Editing Magnetic field (section)

==Relation between H and B==
The formulas derived for the magnetic field above are correct when dealing with the entire current. A magnetic material placed inside a magnetic field, though, generates its own [[bound current]], which can be a challenge to calculate. (This bound current is due to the sum of atomic sized current loops and the [[Spin (physics)|spin]] of the subatomic particles such as electrons that make up the material.) The {{math|'''H'''}}-field as defined above helps factor out this bound current; but to see how, it helps to introduce the concept of ''magnetization'' first.

===Magnetization===
{{Main|Magnetization}}

The ''magnetization'' vector field {{math|'''M'''}} represents how strongly a region of material is magnetized. It is defined as the net [[magnetic dipole moment]] per unit volume of that region. The magnetization of a uniform magnet is therefore a material constant, equal to the magnetic moment {{math|'''m'''}} of the magnet divided by its volume. Since the SI unit of magnetic moment is A⋅m<sup>2</sup>, the SI unit of magnetization {{math|'''M'''}} is ampere per meter, identical to that of the {{math|'''H'''}}-field.

The magnetization {{math|'''M'''}} field of a region points in the direction of the average magnetic dipole moment in that region. Magnetization field lines, therefore, begin near the magnetic south pole and ends near the magnetic north pole. (Magnetization does not exist outside the magnet.)

In the Amperian loop model, the magnetization is due to combining many tiny Amperian loops to form a resultant current called ''[[bound current]]''. This bound current, then, is the source of the magnetic {{math|'''B'''}} field due to the magnet. Given the definition of the magnetic dipole, the magnetization field follows a similar law to that of Ampere's law:<ref>{{harvnb|Griffiths|1999|pp=266–268}}</ref>
<math display="block">\oint \mathbf{M} \cdot \mathrm{d}\boldsymbol{\ell} = I_\mathrm{b},</math>
where the integral is a line integral over any closed loop and {{math|''I''<sub>b</sub>}} is the bound current enclosed by that closed loop.

In the magnetic pole model, magnetization begins at and ends at magnetic poles. If a given region, therefore, has a net positive "magnetic pole strength" (corresponding to a north pole) then it has more magnetization field lines entering it than leaving it. Mathematically this is equivalent to:
<math display="block">\oint_S \mu_0 \mathbf{M} \cdot \mathrm{d}\mathbf{A} = - q_\mathrm{M},</math>
where the integral is a closed surface integral over the closed surface {{math|''S''}} and {{math|''q''<sub>M</sub>}} is the "magnetic charge" (in units of [[magnetic flux]]) enclosed by {{math|''S''}}. (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because the magnetization field moves from south to north.

===H-field and magnetic materials===
[[File:VFPt magnets BHM.svg|thumb|Comparison of {{math|'''B'''}}, {{math|'''H'''}} and {{math|'''M'''}} inside and outside a cylindrical bar magnet.]]
{{see also|Demagnetizing field}}
In SI units, the H-field is related to the B-field by
<math display="block">\mathbf{H}\ \equiv \ \frac{\mathbf{B}}{\mu_0} - \mathbf{M}.</math>

In terms of the H-field, Ampere's law is
<math display="block">\oint \mathbf{H} \cdot \mathrm{d}\boldsymbol{\ell} = \oint \left(\frac{\mathbf{B}}{\mu_0} - \mathbf{M}\right) \cdot \mathrm{d}\boldsymbol{\ell} = I_\mathrm{tot} - I_\mathrm{b} = I_\mathrm{f},</math>
where {{math|I<sub>f</sub>}} represents the 'free current' enclosed by the loop so that the line integral of {{math|'''H'''}} does not depend at all on the bound currents.<ref name=Slater>{{cite book |title=Electromagnetism |author1=John Clarke Slater |author2=Nathaniel Herman Frank |url=https://books.google.com/books?id=GYsphnFwUuUC&pg=PA69 |page=69 |isbn=978-0-486-62263-7 |year=1969 |publisher=Courier Dover Publications |edition=first published in 1947 }}</ref>

For the differential equivalent of this equation see [[#Maxwell's equations|Maxwell's equations]]. Ampere's law leads to the boundary condition
<math display="block">\left(\mathbf{H_1^\parallel} - \mathbf{H_2^\parallel}\right) = \mathbf{K}_\mathrm{f} \times \hat{\mathbf{n}},</math>
where {{math|'''K'''<sub>f</sub>}} is the surface free current density and the unit normal <math>\hat{\mathbf{n}}</math> points in the direction from medium 2 to medium 1.<ref>{{harvnb|Griffiths|1999|p=332}}</ref>

Similarly, a [[surface integral]] of {{math|'''H'''}} over any [[closed surface]] is independent of the free currents and picks out the "magnetic charges" within that closed surface:
<math display="block">\oint_S \mu_0 \mathbf{H} \cdot \mathrm{d}\mathbf{A} = \oint_S (\mathbf{B} - \mu_0 \mathbf{M}) \cdot \mathrm{d}\mathbf{A} = 0 - (-q_\mathrm{M}) = q_\mathrm{M},</math>

which does not depend on the free currents.

The {{math|'''H'''}}-field, therefore, can be separated into two<ref group="note">A third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below.</ref> independent parts:
<math display="block">\mathbf{H} = \mathbf{H}_0 + \mathbf{H}_\mathrm{d}, </math>

where {{math|'''H'''<sub>0</sub>}} is the applied magnetic field due only to the free currents and {{math|'''H'''<sub>d</sub>}} is the [[demagnetizing field]] due only to the bound currents.

The magnetic {{math|'''H'''}}-field, therefore, re-factors the bound current in terms of "magnetic charges". The {{math|'''H'''}} field lines loop only around "free current" and, unlike the magnetic {{math|'''B'''}} field, begins and ends near magnetic poles as well.

===Magnetism===
{{Main|Magnetism}}
Most materials respond to an applied {{math|'''B'''}}-field by producing their own magnetization {{math|'''M'''}} and therefore their own {{math|'''B'''}}-fields. Typically, the response is weak and exists only when the magnetic field is applied. The term ''magnetism'' describes how materials respond on the microscopic level to an applied magnetic field and is used to categorize the magnetic [[phase (matter)|phase]] of a material. Materials are divided into groups based upon their magnetic behavior:
* [[Diamagnetism|Diamagnetic materials]]<ref name=Tilley>{{cite book |title=Understanding Solids |author=RJD Tilley |page=[https://archive.org/details/understandingsol0000till/page/368 368] |isbn=978-0-470-85275-0 |publisher=Wiley |year=2004 |url=https://archive.org/details/understandingsol0000till |url-access=registration }}</ref> produce a magnetization that opposes the magnetic field.
* [[Paramagnetism|Paramagnetic materials]]<ref name=Tilley/> produce a magnetization in the same direction as the applied magnetic field.
* [[Ferromagnetism|Ferromagnetic materials]] and the closely related [[Ferrimagnetism|ferrimagnetic materials]] and [[Antiferromagnetism|antiferromagnetic materials]]<ref name=Chikazumi>{{cite book |title=Physics of ferromagnetism|author1=Sōshin Chikazumi |author2=Chad D. Graham |url=https://books.google.com/books?id=AZVfuxXF2GsC |page=118 |edition=2|year=1997 |publisher=Oxford University Press |isbn=978-0-19-851776-4}}</ref><ref name=Aharoni>{{cite book |title=Introduction to the theory of ferromagnetism |page=27 |author=Amikam Aharoni |url=https://books.google.com/books?id=9RvNuIDh0qMC&pg=PA27 |edition=2 |year=2000 |publisher=Oxford University Press |isbn=978-0-19-850808-3}}</ref> can have a magnetization independent of an applied B-field with a complex relationship between the two fields.
* [[Superconductor]]s (and [[ferromagnetic superconductor]]s)<ref name=Bennemann>{{cite book |title=Superconductivity |editor1=K. H. Bennemann |editor2=John B. Ketterson |chapter-url=https://books.google.com/books?id=PguAgEQTiQwC&pg=PA640 |page=640 |isbn=978-3-540-73252-5 |year=2008 |publisher=Springer |chapter=Unconventional superconductivity in novel materials |author=M Brian Maple |display-authors=etal }}</ref><ref name=Lewis>{{cite book |title=Superconductivity research at the leading edge |author=Naoum Karchev |editor1=Paul S. Lewis |editor2=D. Di (CON) Castro |chapter-url=https://books.google.com/books?id=3AFo_yxBkD0C&pg=PA169 |page=169 |isbn=978-1-59033-861-2 |publisher=Nova Publishers |year=2003 |chapter=Itinerant ferromagnetism and superconductivity}}</ref> are materials that are characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures and magnetic fields (the so-named [[Type II superconductor#Mixed state|mixed state]]) under which they exhibit a complex hysteretic dependence of {{math|'''M'''}} on {{math|'''B'''}}.

In the case of paramagnetism and diamagnetism, the magnetization {{math|'''M'''}} is often proportional to the applied magnetic field such that:
<math display="block">\mathbf{B} = \mu \mathbf{H},</math>
where {{math|''μ''}} is a material dependent parameter called the [[permeability (electromagnetism)|permeability]]. In some cases the permeability may be a second rank [[tensor]] so that {{math|'''H'''}} may not point in the same direction as {{math|'''B'''}}. These relations between {{math|'''B'''}} and {{math|'''H'''}} are examples of [[constitutive equation]]s. However, superconductors and ferromagnets have a more complex {{math|'''B'''}}-to-{{math|'''H'''}} relation; see [[hysteresis#Magnetic hysteresis|magnetic hysteresis]].