View source for Permeability (electromagnetism)

{{Short description|Ability of magnetization}}
{{MagneticCircuitSegments}}

{{electromagnetism|cTopic=Magnetostatics}}

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 [[Mu (letter)|''μ'']].  It is the ratio of the [[Magnetic field|magnetic induction]] <math>B</math> to the magnetizing field <math>H</math> as a function of the field <math>H</math> in a material.  The term was coined by  [[William Thomson, 1st Baron Kelvin]] in 1872,<ref>[https://books.google.com/books?id=aApVAAAAMAAJ&pg=PA484 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 [[Henry (unit)|henries]] per [[Metre|meter]] (H/m), or equivalently in [[newton (unit)|newtons]] per [[ampere]] squared (N/A<sup>2</sup>). The permeability constant ''μ''<sub>0</sub>, 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 quantity|dimensionless]] proportionality factor that indicates the degree of magnetization of a material in response to an applied magnetic field.

==Explanation==
In the [[Maxwell's equations#Macroscopic formulation|macroscopic formulation of electromagnetism]], there appear two different kinds of [[magnetic field]]:
* the [[Magnetic field#The H-field|magnetizing field]] '''H''' which is generated around electric currents and [[displacement current]]s, and also [[demagnetizing field|emanates from the poles of magnets]].  The SI units of '''H''' are [[ampere]]s/meter.
* the [[Magnetic field#The B-field|magnetic flux density]] '''B''' which acts back on the electrical domain, by [[Lorentz force|curving the motion of charges]] and causing [[electromagnetic induction]]. The SI units of '''B''' are [[volt]]-seconds/square meter ([[tesla (unit)|tesla]]s).

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">{{cite book | author=Jackson, John David | title=Classical Electrodynamics | edition=3nd | location=New York | publisher=Wiley | year=1998 | isbn=978-0-471-30932-1 | pages=193}}</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 ''μ''<sub>0</sub>. The SI units of ''μ'' are volt-seconds/ampere-meter, equivalently [[henry (unit)|henry]]/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 magnet]]s), 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 core]]s. 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 [[Maxwell's equations|microscopic formulation of electromagnetism]], where there is no concept of an '''H''' field, the vacuum permeability ''μ''<sub>0</sub> 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 {{anchor|relative_permeability}}==
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 ''μ''<sub>0</sub>:

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

where <math>\mu_0 \approx </math> 4{{pi}}&nbsp;×&nbsp;10<sup>−7</sup>&nbsp;H/m is the [[Vacuum permeability|magnetic permeability of free space]].<ref>[https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf The International System of Units], page 132, The ampere. [[The International Bureau of Weights and Measures|BIPM]].</ref> In terms of relative permeability, the [[magnetic susceptibility]] is

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

The number ''χ''<sub>m</sub> is a [[dimensionless quantity]], sometimes called ''volumetric'' or ''bulk'' susceptibility, to distinguish it from ''χ''<sub>p</sub> (''magnetic mass'' or ''specific'' susceptibility) and ''χ''<sub>M</sub> (''molar'' or ''molar mass'' susceptibility).

==Diamagnetism==
{{Main|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 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 ''μ''<sub>0</sub> (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 [[superconductor]]s exhibit a strong effect.

==Paramagnetism==
{{Main|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 [[1 (number)|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 [[ferromagnetism|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:<ref>{{Cite journal | last1 = Kales | first1 = M. L. | title = Modes in Wave Guides Containing Ferrites | doi = 10.1063/1.1721335 | journal = Journal of Applied Physics | volume = 24 | issue = 5 | pages = 604–608 | year = 1953 |bibcode = 1953JAP....24..604K }}</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 materials==
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&nbsp;T) of 2,000 and a maximum of 35,000<ref name="kaye-laby">G.W.C. Kaye & T.H. Laby, Table of Physical and Chemical Constants, 14th ed, Longman, "Si Steel"</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).
{| class="wikitable sortable"
|+ Magnetic susceptibility and permeability data for selected materials
|-
! Medium
! class="unsortable"      | Susceptibility,<br>volumetric, SI, ''χ''<sub>m</sub>
! data-sort-type="number" | Relative permeability, <br />{{abbr|max.|maximum}}, ''μ''/''μ''<sub>0</sub>
! class="unsortable"      | Permeability, <br/>''μ'' (H/m)
! class="unsortable"      | Magnetic <br/>field
! class="unsortable"      | Frequency, {{abbr|max.|maximum}}
|-
| [[Vacuum]]
| 0
| 1, exactly<ref>by definition</ref>
| 1.25663706212&nbsp;×&nbsp;10<sup>−6</sup> (''μ''<sub>0</sub>)
|
|
|-
| [[Metglas]] 2714A (annealed)
|
| {{val|1000000}}<ref name="Metglas">{{cite web |url=http://www.metglas.com/products/page5_1_2_6.htm |title="Metglas Magnetic Alloy 2714A", ''Metglas'' |publisher=Metglas.com |access-date=2011-11-08 |url-status=dead |archive-url=https://web.archive.org/web/20120206100947/http://www.metglas.com/products/page5_1_2_6.htm |archive-date=2012-02-06 }}</ref>
| {{val|1.26|e=0}}
| At 0.5 T
| 100&nbsp;kHz
|-
| [[Iron]] (99.95% pure Fe annealed in H)
|
| {{val|200000}}<ref name="Iron">{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/tables/magprop.html#c2 |title="Magnetic Properties of Ferromagnetic Materials", ''Iron'' |publisher=C.R Nave Georgia State University |access-date=2013-12-01}}</ref>
| {{val|2.5|e=-1}}
|
|
|-
| [[Permalloy]]
|
| {{val|100000}}<ref name="Jiles">{{cite book
| last = Jiles 
| first = David
| title = Introduction to Magnetism and Magnetic Materials
| publisher = CRC Press 
| year = 1998
| page= 354
| url = https://books.google.com/books?id=axyWXjsdorMC&q=mu+metal&pg=PA354
| isbn = 978-0-412-79860-3}}</ref>
| {{val|1.25|e=-1}}
| At 0.002 T
|
|-
| [https://web.archive.org/web/20200805140106/https://www.magnetec.de/en/materials-products/ NANOPERM®]
|
| {{val|80000}}<ref name="Nanoperm">{{cite web|url=http://www.magnetec.de/eng/pdf/werkstoffkennlinien_nano_e.pdf |title="Typical material properties of NANOPERM", ''Magnetec'' |access-date=2011-11-08}}</ref>
| {{val|1.0|e=-1}}
| At 0.5 T
| 10&nbsp;kHz
|-
| [[Mu-metal]]
|
| {{val|50000}}<ref name="nickal">{{cite web|url=http://www.nickel-alloys.net/nickelalloys.html |title=Nickel Alloys-Stainless Steels, Nickel Copper Alloys, Nickel Chromium Alloys, Low Expansion Alloys |publisher=Nickel-alloys.net |access-date=2011-11-08}}</ref>
| {{val|6.3|e=-2}}
|
|
|-
| [[Mu-metal]]
|
| {{val|20000}}<ref name="hyper">{{cite web|url=http://hyperphysics.phy-astr.gsu.edu/hbase/solids/ferro.html |title="Relative Permeability", ''Hyperphysics'' |publisher=Hyperphysics.phy-astr.gsu.edu |access-date=2011-11-08}}</ref>
| {{val|2.5|e=-2}}
| At 0.002 T
|
|-
| Cobalt-iron <br />(high permeability strip material)
|
| {{val|18000}}<ref name="vacuumschmeltze">{{cite web |url=http://www.vacuumschmelze.com/fileadmin/Medienbiliothek_2010/Downloads/HT/2013-03-27_Soft_Magnetic_Cobalt-_Iron_Alloys_final_version.pdf |title="Soft Magnetic Cobalt-Iron Alloys", ''Vacuumschmeltze'' |publisher=www.vacuumschmeltze.com |access-date=2013-08-03 |url-status=dead |archive-url=http://arquivo.pt/wayback/20160523203358/http%3A//www.vacuumschmelze.com/fileadmin/Medienbiliothek_2010/Downloads/HT/2013%2D03%2D27_Soft_Magnetic_Cobalt%2D_Iron_Alloys_final_version.pdf |archive-date=2016-05-23 }}</ref>
| {{val|2.3|e=-2}}
|
|
|-
| [[Iron]] (99.8% pure)
|
| {{val|5000}}<ref name="Iron" />
| {{val|6.3|e=-3}}
|
|
|-
| [[Electrical steel]]
|
| 2000 - 35000<ref name="kaye-laby"/><ref name="ellingson">{{cite web |url=https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Book%3A_Electromagnetics_II_(Ellingson)/11%3A_Constitutive_Parameters_of_Some_Common_Materials/11.02%3A_Permeability_of_Some_Common_Materials |title="Permeability of Some Common Materials"| access-date=2022-12-09 }}</ref>
| {{val|5.0|e=-3}}
| At 0.002 T
|
|-
| [[Stainless steel#Types|Ferritic stainless steel]] (annealed)
|
| 1000 – 1800<ref name="Carpenter">{{cite web|url=https://www.cartech.com/en/alloy-techzone/technical-information/technical-articles/magnetic-properties-of-stainless-steels|title=Magnetic Properties of Stainless Steels|year=2013|publisher=Carpenter Technology Corporation|author=Carpenter Technology Corporation}}</ref>
| {{val|1.26|e=-3}} – {{val|2.26|e=-3}}
|
|
|-
| [[Martensitic stainless steel]] (annealed)
|
| 750 – 950<ref name="Carpenter" />
| {{val|9.42|e=-4}} – {{val|1.19|e=-3}}
|
|
|-
| [[Ferrite (magnet)#Soft ferrites|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>
| {{val|4.4|e=-4}} – {{val|2.51|e=-2}}
| At 0.25 mT
| {{abbr|{{abbr|approx.|approximately}}|approximately}} 100&nbsp;Hz – 4&nbsp;MHz
|-
| [[Ferrite (magnet)#Soft ferrites|Ferrite]] (nickel zinc)
|
| 10 – 2300<ref>According to Siemens Matsushita SIFERRIT data. https://www.thierry-lequeu.fr/data/SIFERRIT.pdf</ref>
| {{val|1.26|e=-5}} – {{val|2.89|e=-3}}
| At ≤ 0.25 mT
| {{abbr|approx.|approximately}} 1&nbsp;kHz – 400&nbsp;MHz{{Citation needed|date=February 2012}}
|-
| [[Ferrite (magnet)#Soft ferrites|Ferrite]] (magnesium manganese zinc)
|
| 350 – 500<ref>According to PRAMET Šumperk fonox data. https://www.doe.cz/wp-content/uploads/fonox.pdf</ref>
| {{val|4.4|e=-4}} – {{val|6.28|e=-4}}
| At 0.25 mT
|
|-
| [[Ferrite (magnet)#Soft ferrites|Ferrite]] (cobalt nickel zinc)
|
| 40 – 125<ref>According to Ferronics Incorporated data. http://www.ferronics.com/catalog/ferronics_catalog.pdf</ref>
| {{val|5.03|e=-5}} – {{val|1.57|e=-4}}
| At 0.001 T
| {{abbr|approx.|approximately}} 2&nbsp;MHz – 150&nbsp;MHz
|-
| Mo-Fe-Ni powder compound <br />(molypermalloy powder, MPP)
|
| 14 – 550<ref>According to Magnetics MPP-molypermalloy powder data. https://www.mag-inc.com/Products/Powder-Cores/MPP-Cores</ref>
| {{val|1.76|e=-5}} – {{val|6.91|e=-4}}
|
| {{abbr|approx.|approximately}} 50&nbsp;Hz – 3&nbsp;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>
| {{val|1.76|e=-5}} – {{val|2.01|e=-4}}
| At 0.001 T
| {{abbr|approx.|approximately}} 50&nbsp;Hz – 2&nbsp;MHz
|-
| Al-Si-Fe powder compound (Sendust)
|
| 14 – 160<ref>According to Micrometals-Arnold Sendust data. https://www.micrometalsarnoldpowdercores.com/products/materials/sendust</ref>
| {{val|1.76|e=-5}} – {{val|2.01|e=-4}}
|
| {{abbr|approx.|approximately}} 50&nbsp;Hz – 5&nbsp;MHz<ref>According to Micrometals-Arnold High Frequency Sendust data. https://www.micrometalsarnoldpowdercores.com/products/materials/sendust-high-frequency</ref>
|-
| Iron powder compound
|
| 14 – 100<ref>{{Cite web|url=https://micrometals.com/materials/pc|title=Micrometals Powder Core Solutions|website=micrometals.com|access-date=2019-08-17}}</ref>
| {{val|1.76|e=-5}} – {{val|1.26|e=-4}}
| At 0.001 T
| {{abbr|approx.|approximately}} 50&nbsp;Hz – 220&nbsp;MHz
|-
| Silicon iron powder compound
|
| 19 – 90<ref>According to Magnetics XFlux data. https://www.mag-inc.com/Products/Powder-Cores/XFlux-Cores</ref><ref>{{Cite web|url=https://micrometals.com/materials/200c|title=Micrometals Powder Core Solutions|website=micrometals.com|access-date=2019-08-18}}</ref>
| {{val|2.39|e=-5}} – {{val|1.13|e=-4}}
|
| {{abbr|approx.|approximately}} 50&nbsp;Hz – 40&nbsp;MHz
|-
| Carbonyl iron powder compound
|
| 4 – 35<ref>{{Cite web|url=https://www.micrometals.com/materials/rf|title=Micrometals Powder Core Solutions|website=www.micrometals.com|access-date=2019-08-17}}</ref>
| {{val|5.03|e=-6}} – {{val|4.4|e=-5}}
| At 0.001 T
| {{abbr|approx.|approximately}} 20&nbsp;kHz – 500&nbsp;MHz
|-
| [[Steel|Carbon steel]]
|
| {{val|100}}<ref name="hyper" />
| {{val|1.26|e=-4}}
| At 0.002 T
|
|-
| [[Nickel]]
|
| 100<ref name="hyper" /> – 600
| {{val|1.26|e=-4}} – {{val|7.54|e=-4}}
| At 0.002 T
|
|-
| [[Martensitic stainless steel]] (hardened)
|
| 40 – 95<ref name="Carpenter" />
| {{val|5.0|e=-5}} – {{val|1.2|e=-4}}
|
|
|-
| [[Stainless steel#Types|Austenitic stainless steel]]
|
| 1.003 – 1.05<ref name="Carpenter" /><ref name="SSAS">{{cite web|url=http://www.bssa.org.uk/cms/File/SSAS2.81-Magnetic%20Properties.pdf|title=Magnetic Properties of Stainless Steel|author=British Stainless Steel Association|publisher=Stainless Steel Advisory Service|year=2000}}</ref><ref group="note">The permeability of austenitic stainless steel strongly depends on the history of mechanical strain applied to it, e.g. by [[cold forming|cold working]]</ref>
| {{val|1.260|e=-6}} – {{val|8.8|e=-6}}
|
|
|-
| [[Neodymium magnet]]
|
| 1.05<ref>{{cite book|url=https://books.google.com/books?id=_y3LSh1XTJYC&pg=PT232|page=232|title=Design of Rotating Electrical Machines|author1=Juha Pyrhönen |author2=Tapani Jokinen |author3=Valéria Hrabovcová |publisher=John Wiley and Sons|year=2009|isbn=978-0-470-69516-6}}</ref>
| {{val|1.32|e=-6}}
|
|
|-
| [[Platinum]]
|
| {{val|1.000265}}
| {{val|1.256970|e=-6}}
|
|
|-
| [[Aluminum]]
| {{val|2.22|e=-5}}<ref name="clarke">{{cite web|author=Richard A. Clarke |url=http://www.ee.surrey.ac.uk/Workshop/advice/coils/mu/ |title=Magnetic properties of materials, surrey.ac.uk |publisher=Ee.surrey.ac.uk |access-date=2011-11-08}}</ref>
| {{val|1.000022}}
| {{val|1.256665|e=-6}}
|
|
|-
| [[Wood]]
|
| {{val|1.00000043}}<ref name="clarke" />
| {{val|1.25663760|e=-6}}
|
|
|-
| [[Air]]
|
| {{val|1.00000037}}<ref name="Cullity2008">B. D. Cullity and C. D. Graham (2008), Introduction to Magnetic Materials, 2nd edition, 568 pp., p.16</ref>
| {{val|1.25663753|e=-6}}
|
|
|-
| [[Concrete]] (dry)
|
| 1<ref>{{cite web|author=NDT.net |url=http://www.ndt.net/article/ndtce03/papers/v078/v078.htm |title=Determination of dielectric properties of insitu concrete at radar frequencies |publisher=Ndt.net |access-date=2011-11-08}}</ref>
|
|
|
|-
| [[Hydrogen]]
| {{val|-2.2|e=-9}}<ref name="clarke" />
| {{val|1.0000000}}
| {{val|1.2566371|e=-6}}
|
|
|-
| [[Teflon]]
|
| {{val|1.0000}}
| {{val|1.2567|e=-6}}<ref name="hyper"/>
| 
|
|-
| [[Sapphire]]
| {{val|-2.1|e=-7}}
| {{val|0.99999976}}
| {{val|1.2566368|e=-6}}
|
|
|-
| [[Copper]]
| {{val|-6.4|e=-6}} or <br/>{{val|-9.2|e=-6}}<ref name="clarke" />
| {{val|0.999994}}
| {{val|1.256629|e=-6}}
|
|
|-
| [[Water]]
| {{val|-8.0|e=-6}}
| {{val|0.999992}}
| {{val|1.256627|e=-6}}
|
|
|-
| [[Bismuth]]
| {{val|-1.66|e=-4}}
| {{val|0.999834}}
| {{val|1.25643|e=-6}}
|
|
|-
| [[Pyrolytic carbon]]
|
| {{val|0.9996}}
| {{val|1.256|e=-6}}
|
|
|-
| [[Superconductor]]s
| −1
| 0
| 0
|
|
|}

[[File: Permeability of ferromagnet by Zureks.svg|thumb|Magnetisation curve for ferromagnets (and ferrimagnets) and corresponding permeability]]

A good [[Magnetic core#Magnetic core materials|magnetic core material]] must have high permeability.<ref>{{cite web|
url=http://www.ti.com/lit/ml/slup124/slup124.pdf|
title=Magnetics Design 2 – Magnetic Core Characteristics|
author=Dixon, L H|
publisher=Texas Instruments|
year=2001}}</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 number|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.<ref name="getzlaff">M. Getzlaff, ''Fundamentals of magnetism'', Berlin: Springer-Verlag, 2008.</ref> These fields can be written as [[phasor (electronics)|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 also==
* [[Antiferromagnetism]]
* [[Diamagnetism]]
* [[Electromagnet]]
* [[Ferromagnetism]]
* [[Magnetic reluctance]]
* [[Paramagnetism]]
* [[Permittivity]]
* [[SI electromagnetism units]]

==Notes==
<references group="note" />

==References==
{{Reflist}}

==External links==
* [http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html Electromagnetism] - a chapter from an online textbook
* [https://www.fxsolver.com/browse/?q=permeability&p=-1 Permeability calculator]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/solids/ferro.html Relative Permeability]
* [http://www.ee.surrey.ac.uk/Workshop/advice/coils/mu/ Magnetic Properties of Materials]
* RF Cafe's [http://www.rfcafe.com/references/electrical/cond-high-freq.htm Conductor Bulk Resistivity & Skin Depths]

{{Authority control}}

{{DEFAULTSORT:Permeability (Electromagnetism)}}
[[Category:Electric and magnetic fields in matter]]
[[Category:Physical quantities]]