moment

Every electron carries two sources of magnetic moment. Its orbital motion around the nucleus behaves like a tiny current loop, giving μ_L = -e/(2m_e)·L. Its intrinsic spin — with no classical analogue at all — contributes μ_S = -g_S·e/(2m_e)·S, where the Landé g-factor g_S ≈ 2.002. Hund's rules determine how electrons fill an atom's shells to maximise total spin first, then orbital momentum. Iron ([Ar] 3d⁶4s²) ends up with four unpaired spins and a moment of 4 Bohr magnetons per atom — cobalt has 3, nickel 0.6, and gadolinium the highest of any element at 7.

Dia-, para- and ferromagnetism

Every material is weakly diamagnetic — an applied field induces orbital currents that oppose it (Lenz's law), giving a tiny negative susceptibility. Materials with unpaired electrons but no cooperative ordering are paramagnetic: moments align partially with the field, and susceptibility falls with temperature as Curie's law χ = C/T. In ferromagnets — iron, cobalt, nickel — a strong exchange interaction between neighbouring spins causes spontaneous parallel alignment even with no applied field, below a critical Curie temperature (1044 K for iron).

Exchange interaction: why spins line up

The Heisenberg exchange Hamiltonian, H = −2J·Σ S_i·S_j, sets the sign of J from the interplay of the Pauli exclusion principle and Coulomb repulsion: electrons with parallel spins must occupy different spatial orbitals, keeping them farther apart and lowering their Coulomb energy. When that effect wins, J > 0 and parallel alignment is favoured — ferromagnetism. Iron, cobalt and nickel all satisfy the Stoner criterion for band ferromagnetism, I(E_F)·D(E_F) > 1, thanks to a high density of states at the Fermi level in the 3d transition metals.

Domains, hysteresis and hard versus soft magnets

A ferromagnet below its Curie temperature still splits into magnetic domains — regions of uniform magnetisation pointing different directions — because this closes flux internally and lowers total magnetostatic energy. Domains are separated by Bloch walls where magnetisation rotates 180° over roughly 10-100 nm. Under an applied field, favourably oriented domains grow while walls pin at grain boundaries and defects, producing irreversible hysteresis. Soft magnets (electrical steel, permalloy) have low coercivity and a small hysteresis loop, ideal for transformer cores that must switch cheaply. Hard magnets (AlNiCo, SmCo₅, and especially Nd₂Fe₁₄B with an energy product up to 520 kJ/m³) resist demagnetisation and hold their field permanently — the basis of EV motors, headphones and hard-disk read heads.

Frequently asked questions

Try it live

Everything above runs in your browser — open Magnetic Field Lines, place north and south poles on the canvas, and watch field lines, particle streams and heat maps form in real time.