# Permeability (electromagnetism)

In electromagnetism, permeability is the measure of the resistance of a material against the formation of a magnetic field, otherwise known as distributed inductance in transmission line theory. Hence, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically represented by the (italicized) Greek letter μ. The term was coined in September 1885 by Oliver Heaviside. The reciprocal of magnetic permeability is magnetic reluctivity.

In SI units, permeability is measured in henries per meter (H/m), or equivalently in newtons per ampere squared (N⋅A−2). The permeability constant μ0, also known as the magnetic constant or the permeability of free space, is a measure of the amount of resistance encountered when forming a magnetic field in a classical vacuum. Until 20 May 2019, the magnetic constant had the exact (defined) value μ0 = 4π × 10−7 H/m ≈ 12.57×10−7 H/m.

On 20 May 2019, a revision to the SI system went into effect, making the vacuum permeability no longer a constant but rather a value that needs to be determined experimentally; 4π × 1.00000000082(20)×10−7 H⋅m−1 is a recently measured value in the new system. It is proportional to the dimensionless fine-structure constant with no other dependencies.

A closely related property of materials is magnetic susceptibility, which is a dimensionless proportionality factor that indicates the degree of magnetization of a material in response to an applied magnetic field.

## Explanation

In electromagnetism, the auxiliary magnetic field H represents how a magnetic field B influences the organization of magnetic dipoles in a given medium, including dipole migration and magnetic dipole reorientation. Its relation to permeability is

$\mathbf {B} =\mu \mathbf {H}$ where the permeability, μ, is a scalar if the medium is isotropic or a second rank tensor for an anisotropic medium.

In general, permeability is not a constant, as it can vary with the position in the medium, the frequency of the applied magnetic field, humidity, temperature, and other parameters. In a nonlinear medium, the permeability can depend on the strength of the magnetic field. Permeability as a function of frequency can take on real or complex values. In ferromagnetic materials, the relationship between B and H exhibits both non-linearity and hysteresis: B is not a single-valued function of H, but depends also on the history of the material. For these materials, it is sometimes useful to consider the incremental permeability defined as

$\Delta \mathbf {B} =\mu _{\Delta }\Delta \mathbf {H} .$ This definition is useful in local linearizations of non-linear material behaviour, for example in a Newton–Raphson iterative solution scheme that computes the changing saturation of a magnetic circuit.

Permeability is the inductance per unit length. In SI units, permeability is measured in henries per metre (H⋅m−1 = J/(A2⋅m) = N⋅A−2). The auxiliary magnetic field H has dimensions current per unit length and is measured in units of amperes per metre (A⋅m−1). The product μH thus has dimensions inductance times current per unit area (H⋅A/m2). But inductance is magnetic flux per unit current, so the product has dimensions magnetic flux per unit area, that is, magnetic flux density. This is the magnetic field B, which is measured in webers (volt-seconds) per square-metre (V⋅s/m2), or teslas (T).

B is related to the Lorentz force on a moving charge q:

$\mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ).$ The charge q is given in coulombs (C), the velocity v in meters per second (m/s), so that the force F is in newtons (N):

$[q\mathbf {v} \times \mathbf {B} ]=\mathrm {C\cdot {\dfrac {m}{s}}\cdot {\dfrac {V\cdot s}{m^{2}}}} =\mathrm {\dfrac {C\cdot (J/C)}{m}} =\mathrm {{\dfrac {J}{m}}=N}$ H is related to the magnetic dipole density. A magnetic dipole is a closed circulation of electric current. The dipole moment has dimensions current times area, units ampere square-metre (A⋅m2), and magnitude equal to the current around the loop times the area of the loop. The H field at a distance from a dipole has magnitude proportional to the dipole moment divided by distance cubed, which has dimensions current per unit length.

## Relative permeability and magnetic susceptibility

Relative permeability, denoted by the symbol $\mu _{\mathrm {r} }$ , is the ratio of the permeability of a specific medium to the permeability of free space μ0:

$\mu _{\mathrm {r} }={\frac {\mu }{\mu _{0}}},$ where $\mu _{0}\approx$ 4π × 10−7 N⋅A−2 is the magnetic permeability of free space. In terms of relative permeability, the magnetic susceptibility is

$\chi _{m}=\mu _{r}-1.$ The number χm is a dimensionless quantity, sometimes called volumetric or bulk susceptibility, to distinguish it from χp (magnetic mass or specific susceptibility) and χM (molar or molar mass susceptibility).

## Diamagnetism

Diamagnetism is the property of an object which causes it to create a magnetic field in opposition of an externally applied magnetic field, thus causing a repulsive effect. Specifically, an external magnetic field alters the orbital velocity of electrons around their nuclei, thus changing the magnetic dipole moment in the direction opposing the external field. Diamagnets are materials with a magnetic permeability less than μ0 (a relative permeability less than 1).

Consequently, diamagnetism is a form of magnetism that a substance exhibits only in the presence of an externally applied magnetic field. It is generally a quite weak effect in most materials, although superconductors exhibit a strong effect.

## Paramagnetism

Paramagnetism is a form of magnetism which occurs only in the presence of an externally applied magnetic field. Paramagnetic materials are attracted to magnetic fields, hence have a relative magnetic permeability greater than one (or, equivalently, a positive magnetic susceptibility).

The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect. Unlike ferromagnets, paramagnets do not retain any magnetization in the absence of an externally applied magnetic field, because thermal motion causes the spins to become randomly oriented without it. Thus the total magnetization will drop to zero when the applied field is removed. Even in the presence of the field, there is only a small induced magnetization because only a small fraction of the spins will be oriented by the field. This fraction is proportional to the field strength and this explains the linear dependency. The attraction experienced by ferromagnets is non-linear and much stronger so that it is easily observed, for instance, in magnets on one's refrigerator.

## Gyromagnetism

For gyromagnetic media (see Faraday rotation) the magnetic permeability response to an alternating electromagnetic field in the microwave frequency domain is treated as a non-diagonal tensor expressed by:

{\begin{aligned}\mathbf {B} (\omega )&={\begin{vmatrix}\mu _{1}&-i\mu _{2}&0\\i\mu _{2}&\mu _{1}&0\\0&0&\mu _{z}\end{vmatrix}}\mathbf {H} (\omega )\end{aligned}} ## Values for some common materials

The following table should be used with caution as the permeability of ferromagnetic materials varies greatly with field strength. For example, 4% Si steel has an initial relative permeability (at or near 0 T) of 2,000 and a maximum of 35,000 and, indeed, the relative permeability of any material at a sufficiently high field strength trends toward 1 (at magnetic saturation).

Magnetic susceptibility and permeability data for selected materials
Medium Susceptibility,
volumetric, SI, χm
Permeability, μ (H/m) Relative permeability, max., μ/μ0 Magnetic field Frequency,
max.
Metglas 2714A (annealed) 1.26×100 1000000 At 0.5 T 100 kHz
Iron (99.95% pure Fe annealed in H) 2.5×10−1 200000
Permalloy 8000 1.25×10−1 100000 At 0.002 T
NANOPERM® 1.0×10−1 80000 At 0.5 T 10 kHz
Mu-metal 6.3×10−2 50000
Mu-metal 2.5×10−2 20000 At 0.002 T
Cobalt-iron (high permeability strip material) 2.3×10−2 18000
Iron (99.8% pure) 6.3×10−3 5000
Electrical steel 5.0×10−3 4000 At 0.002 T
Ferritic stainless steel (annealed) 1.26×10−32.26×10−3 1000 – 1800
Martensitic stainless steel (annealed) 9.42×10−41.19×10−3 750 – 950
Ferrite (manganese zinc) 4.4×10−42.51×10−2 350 – 20 000 At 0.25 mT Approx. 100 Hz – 4 MHz
Ferrite (nickel zinc) 1.26×10−52.89×10−3 10 – 2300 At ≤ 0.25 mT Approx. 1 kHz – 400 MHz
Ferrite

(magnesium manganese zinc)

4.4×10−46.28×10−4 350 - 500 At 0.25 mT Deflection yoke cores for colour TV-sets
Ferrite (cobalt nickel zinc) 5.03×10−51.57×10−4 40 – 125 At 0.001 T Approx. 2 MHz – 150 MHz
Mo-Fe-Ni powder compound

(MPP - molypermalloy powder)

1.76×10−56.91×10−4 14 – 550 Approx. 50 Hz – 3 MHz
Nickel iron powder compound 1.76×10−52.01×10−4 14 – 160 At 0.001 T Approx. 50 Hz – 2 MHz
Al-Si-Fe powder compound (Sendust) 1.76×10−52.01×10−4 14 – 160 Approx. 50 Hz – 5 MHz
Iron powder compound 1.76×10−51.26×10−4 14 – 100 At 0.001 T Approx. 50 Hz – 220 MHz
Silicon iron powder compound 2.39×10−51.13×10−4 19 – 90 Approx. 50 Hz – 40 MHz
Carbonyl iron powder compound 5.03×10−64.4×10−5 4 – 35 At 0.001 T Approx. 20 kHz – 500 MHz
Carbon steel 1.26×10−4 100 At 0.002 T
Nickel 1.26×10−47.54×10−4 100 – 600 At 0.002 T
Martensitic stainless steel (hardened) 5.0×10−51.2×10−4 40 – 95
Austenitic stainless steel 1.260×10−68.8×10−6 1.003 – 1.05[note 1]
Neodymium magnet 1.32×10−6 1.05
Platinum 1.256970×10−6 1.000265
Aluminum 2.22×10−5 1.256665×10−6 1.000022
Wood 1.25663760×10−6 1.00000043
Air 1.25663753×10−6 1.00000037
Concrete (dry) 1
Vacuum 0 4π × 10−7 (μ0) 1, exactly
Hydrogen −2.2×10−9 1.2566371×10−6 1.0000000
Teflon 1.2567×10−6 1.0000
Sapphire −2.1×10−7 1.2566368×10−6 0.99999976
Copper −6.4×10−6 or
−9.2×10−6
1.256629×10−6 0.999994
Water −8.0×10−6 1.256627×10−6 0.999992
Bismuth −1.66×10−4 1.25643×10−6 0.999834
Pyrolytic carbon 1.256×10−6 0.9996
Superconductors −1 0 0

A good magnetic core material must have high permeability.

For passive magnetic levitation a relative permeability below 1 is needed (corresponding to a negative susceptibility).

Permeability varies with a magnetic field. Values shown above are approximate and valid only at the magnetic fields shown. They are given for a zero frequency; in practice, the permeability is generally a function of the frequency. When the frequency is considered, the permeability can be complex, corresponding to the in-phase and out of phase response.

## Complex permeability

A useful tool for dealing with high frequency magnetic effects is the complex permeability. While at low frequencies in a linear material the magnetic field and the auxiliary magnetic field are simply proportional to each other through some scalar permeability, at high frequencies these quantities will react to each other with some lag time. These fields can be written as phasors, such that

$H=H_{0}e^{j\omega t}\qquad B=B_{0}e^{j\left(\omega t-\delta \right)}$ where $\delta$ is the phase delay of $B$ from $H$ .

Understanding permeability as the ratio of the magnetic flux density to the magnetic field, the ratio of the phasors can be written and simplified as

$\mu ={\frac {B}{H}}={\frac {B_{0}e^{j\left(\omega t-\delta \right)}}{H_{0}e^{j\omega t}}}={\frac {B_{0}}{H_{0}}}e^{-j\delta },$ so that the permeability becomes a complex number.

By Euler's formula, the complex permeability can be translated from polar to rectangular form,

$\mu ={\frac {B_{0}}{H_{0}}}\cos(\delta )-j{\frac {B_{0}}{H_{0}}}\sin(\delta )=\mu '-j\mu ''.$ The ratio of the imaginary to the real part of the complex permeability is called the loss tangent,

$\tan(\delta )={\frac {\mu ''}{\mu '}},$ which provides a measure of how much power is lost in material versus how much is stored.