Editing Magnet (section)

== Units and calculations ==
{{Main|Magnetostatics}}

For most engineering applications, MKS (rationalized) or [[SI]] (Système International) units are commonly used. Two other sets of units, [[Gaussian units|Gaussian]] and [[CGS#Electromagnetic units (EMU)|CGS-EMU]], are the same for magnetic properties and are commonly used in physics.{{citation needed|date=March 2016}}

In all units, it is convenient to employ two types of magnetic field, '''B''' and '''H''', as well as the [[magnetization]] '''M''', defined as the magnetic moment per unit volume.

# The magnetic induction field '''B''' is given in SI units of teslas (T). '''B''' is the magnetic field whose time variation produces, by Faraday's Law, circulating electric fields (which the power companies sell). '''B''' also produces a deflection force on moving charged particles (as in TV tubes). The tesla is equivalent to the magnetic flux (in webers) per unit area (in meters squared), thus giving '''B''' the unit of a flux density. In CGS, the unit of '''B''' is the gauss (G). One tesla equals 10<sup>4</sup>&nbsp;G.
# The magnetic field '''H''' is given in SI units of ampere-turns per meter (A-turn/m). The ''turns'' appear because when '''H''' is produced by a current-carrying wire, its value is proportional to the number of turns of that wire. In CGS, the unit of '''H''' is the oersted (Oe). One A-turn/m equals 4π×10<sup>−3</sup> Oe.
# The magnetization '''M''' is given in SI units of amperes per meter (A/m). In CGS, the unit of '''M''' is the oersted (Oe). One A/m equals 10<sup>−3</sup>&nbsp;emu/cm<sup>3</sup>. A good permanent magnet can have a magnetization as large as a million amperes per meter.
# In SI units, the relation '''B'''&nbsp;= ''μ''<sub>0</sub>('''H'''&nbsp;+&nbsp;'''M''') holds, where ''μ''<sub>0</sub> is the permeability of space, which equals 4π×10<sup>−7</sup>&nbsp;T•m/A. In CGS, it is written as '''B'''&nbsp;= '''H'''&nbsp;+&nbsp;4π'''M'''. (The pole approach gives ''μ''<sub>0</sub>'''H''' in SI units. A ''μ''<sub>0</sub>'''M''' term in SI must then supplement this ''μ''<sub>0</sub>'''H''' to give the correct field within '''B''', the magnet. It will agree with the field '''B''' calculated using Ampèrian currents).

Materials that are not permanent magnets usually satisfy the relation '''M'''&nbsp;= ''χ'''''H''' in SI, where ''χ'' is the (dimensionless) magnetic susceptibility. Most non-magnetic materials have a relatively small ''χ'' (on the order of a millionth), but soft magnets can have ''χ'' on the order of hundreds or thousands. For materials satisfying '''M'''&nbsp;= ''χ'''''H''', we can also write '''B'''&nbsp;= ''μ''<sub>0</sub>(1&nbsp;+&nbsp;''χ'')'''H'''&nbsp;= ''μ''<sub>0</sub>''μ''<sub>r</sub>'''H'''&nbsp;= ''μ'''''H''', where ''μ''<sub>r</sub>&nbsp;= 1&nbsp;+&nbsp;''χ'' is the (dimensionless) relative permeability and μ&nbsp;=μ<sub>0</sub>μ<sub>r</sub> is the magnetic permeability. Both hard and soft magnets have a more complex, history-dependent, behavior described by what are called [[Magnetic hysteresis|hysteresis loops]], which give either '''B''' vs. '''H''' or '''M''' vs. '''H'''. In CGS, '''M'''&nbsp;= ''χ'''''H''', but ''χ''<sub>SI</sub>&nbsp;= 4''πχ''<sub>CGS</sub>, and μ&nbsp;=&nbsp;μ<sub>r</sub>.

Caution: in part because there are not enough Roman and Greek symbols, there is no commonly agreed-upon symbol for magnetic pole strength and magnetic moment. The symbol ''m'' has been used for both pole strength (unit A•m, where here the upright m is for meter) and for magnetic moment (unit A•m<sup>2</sup>). The symbol ''μ'' has been used in some texts for magnetic permeability and in other texts for magnetic moment. We will use ''μ'' for magnetic permeability and ''m'' for magnetic moment. For pole strength, we will employ ''q''<sub>''m''</sub>. For a bar magnet of cross-section ''A'' with uniform magnetization ''M'' along its axis, the pole strength is given by ''q<sub>m</sub>''&nbsp;= ''MA'', so that ''M'' can be thought of as a pole strength per unit area.

=== Fields of a magnet ===
[[Image:VFPt four magnets.svg|thumb|Field lines of cylindrical magnets with various aspect ratios]]
Far away from a magnet, the magnetic field created by that magnet is almost always described (to a good approximation) by a [[dipole|dipole field]] characterized by its total magnetic moment. This is true regardless of the shape of the magnet, so long as the magnetic moment is non-zero. One characteristic of a dipole field is that the strength of the field falls off inversely with the cube of the distance from the magnet's center.

Closer to the magnet, the magnetic field becomes more complicated and more dependent on the detailed shape and magnetization of the magnet. Formally, the field can be expressed as a [[multipole expansion]]: A dipole field, plus a [[quadrupole|quadrupole field]], plus an octupole field, etc.

At close range, many different fields are possible. For example, for a long, skinny bar magnet with its north pole at one end and south pole at the other, the magnetic field near either end falls off inversely with [[Inverse-square law|the square of the distance]] from that pole.

=== Calculating the magnetic force ===
{{main|Force between magnets}}

====Pull force of a single magnet====
The strength of a given magnet is sometimes given in terms of its ''pull force'' — its ability to pull [[Ferromagnetism|ferromagnetic]] objects.<ref>{{Cite web|url=https://www.kjmagnetics.com/blog.asp?p=how-much-will-a-magnet-hold|title=How Much Will a Magnet Hold?|website=www.kjmagnetics.com|access-date=2020-01-20}}</ref> The pull force exerted by either an electromagnet or a permanent magnet with no air gap (i.e., the ferromagnetic object is in direct contact with the pole of the magnet<ref>{{Cite web|url=https://www.duramag.com/techtalk/tech-briefs/magnetic-pull-force-explained/|title=Magnetic Pull Force Explained - What is Magnet Pull Force? {{!}} Dura Magnetics USA|date=19 October 2016|language=en-US|access-date=2020-01-20}}</ref>) is given by the [[Maxwell equation]]:<ref>{{Cite book|title=Materials Handbook: A Concise Desktop Reference|last=Cardarelli|first=François|edition=Second|year=2008|page=493|publisher=Springer|url=https://books.google.com/books?id=PvU-qbQJq7IC&pg=PA493|isbn=9781846286681|url-status=live|archive-url=https://web.archive.org/web/20161224182756/https://books.google.com/books?id=PvU-qbQJq7IC&pg=PA493|archive-date=2016-12-24}}</ref>
:<math>F={{B^2 A}\over{2 \mu_{0}}}</math>,

where:
* ''F'' is force (SI unit: [[newton (unit)|newton]])
* ''A'' is the cross section of the area of the pole (in square meters)
* ''B'' is the magnetic induction exerted by the magnet.

This result can be easily derived using [[Force between magnets#Gilbert model|Gilbert model]], which assumes that the pole of magnet is charged with [[magnetic monopole]]s that induces the same in the ferromagnetic object.

If a magnet is acting vertically, it can lift a mass ''m'' in kilograms given by the simple equation:

:<math>m={{B^2 A}\over{2 \mu_{0} g}},</math>

where g is the [[gravitational acceleration]].

==== Force between two magnetic poles ====
{{further|Magnetic moment#Forces between two magnetic dipoles}}

[[Classical mechanics|Classically]], the force between two magnetic poles is given by:<ref>{{cite web |url=http://geophysics.ou.edu/solid_earth/notes/mag_basic/mag_basic.html |title=Basic Relationships |publisher=Geophysics.ou.edu |access-date=2009-10-19 |url-status=dead |archive-url=https://web.archive.org/web/20100709205321/http://geophysics.ou.edu/solid_earth/notes/mag_basic/mag_basic.html |archive-date=2010-07-09 }}</ref>

:<math>F={{\mu q_{m1} q_{m2}}\over{4\pi r^2}}</math>

where
:''F'' is force (SI unit: [[newton (unit)|newton]])
:''q''<sub>''m''1</sub> and ''q''<sub>''m''2</sub> are the magnitudes of magnetic poles (SI unit: [[ampere-meter]])
:''μ'' is the [[permeability (electromagnetism)|permeability]] of the intervening medium (SI unit: [[tesla (unit)|tesla]] [[meter]] per [[ampere]], henry per meter or newton per ampere squared)
:''r'' is the separation (SI unit: meter).

The pole description is useful to the engineers designing real-world magnets, but real magnets have a pole distribution more complex than a single north and south. Therefore, implementation of the pole idea is not simple. In some cases, one of the more complex formulae given below will be more useful.

==== Force between two nearby magnetized surfaces of area ''A'' ====

The mechanical force between two nearby magnetized surfaces can be calculated with the following equation. The equation is valid only for cases in which the effect of [[fringing]] is negligible and the volume of the air gap is much smaller than that of the magnetized material:<ref name="tri-c">{{cite web|url=http://instruct.tri-c.edu/fgram/web/Mdipole.htm |title=Magnetic Fields and Forces |access-date=2009-12-24 |url-status=dead |archive-url=https://web.archive.org/web/20120220030524/http://instruct.tri-c.edu/fgram/web/Mdipole.htm |archive-date=2012-02-20 }}</ref><ref>{{cite web|url=http://info.ee.surrey.ac.uk/Workshop/advice/coils/force.html|title=The force produced by a magnetic field|access-date=2010-03-09|url-status=live|archive-url=https://web.archive.org/web/20100317153105/http://info.ee.surrey.ac.uk/Workshop/advice/coils/force.html|archive-date=2010-03-17}}</ref>
:<math>F=\frac{\mu_0 H^2 A}{2} = \frac{B^2 A}{2 \mu_0}</math>
where:
:''A'' is the area of each surface, in m<sup>2</sup>
:''H'' is their magnetizing field, in A/m
:''μ''<sub>0</sub> is the permeability of space, which equals 4π×10<sup>−7</sup>&nbsp;T•m/A
:''B'' is the flux density, in T.

==== Force between two bar magnets ====

The force between two identical cylindrical bar magnets placed end to end at large distance <math>z\gg R</math> is approximately:{{Dubious|date=September 2017}},<ref name="tri-c"/>
:<math>F \simeq \left[\frac {B_0^2 A^2 \left( L^2+R^2 \right)} {\pi\mu_0L^2}\right] \left[{\frac 1 {z^2}} + {\frac 1 {(z+2L)^2}} - {\frac 2 {(z+L)^2}} \right]</math>

where:

:''B<sub>0</sub>'' is the magnetic flux density very close to each pole, in T,
:''A'' is the area of each pole, in m<sup>2</sup>,
:''L'' is the length of each magnet, in m,
:''R'' is the radius of each magnet, in m, and
:''z'' is the separation between the two magnets, in m.

:<math>B_0 \,=\, \frac{\mu_0}{2}M</math> relates the flux density at the pole to the magnetization of the magnet.

Note that all these formulations are based on Gilbert's model, which is usable in relatively great distances. In other models (e.g., Ampère's model), a more complicated formulation is used that sometimes cannot be solved analytically. In these cases, [[numerical methods]] must be used.

==== Force between two cylindrical magnets ====

For two cylindrical magnets with radius <math> R </math> and length <math>L</math>, with their magnetic dipole aligned, the force can be asymptotically approximated at large distance <math>z\gg R</math> by,<ref>{{cite journal|author1=David Vokoun |author2=Marco Beleggia |author3=Ludek Heller |author4=Petr Sittner |title= Magnetostatic interactions and forces between cylindrical permanent magnets|journal= Journal of Magnetism and Magnetic Materials|volume =321|issue =22|year=2009|pages=3758–3763|doi=10.1016/j.jmmm.2009.07.030|bibcode = 2009JMMM..321.3758V|doi-access=free}}</ref>

:<math>
F(z) \simeq \frac{\pi\mu_0}{4} M^2 R^4 \left[\frac{1}{z^2} + \frac{1}{(z+2L)^2} - \frac{2}{(z + L)^2}\right]
</math>

where <math>M</math> is the magnetization of the magnets and <math>z</math> is the gap between the magnets.
A measurement of the magnetic flux density very close to the magnet <math> B_0 </math> is related to <math>M</math> approximately by the formula
<!-- Note: most of this here is only approximately true. For instance B_0 is not constant over the surface and the approximation becomes very wrong when the magnet is short i.e. L<R. -->

:<math>
B_0 = \frac{\mu_0}{2} M
</math>

The effective magnetic dipole can be written as

:<math>
m = M V
</math>

Where <math>V</math> is the volume of the magnet. For a cylinder, this is <math>V = \pi R^2 L</math>.

When <math>z\gg L</math>, the point dipole approximation is obtained,

:<math>
F(x) = \frac{3\pi\mu_0}{2} M^2 R^4 L^2\frac{1}{z^4} = \frac{3\mu_0}{2\pi} M^2 V^2\frac{1}{z^4} = \frac{3\mu_0}{2\pi} m_1 m_2\frac{1}{z^4}
</math>

which matches the expression of the force between two magnetic dipoles.