Magnetic permeability

Magnetic permeability μ is the constant of proportionality between the B-field and the H-field inside a linear magnetic material: B = μH. In vacuum, μ equals the vacuum permeability μ₀ = 4π × 10⁻⁷ T·m/A (exact until the 2019 SI redefinition, now a measured value close to that). Inside matter, μ is larger or smaller depending on whether the material's magnetization adds to or subtracts from the applied field: μ = μ₀(1 + χ) = μ₀ μ_r.

The relative permeability μ_r is a dimensionless number that's often more convenient to work with. Most non-magnetic materials have μ_r ≈ 1 to six decimal places — air, glass, copper, water are all effectively vacuum from the magnetic point of view. Paramagnets like aluminium have μ_r a tiny bit above 1, diamagnets like bismuth a tiny bit below. Ferromagnets are the outliers: soft iron can reach μ_r ≈ 5,000, mu-metal ≈ 50,000, supermalloy ≈ 800,000. These are the materials engineers use to channel magnetic flux — inside a transformer core or a motor stator, they behave like perfect conductors of the B-field, squeezing flux lines into whatever shape the iron takes.

Permeability is only a useful concept in the linear regime. Ferromagnets are nonlinear: the M-vs-H curve saturates, and the B-vs-H curve shows hysteresis. In this regime μ is not a single number; it's either a differential permeability dB/dH that varies along the loop, or an effective permeability defined at one particular operating point. Manufacturers of magnetic materials quote both, plus the saturation B_sat and the coercive H_c, to characterise the material completely.