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Wikipedia: permeability (electromagnetism)

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. The 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 $μ 0$ is known as the magnetic constant or the permeability of vacuum, and has the exact or defined value $μ 0$ = 4π×10⁻⁷ N·A⁻².

Ferromagnets

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

Permeability in linear materials owes its existence to the approximation:

$\mathbf{M}=\chi_m\mathbf{H}$

where $\chi_m \,$ is a dimensionless scalar called the magnetic susceptibility.

According to the definition of the auxiliary field, H

$\mathbf{B}=\mu_0 (\mathbf{H}+\mathbf{M})=\mu_0 (1 + \chi_m)\mathbf{H} = \mu \mathbf{H}$

where

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

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

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

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

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

Relative permeability

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:

$\mu_{r} = \frac{\mu}{\mu_{0}}$

In terms of relative permeability, the magnetic susceptibility is:

$\chi_m = \mu_r - 1 \,$

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 x10^-6
Mu-metal	20,000 ^[1]	25,000 N/A²	at 0.002 T
Permalloy	8000 ^[1]	10,000 N/A²	at 0.002 T
Transformer iron with ρ=0.01 µΩ·m	4000 ^[1]	5000 N/A²	at 0.002 T
Steel	700 ^[1]	875 N/A²	at 0.002 T
Nickel	100 ^[1]	125 N/A²	at 0.002 T
soft ferrite with ρ=0.1 Ωm	source, ferroxcube	5000 N/A²	< 0.1 mT
soft ferrite with ρ=10 Ωm	source, ferroxcube	2500 N/A²	< 0.1 mT
Platinum	2.65 × 10⁻⁴	1.2569701 N/A²
Aluminum	2.22 × 10⁻⁵ ^[2]	1.2566650 N/A²
Hydrogen	8 × 10⁻⁹ or 2.2 × 10⁻⁹ ^[2]	1.2566371 N/A²
Vacuum	0	1.2566371 N/A²
Sapphire	−2.1 × 10⁻⁷	1.2566368 N/A²
Copper	−6.4 × 10⁻⁶ or −9.2 × 10⁻⁶ ^[2]	1.2566290 N/A²
Water	−8.0 × 10⁻⁶	1.2566270 N/A²

Permeability varies with magnetic field. Values shown above are approximate and valid only at the magnetic fields shown. Moreover, they are given for a zero frequency; in practice, the permeability is generally a function of the frequency.

Note that the magnetic permeability $μ 0$ has an exact value in SI units (i.e. there is no error bar or uncertainty in its value), a fact that is intimately related to the next formula:
$\varepsilon_0\mu_0 = \frac{1}{c^2}$ , and that the definition that the speed of light is exactly 299,792,458 meters/second. The agreed upon international definitions and best determinations of the values of the fundamental physical constants in SI are given by the CODATA database supported on the web by NIST

magnetic permeability

Ferromagnets

Relative permeability

References

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