Editing Permeability (electromagnetism) (section)

== 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 per 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 per [[square meter]], a ratio equivalent to one [[Tesla (unit)|tesla]].

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 per ampere-meter, equivalently [[henry (unit)|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 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]].