Permeability (electromagnetism)

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In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter μ. It is the ratio of the magnetic induction <math>B</math> to the magnetizing field <math>H</math> in a material. The term was coined by William Thomson, 1st Baron Kelvin in 1872,<ref>Magnetic Permeability, and Analogues in Electro-static Induction, Conduction of Heat, and Fluid Motion, March 1872.</ref> and used alongside permittivity by Oliver Heaviside in 1885. The reciprocal of permeability is magnetic reluctivity.

In SI units, permeability is measured in henries per meter (H/m), or equivalently in newtons per ampere squared (N/A2). The permeability constant μ0, also known as the magnetic constant or the permeability of free space, is the proportionality between magnetic induction and magnetizing force when forming a magnetic field in a classical vacuum.

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.

ExplanationEdit

In the macroscopic formulation of electromagnetism, there appear two different kinds of magnetic field:

The concept of permeability arises since in many materials (and in vacuum), there is a simple relationship between H and B at any location or time, in that the two fields are precisely proportional to each other:<ref name="jackson">Template:Cite book</ref>

<math>\mathbf{B}=\mu \mathbf{H},</math>

where the proportionality factor μ is the permeability, which depends on the material. The permeability of vacuum (also known as permeability of free space) is a physical constant, denoted μ0. The SI units of μ are volt-seconds per ampere-meter, equivalently henry per meter. Typically μ would be a scalar, but for an anisotropic material, μ could be a second rank tensor.

However, inside strong magnetic materials (such as iron, or permanent magnets), there is typically no simple relationship between H and B. The concept of permeability is then nonsensical or at least only applicable to special cases such as unsaturated magnetic cores. Not only do these materials have nonlinear magnetic behaviour, but often there is significant magnetic hysteresis, so there is not even a single-valued functional relationship between B and H. However, considering starting at a given value of B and H and slightly changing the fields, it is still possible to define an incremental permeability as:<ref name="jackson"/>

<math>\Delta\mathbf{B}=\mu \, \Delta\mathbf{H}.</math>

assuming B and H are parallel.

In the microscopic formulation of electromagnetism, where there is no concept of an H field, the vacuum permeability μ0 appears directly (in the SI Maxwell's equations) as a factor that relates total electric currents and time-varying electric fields to the B field they generate. In order to represent the magnetic response of a linear material with permeability μ, this instead appears as a magnetization M that arises in response to the B field: <math>\mathbf{M} = \left(\mu_0^{-1} - \mu^{-1}\right) \mathbf{B}</math>. The magnetization in turn is a contribution to the total electric current—the magnetization current.

Relative permeability and magnetic susceptibility Edit

Relative permeability, denoted by the symbol <math>\mu_\mathrm{r}</math>, is the ratio of the permeability of a specific medium to the permeability of free space μ0:

<math>\mu_\mathrm{r} = \frac \mu {\mu_0},</math>

where <math>\mu_0 \approx </math> 4Template:Pi × 10−7 H/m is the magnetic permeability of free space.<ref>The International System of Units, page 132, The ampere. BIPM.</ref> In terms of relative permeability, the magnetic susceptibility is

<math>\chi_m = \mu_r - 1.</math>

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).

DiamagnetismEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 atom's 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.

ParamagnetismEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 it is 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.

GyromagnetismEdit

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:<ref>Template:Cite journal</ref>

<math>\begin{align}

\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{align}</math>

Values for some common materialsEdit

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 <ref name="kaye-laby">G.W.C. Kaye & T.H. Laby, Table of Physical and Chemical Constants, 14th ed, Longman, "Si Steel"</ref><ref>https://publikationen.bibliothek.kit.edu/1000066142/4047647 for the 38,000 figure 5.2</ref> 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, χm
Relative permeability,
Template:Abbr, μ/μ0
Permeability,
μ (H/m)
Magnetic
field
Frequency, Template:Abbr
Vacuum 0 1, exactly<ref>by definition</ref> Template:Physconst
Metglas 2714A (annealed) Template:Val<ref name="Metglas">{{#invoke:citation/CS1|citation CitationClass=web

}}</ref>

Template:Val At 0.5 T 100 kHz
Iron (99.95% pure Fe annealed in H) Template:Val<ref name="Iron">{{#invoke:citation/CS1|citation CitationClass=web

}}</ref>

Template:Val
Permalloy Template:Val<ref name="Jiles">Template:Cite book</ref> Template:Val At 0.002 T
NANOPERM® Template:Val<ref name="Nanoperm">{{#invoke:citation/CS1|citation CitationClass=web

}}</ref>

Template:Val At 0.5 T 10 kHz
Mu-metal Template:Val<ref name="nickal">{{#invoke:citation/CS1|citation CitationClass=web

}}</ref>

Template:Val
Mu-metal Template:Val<ref name="hyper">{{#invoke:citation/CS1|citation CitationClass=web

}}</ref>

Template:Val At 0.002 T
Cobalt-iron
(high permeability strip material)
Template:Val<ref name="vacuumschmeltze">{{#invoke:citation/CS1|citation CitationClass=web

}}</ref>

Template:Val
Iron (99.8% pure) Template:Val<ref name="Iron" /> Template:Val
Electrical steel citation CitationClass=web

}}</ref><ref>https://publikationen.bibliothek.kit.edu/1000066142/4047647 for 38000 at 1 T figure 5.2</ref>

Template:Val At 0.002 T, 1 T
Ferritic stainless steel (annealed) citation CitationClass=web

}}</ref>

Template:ValTemplate:Val
Martensitic stainless steel (annealed) 750 – 950<ref name="Carpenter" /> Template:ValTemplate:Val
Ferrite (manganese zinc) 350 – 20 000<ref>According to Ferroxcube (formerly Philips) Soft Ferrites data. https://www.ferroxcube.com/zh-CN/download/download/21</ref> Template:ValTemplate:Val At 0.25 mT Template:Abbr 100 Hz – 4 MHz
Ferrite (nickel zinc) 10 – 2300<ref>According to Siemens Matsushita SIFERRIT data. https://www.thierry-lequeu.fr/data/SIFERRIT.pdf</ref> Template:ValTemplate:Val At ≤ 0.25 mT Template:Abbr 1 kHz – 400 MHzTemplate:Citation needed
Ferrite (magnesium manganese zinc) 350 – 500<ref>According to PRAMET Šumperk fonox data. https://www.doe.cz/wp-content/uploads/fonox.pdf</ref> Template:ValTemplate:Val At 0.25 mT
Ferrite (cobalt nickel zinc) 40 – 125<ref>According to Ferronics Incorporated data. http://www.ferronics.com/catalog/ferronics_catalog.pdf</ref> Template:ValTemplate:Val At 0.001 T Template:Abbr 2 MHz – 150 MHz
Mo-Fe-Ni powder compound
(molypermalloy powder, MPP)
14 – 550<ref>According to Magnetics MPP-molypermalloy powder data. https://www.mag-inc.com/Products/Powder-Cores/MPP-Cores</ref> Template:ValTemplate:Val Template:Abbr 50 Hz – 3 MHz
Nickel iron powder compound 14 – 160<ref>According to MMG IOM Limited High Flux data. http://www.mmgca.com/catalogue/MMG-Sailcrest.pdf</ref> Template:ValTemplate:Val At 0.001 T Template:Abbr 50 Hz – 2 MHz
Al-Si-Fe powder compound (Sendust) 14 – 160<ref>According to Micrometals-Arnold Sendust data. https://www.micrometalsarnoldpowdercores.com/products/materials/sendust</ref> Template:ValTemplate:Val Template:Abbr 50 Hz – 5 MHz<ref>According to Micrometals-Arnold High Frequency Sendust data. https://www.micrometalsarnoldpowdercores.com/products/materials/sendust-high-frequency</ref>
Iron powder compound citation CitationClass=web

}}</ref>

Template:ValTemplate:Val At 0.001 T Template:Abbr 50 Hz – 220 MHz
Silicon iron powder compound citation CitationClass=web

}}</ref>

Template:ValTemplate:Val Template:Abbr 50 Hz – 40 MHz
Carbonyl iron powder compound citation CitationClass=web

}}</ref>

Template:ValTemplate:Val At 0.001 T Template:Abbr 20 kHz – 500 MHz
Carbon steel Template:Val<ref name="hyper" /> Template:Val At 0.002 T
Nickel 100<ref name="hyper" /> – 600 Template:ValTemplate:Val At 0.002 T
Martensitic stainless steel (hardened) 40 – 95<ref name="Carpenter" /> Template:ValTemplate:Val
Austenitic stainless steel citation CitationClass=web

}}</ref>Template:Efn

Template:ValTemplate:Val
Neodymium magnet 1.05<ref>Template:Cite book</ref> Template:Val
Platinum Template:Val Template:Val
Aluminum Template:Val<ref name="clarke">{{#invoke:citation/CS1|citation CitationClass=web

}}</ref>

Template:Val Template:Val
Wood Template:Val<ref name="clarke" /> Template:Val
Air Template:Val<ref name="Cullity2008">B. D. Cullity and C. D. Graham (2008), Introduction to Magnetic Materials, 2nd edition, 568 pp., p.16</ref> Template:Val
Concrete (dry) citation CitationClass=web

}}</ref>

Hydrogen Template:Val<ref name="clarke" /> Template:Val Template:Val
Teflon Template:Val Template:Val<ref name="hyper"/>
Sapphire Template:Val Template:Val Template:Val
Copper Template:Val or
Template:Val<ref name="clarke" />
Template:Val Template:Val
Water Template:Val Template:Val Template:Val
Bismuth Template:Val Template:Val Template:Val
Pyrolytic carbon Template:Val Template:Val
Superconductors −1 0 0
File:Permeability of ferromagnet by Zureks.svg
Magnetisation curve for ferromagnets (and ferrimagnets) and corresponding permeability

A good magnetic core material must have high permeability.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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 permeabilityEdit

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.<ref name="getzlaff">M. Getzlaff, Fundamentals of magnetism, Berlin: Springer-Verlag, 2008.</ref> These fields can be written as phasors, such that

<math>H = H_0 e^{j \omega t} \qquad B = B_0 e^{j\left(\omega t - \delta \right)}</math>

where <math>\delta</math> is the phase delay of <math>B</math> from <math>H</math>.

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

<math>\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},</math>

so that the permeability becomes a complex number.

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

<math>\mu = \frac{B_0}{H_0}\cos(\delta) - j \frac{B_0}{H_0}\sin(\delta) = \mu' - j \mu.</math>

The ratio of the imaginary to the real part of the complex permeability is called the loss tangent,

<math>\tan(\delta) = \frac{\mu}{\mu'},</math>

which provides a measure of how much power is lost in material versus how much is stored.

See alsoEdit

NotesEdit

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ReferencesEdit

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External linksEdit

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