Permeability (electromagnetism)

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In electromagnetism, permeability is the degree of magnetization of a material that responds linearly to an applied magnetic field. Magnetic permeability is represented by the Greek letter μ. This term was coined in September, 1885 by Oliver Heaviside.

In SI units, permeability is measured in henries per metre, or newtons per ampere squared. The constant value <math> \mu_0 </math> is known as the magnetic constant or the permeability of vacuum, and has the exact value 4π×10⁻⁷ N·A⁻².

Some materials, called ferromagnetic or ferromagnets, are highly magnetic by nature, relative to most materials. They are composed of a large number of very small magnetic units working together called domains. Domains are not always aligned, and they often act against each other to reduce the strength of the net magnetic field.

If one puts the ferromagnetic material into an externally applied magnetic field, the domains tend to line up, so that the sum of the fields from the ferromagnet and the resulting magnetic field is higher in magnitude than the applied magnetic field alone.

Permeability in linear materials owes its existence to the approximation:

<math> \mathbf{M}=\chi_m\mathbf{H}</math>

where <math>\chi_m \,</math> is a dimensionless scalar called the magnetic susceptibility.

According to the definition of the auxiliary field, H

<math>\mathbf{B}=\mu_0 (\mathbf{H}+\mathbf{M})=\mu_0 (1 + \chi_m)\mathbf{H} = \mu \mathbf{H}</math>

where

μ is the material's permeability, measured in henries per metre

B is the magnetic flux density (also called the magnetic induction) in the material, measured in teslas

H is the auxiliary field strength, measured in amperes per metre

M is the material's magnetization, measured in amperes per metre

The permittivity of free space (the electric constant) and the magnetic constant are related to the speed of light (c) by the formula: <math>\varepsilon_0\mu_0 = \frac{1}{c^2}</math>

Relative permeability, sometimes denoted by the symbol μ_r, is the ratio of the permeability of a specific medium to the permeability of free space μ₀:

In terms of relative permeability, the magnetic susceptibility is:

χ_m, a dimensionless quantity, is sometimes called volumetric or bulk susceptibility, to distinguish it from χ_p (magnetic mass or specific susceptibility) and χ_M (molar or molar mass susceptibility).

Magnetic permeability & susceptibility for selected materials
Medium	Susceptibility	Permeability
Mumetal	20,000 <ref name="hyper">"Relative Permeability", Hyperphysics</ref>	25,000 µN/A²	at 0.002 T
Permalloy	8000 <ref name="hyper">"Relative Permeability", Hyperphysics</ref>	10,000 µN/A²	at 0.002 T
Transformer iron with ρ=0.01 µΩ·m	4000 <ref name="hyper">"Relative Permeability", Hyperphysics</ref>	5000 µN/A²	at 0.002 T
Steel	700 <ref name="hyper" />	875 µN/A²	at 0.002 T
Nickel	100 <ref name="hyper" />	125 µN/A²	at 0.002 T
soft ferrite with ρ=0.1 Ωm	source, ferroxcube	5000 µN/A²	< 0.1 mT
soft ferrite with ρ=10 Ωm	source, ferroxcube	2500 µN/A²	< 0.1 mT
Platinum	2.65 × 10⁻⁴	1.2569701 µN/A²
Aluminum	2.22 × 10⁻⁵ <ref name="clarke">Clarke, R. Magnetic properties of materials, surrey.ac.uk</ref>	1.2566650 µN/A²
Hydrogen	8 × 10⁻⁹ or 2.2 × 10⁻⁹ <ref name="clarke" />	1.2566371 µN/A²
Vacuum	0	1.2566371 µN/A²
Sapphire	−2.1 × 10⁻⁷	1.2566368 µN/A²
Copper	−6.4 × 10⁻⁶ or −9.2 × 10⁻⁶ <ref name="clarke" />	1.2566290 µN/A²
Water	−8.0 × 10⁻⁶	1.2566270 µN/A²