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The magnetization $\mathbf{M}$ is magnetic dipole moment per unit volume. In magnetostatics it producesThe magnetic field intensity $\mathbf{H}=\mathbf{B}/{\mu }_0-\mathbf{M}$ obeysso $\mathbf{H}$ isolates free currents while $\mathbf{M}$ accounts for bound currents. For a linear isotropic material, $\mathbf{B}=\mu \mathbf{H}$.

Ampere's law for a circular loop around the free line current givesin both media. ThereforeAt $r=R$, the net bound surface-current density isHence its magnitude isand for a positive current along $+{\mathbf{e}}_z$ it points along $-{\mathbf{e}}_z$ when ${\mu }_1>{\mu }_2$, reversing when ${\mu }_1<{\mu }_2$.

For the partition function $Z={\sum }_{\{S\}}{e}^{-\beta E}$, differentiation with respect to the external field givesTherefore the magnetization per site is

A pure thermodynamic phase is an extremal equilibrium Gibbs state rather than a probabilistic mixture of distinct macroscopic phases. On an ordered Ising branch, taking the thermodynamic limit before removing a selecting conjugate field gives a nonzero magnetization. A symmetric mixture can instead have zero mean but a nondecaying two-point contribution. The connected correlation function and correlation length describing ordinary critical fluctuations should be evaluated in a chosen pure phase when this distinction matters.

The spin magnetization is the average signed spin of a discrete spin system. For $S_i\in \{-1,1\}$, $k$ positive spins give $M=-1+2k/n$. Multiplying by the magnetic moment per spin and the spin density relates this dimensionless average to physical magnetization.

The staggered magnetization weights the local magnetization with opposite signs on the two sublattices of a bipartite graph. It is nonzero in a Néel state even when the uniform magnetization vanishes.