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Permeability (electromagnetism)

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Ability of magnetization

This article is about the magnetic constant. For the analogous electric constant, seepermittivity.

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Inelectromagnetism,permeability is the measure ofmagnetization produced in a material in response to an appliedmagnetic field. Permeability is typically represented by the (italicized) Greek letterμ. It is the ratio of themagnetic induction $B {\displaystyle B}$ to the magnetizing field $H {\displaystyle H}$ in a material. The term was coined byWilliam Thomson, 1st Baron Kelvin in 1872,^[1] and used alongsidepermittivity byOliver Heaviside in 1885. The reciprocal of permeability ismagnetic reluctivity.

InSI units, permeability is measured inhenries permeter (H/m), or equivalently innewtons perampere squared (N/A²). The permeability constantμ₀, also known as themagnetic constant or the permeability of free space, is the proportionality between magnetic induction and magnetizing force when forming a magnetic field in a classicalvacuum.

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

Explanation

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In themacroscopic formulation of electromagnetism, there appear two different kinds ofmagnetic field:

themagnetizing fieldH which is generated around electric currents anddisplacement currents, and alsoemanates from the poles of magnets. The SI units ofH areamperes per meter.
themagnetic flux densityB which acts back on the electrical domain, bycurving the motion of charges and causingelectromagnetic induction. The SI units ofB arevolt-seconds persquare meter, a ratio equivalent to onetesla.

The concept of permeability arises since in many materials (and in vacuum), there is a simple relationship betweenH andB at any location or time, in that the two fields are precisely proportional to each other:^[2]

\mathbf {B} =\mu \mathbf {H} ,

where the proportionality factorμ is the permeability, which depends on the material. Thepermeability of vacuum (also known as permeability of free space) is a physical constant, denotedμ₀. The SI units ofμ are volt-seconds per ampere-meter, equivalentlyhenry per meter. Typicallyμ would be a scalar, but for an anisotropic material,μ could be a second ranktensor.

However, inside strong magnetic materials (such as iron, orpermanent magnets), there is typically no simple relationship betweenH andB. The concept of permeability is then nonsensical or at least only applicable to special cases such as unsaturatedmagnetic cores. Not only do these materials have nonlinear magnetic behaviour, but often there is significantmagnetic hysteresis, so there is not even a single-valued functional relationship betweenB andH. However, considering starting at a given value ofB andH and slightly changing the fields, it is still possible to define anincremental permeability as:^[2]

\Delta \mathbf {B} =\mu \,\Delta \mathbf {H} .

assumingB andH are parallel.

In themicroscopic formulation of electromagnetism, where there is no concept of anH field, the vacuum permeabilityμ₀ appears directly (in the SI Maxwell's equations) as a factor that relates total electric currents and time-varying electric fields to theB field they generate. In order to represent the magnetic response of a linear material with permeabilityμ, this instead appears as amagnetizationM that arises in response to theB field: $\mathbf {M} =\left(\mu _{0}^{-1}-\mu ^{-1}\right)\mathbf {B}$ . The magnetization in turn is a contribution to the total electric current—themagnetization current.

Relative permeability and magnetic susceptibility

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Relative permeability, denoted by the symbol $\mu _{\mathrm {r} }$ , is the ratio of the permeability of a specific medium to the permeability of free spaceμ₀:

\mu _{\mathrm {r} }={\frac {\mu }{\mu _{0}}},

where $\mu _{0}\approx$ 4π × 10⁻⁷ H/m is themagnetic permeability of free space.^[3] In terms of relative permeability, themagnetic susceptibility is

\chi _{m}=\mu _{r}-1.

The numberχ_m is adimensionless quantity, sometimes calledvolumetric orbulk susceptibility, to distinguish it fromχ_p (magnetic mass orspecific susceptibility) andχ_M (molar ormolar mass susceptibility).

Diamagnetism

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Main article:Diamagnetism

Diamagnetism is the property of an object which causes it to create amagnetic 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 atom's nuclei, thus changing themagnetic dipole moment in the direction opposing the external field. Diamagnets are materials with amagnetic permeability less thanμ₀ (a relative permeability less than 1).

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

Paramagnetism

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Main article:Paramagnetism

Paramagnetism is a form ofmagnetism 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 thanone (or, equivalently, a positivemagnetic susceptibility).

The magnetic moment induced by the applied field islinear in the field strength, and it is ratherweak. It typically requires a sensitive analytical balance to detect the effect. Unlikeferromagnets, paramagnets do not retain any magnetization in the absence of an externally applied magnetic field, becausethermal motion causes the spins to becomerandomly 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 smallinduced 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

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For gyromagnetic media (seeFaraday rotation) the magnetic permeability response to an alternating electromagnetic field in the microwave frequency domain is treated as a non-diagonal tensor expressed by:^[4]

{\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

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The following table should be used with caution as the permeability of ferromagnetic materials varies greatly with field strength and specific composition and fabrication. For example, 4% electrical steel has an initial relative permeability (at or near 0 T) of 2,000 and a maximum of 38,000 at T = 1^[5]^[6] and different range of values at different percent of Si and manufacturing process, 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,χ_m	Relative permeability, max.,μ/μ₀	Permeability, μ (H/m)	Magnetic field	Frequency,max.
Vacuum	0	1, exactly^[7]	1.256637061×10⁻⁶
Metglas 2714A (annealed)		1000000^[8]	1.26×10⁰	At 0.5 T	100 kHz
Iron (99.95% pure Fe annealed in H)		200000^[9]	2.5×10⁻¹
Permalloy		100000^[10]	1.25×10⁻¹	At 0.002 T
NANOPERM®		80000^[11]	1.0×10⁻¹	At 0.5 T	10 kHz
Mu-metal		50000^[12]	6.3×10⁻²
Mu-metal		20000^[13]	2.5×10⁻²	At 0.002 T
Cobalt-iron (high permeability strip material)		18000^[14]	2.3×10⁻²
Iron (99.8% pure)		5000^[9]	6.3×10⁻³
Electrical steel		2000 – 38000^[5]^[15]^[16]	5.0×10⁻³	At 0.002 T, 1 T
Ferritic stainless steel (annealed)		1000 – 1800^[17]	1.26×10⁻³ –2.26×10⁻³
Martensitic stainless steel (annealed)		750 – 950^[17]	9.42×10⁻⁴ –1.19×10⁻³
Ferrite (manganese zinc)		350 – 20 000^[18]	4.4×10⁻⁴ –2.51×10⁻²	At 0.25 mT	approx. 100 Hz – 4 MHz
Ferrite (nickel zinc)		10 – 2300^[19]	1.26×10⁻⁵ –2.89×10⁻³	At ≤ 0.25 mT	approx. 1 kHz – 400 MHz^[citation needed]
Ferrite (magnesium manganese zinc)		350 – 500^[20]	4.4×10⁻⁴ –6.28×10⁻⁴	At 0.25 mT
Ferrite (cobalt nickel zinc)		40 – 125^[21]	5.03×10⁻⁵ –1.57×10⁻⁴	At 0.001 T	approx. 2 MHz – 150 MHz
Mo-Fe-Ni powder compound (molypermalloy powder, MPP)		14 – 550^[22]	1.76×10⁻⁵ –6.91×10⁻⁴		approx. 50 Hz – 3 MHz
Nickel iron powder compound		14 – 160^[23]	1.76×10⁻⁵ –2.01×10⁻⁴	At 0.001 T	approx. 50 Hz – 2 MHz
Al-Si-Fe powder compound (Sendust)		14 – 160^[24]	1.76×10⁻⁵ –2.01×10⁻⁴		approx. 50 Hz – 5 MHz [25]
Iron powder compound		14 – 100^[26]	1.76×10⁻⁵ –1.26×10⁻⁴	At 0.001 T	approx. 50 Hz – 220 MHz
Silicon iron powder compound		19 – 90^[27]^[28]	2.39×10⁻⁵ –1.13×10⁻⁴		approx. 50 Hz – 40 MHz
Carbonyl iron powder compound		4 – 35^[29]	5.03×10⁻⁶ –4.4×10⁻⁵	At 0.001 T	approx. 20 kHz – 500 MHz
Carbon steel		100^[13]	1.26×10⁻⁴	At 0.002 T
Nickel		100^[13] – 600	1.26×10⁻⁴ –7.54×10⁻⁴	At 0.002 T
Martensitic stainless steel (hardened)		40 – 95^[17]	5.0×10⁻⁵ –1.2×10⁻⁴
Austenitic stainless steel		1.003 – 1.05^[17]^[30]^[a]	1.260×10⁻⁶ –8.8×10⁻⁶
Neodymium magnet		1.05^[31]	1.32×10⁻⁶
Platinum		1.000265	1.256970×10⁻⁶
Aluminum	2.22×10⁻⁵^[32]	1.000022	1.256665×10⁻⁶
Wood		1.00000043^[32]	1.25663760×10⁻⁶
Air		1.00000037^[33]	1.25663753×10⁻⁶
Concrete (dry)		1^[34]
Hydrogen	−2.2×10⁻⁹^[32]	1.0000000	1.2566371×10⁻⁶
Teflon		1.0000	1.2567×10⁻⁶^[13]
Sapphire	−2.1×10⁻⁷	0.99999976	1.2566368×10⁻⁶
Copper	−6.4×10⁻⁶ or −9.2×10⁻⁶^[32]	0.999994	1.256629×10⁻⁶
Water	−8.0×10⁻⁶	0.999992	1.256627×10⁻⁶
Bismuth	−1.66×10⁻⁴	0.999834	1.25643×10⁻⁶
Pyrolytic carbon		0.9996	1.256×10⁻⁶
Superconductors	−1	0	0
Cobalt		50 - 1500^[35]

Magnetisation curve for ferromagnets (and ferrimagnets) and corresponding permeability

A goodmagnetic core material must have high permeability.^[36]

Forpassivemagnetic 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 becomplex, corresponding to the in-phase and out of phase response.

Complex permeability

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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.^[37] These fields can be written asphasors, such that