Editing Electromagnet (section)

== Physics ==
[[File:Electromagnetism.svg|thumb|Current (<math>I</math>) through a wire produces a magnetic field (<math>B</math>). The field is oriented according to the [[Right-hand rule#Electromagnetism|right-hand rule]].]]
[[File:Magnetic field of wire loop.svg|thumb|The magnetic field lines of a current-carrying loop of wire pass through the center of the loop, concentrating the field there.]]
[[File:Elecmagnet.png|thumb|Magnetic field generated by passing a current through a coil]]
An electric current flowing in a wire creates a magnetic field around the wire, due to [[Ampere's circuital law|Ampere's law]] ''(see drawing of wire with magnetic field)''. To concentrate the magnetic field in an electromagnet, the wire is wound into a [[electromagnetic coil|coil]] with many turns of wire lying side-by-side.<ref name="Merzouki" /> The magnetic field of all the turns of wire passes through the center of the coil, creating a strong magnetic field there.<ref name="Merzouki" /> A coil forming the shape of a straight tube (a [[helix]]) is called a [[solenoid]].<ref name="Hyperphysics" /><ref name="Merzouki" />

The direction of the magnetic field through a coil of wire can be determined by the [[Right-hand rule#Electromagnetism|right-hand rule]].<ref>{{cite book
 |last1       = Millikin
 |first1      = Robert
 |first2      = Edwin
 |last2       = Bishop
 |title       = Elements of Electricity
 |publisher   = American Technical Society
 |year        = 1917
 |location    = Chicago
 |pages       = [https://archive.org/details/elementselectri00bishgoog/page/n140 125]
 |url         = https://archive.org/details/elementselectri00bishgoog
 }}</ref><ref>{{cite book
 |last        = Fleming
 |first       = John Ambrose
 |title       = Short Lectures to Electrical Artisans, 4th Ed
 |publisher   = E.& F. N. Spon
 |year        = 1892
 |location    = London
 |pages       = 38–40
 |url         = https://books.google.com/books?id=wzdHAAAAIAAJ&pg=PA38
 |url-status  = live
 |archive-url = https://web.archive.org/web/20170111023313/https://books.google.com/books?id=wzdHAAAAIAAJ&pg=PA38
 |archive-date = 2017-01-11
}}</ref> If the fingers of the right hand are curled around the coil in the direction of current flow ([[conventional current]], flow of [[positive charge]]) through the windings, the thumb points in the direction of the field inside the coil. The side of the magnet that the field lines emerge from is defined to be the ''north pole''.

=== Magnetic core ===
'''For definitions of the variables below, see box at end of article.'''

Much stronger magnetic fields can be produced if a [[magnetic core]], made of a [[soft magnetic material|soft]] [[ferromagnetic]] (or [[ferrimagnetic]]) material such as [[iron]], is placed inside the coil.<ref name="Hyperphysics">{{cite web
 |last        = Nave
 |first       = Carl R.
 |title       = Electromagnet
 |website     = Hyperphysics
 |publisher   = Dept. of Physics and Astronomy, Georgia State Univ.
 |date        = 2012
 |url         = http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
 |access-date = September 17, 2014
 |url-status  = live
 |archive-url = https://web.archive.org/web/20140922065602/http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/elemag.html
 |archive-date = September 22, 2014
}}</ref><ref name="Merzouki">{{cite book
 |last1       = Merzouki
 |first1      = Rochdi
 |last2       = Samantaray
 |first2      = Arun Kumar
 |last3       = Pathak
 |first3      = Pushparaj Mani
 |title       = Intelligent Mechatronic Systems: Modeling, Control and Diagnosis
 |publisher   = Springer Science & Business Media
 |date        = 2012
 |pages       = 403–405
 |url         = https://books.google.com/books?id=k81ECeMxyk8C&q=ferromagnetic+electromagnet&pg=PA404
 |isbn        = 978-1447146285
 |url-status  = live
 |archive-url  = https://web.archive.org/web/20161203041435/https://books.google.com/books?id=k81ECeMxyk8C&pg=PA404&dq=ferromagnetic+electromagnet#v=onepage&q=ferromagnetic%20electromagnet&f=false
 |archive-date = 2016-12-03
}}</ref><ref name="Gates">{{cite book
 |last1       = Gates
 |first1      = Earl
 |title       = Introduction to Basic Electricity and Electronics Technology
 |publisher   = Cengage Learning
 |date        = 2013
 |pages       = 184
 |url         = https://books.google.com/books?id=7vE7Esf3WVAC&q=electromagnet+ferromagnetic+electromagnet+coil&pg=PA184
 |isbn        = 978-1133948513
 |url-status  = live
 |archive-url  = https://web.archive.org/web/20170110211309/https://books.google.com/books?id=7vE7Esf3WVAC&pg=PA184&dq=electromagnet+ferromagnetic+electromagnet+coil
 |archive-date = 2017-01-10
}}</ref><ref name="Shipman">{{cite book
 |last1       = Shipman
 |first1      = James
 |last2       = Jerry
 |first2      = Wilson
 |last3       = Todd
 |first3      = Aaron
 |title       = Introduction to Physical Science
 |publisher   = Cengage Learning
 |edition     = 12
 |date        = 2009
 |pages       = 205–206
 |url         = https://books.google.com/books?id=PZs8AAAAQBAJ&q=electromagnet+ferromagnetic+solenoid+coil&pg=PA205
 |isbn        = 978-1111810283
 |url-status  = live
 |archive-url = https://web.archive.org/web/20170111023648/https://books.google.com/books?id=PZs8AAAAQBAJ&pg=PA205&dq=electromagnet+ferromagnetic+solenoid+coil
 |archive-date = 2017-01-11
}}</ref> A core can increase the magnetic field to thousands of times the strength of the field of the coil alone, due to the high [[Permeability (electromagnetism)|magnetic permeability]] <math>\mu</math> of the material.<ref name="Hyperphysics" /><ref name="Merzouki" /> Not all electromagnets use cores, so this is called a ''ferromagnetic-core'' or ''iron-core'' electromagnet.

This phenomenon occurs because the magnetic core's material (often iron or steel) is composed of small regions called [[magnetic domains]] that act like tiny magnets (see [[ferromagnetism]]). Before the current in the electromagnet is turned on, these domains point in random directions, so their tiny magnetic fields cancel each other out, and the core has no large-scale magnetic field. When a current passes through the wire wrapped around the core, its [[magnetic field]] penetrates the core and turns the domains to align in parallel with the field. As they align, all their tiny magnetic fields add to the wire's field, which creates a large magnetic field that extends into the space around the magnet. The core concentrates the field, and the magnetic field passes through the core in lower [[reluctance]] than it would when passing through air.

The larger the current passed through the wire coil, the more the domains align, and the stronger the magnetic field is. Once all the domains are aligned, any additional current only causes a slight increase in the strength of the magnetic field. Eventually, the field strength levels off and becomes nearly constant, regardless of how much current is sent through the windings.<ref name="Merzouki" /> This phenomenon is called [[Saturation (magnetic)|saturation]], and is the main nonlinear feature of ferromagnetic materials.<ref name="Merzouki" /> For most high-permeability core steels, the maximum possible strength of the magnetic field is around 1.6 to 2 [[Tesla (unit)|teslas]] (T).<ref name="Pauley">"''Saturation flux levels of various magnetic materials range up to 24.5 kilogauss''" (2.5 T) p.1  "''Silicon steel saturates at about 17 kilogauss''" (1.7 T) p.3  {{cite journal |last=Pauley |first=Donald E. |date=March 1996 |title=Power Supply Magnetics Part 1: Selecting transformer/inductor core material |url=http://www.arnoldmagnetics.com/WorkArea/DownloadAsset.aspx?id=4396 |url-status=dead |journal=Power Conversion and Intelligent Motion |archive-url=https://web.archive.org/web/20141224075136/http://www.arnoldmagnetics.com/WorkArea/DownloadAsset.aspx?id=4396 |archive-date=December 24, 2014 |access-date=September 19, 2014}}</ref><ref name="MagneticMaterials">The most widely used magnetic core material, 3% silicon steel, has saturation induction of 20 kilogauss (2 T). {{cite web |date=2013 |title=Material Properties, 3% grain-oriented silicon steel |url=http://ludens.cl/Electron/Magnet.html |url-status=live |archive-url=https://web.archive.org/web/20140920230119/http://ludens.cl/Electron/Magnet.html |archive-date=September 20, 2014 |access-date=September 19, 2014 |website=Catalog |publisher=Magnetic Materials Co. |page=16}}</ref><ref name="Short">"''Magnetic steel fully saturates at about 2 T''" {{cite book |last1=Short |first1=Thomas Allen |url=https://books.google.com/books?id=mVW2D_6XB5EC&pg=PA214 |title=Electric Power Distribution Handbook |date=2003 |publisher=CRC Press |isbn=978-0203486504 |pages=214}}</ref> This is why the very strongest electromagnets, such as superconducting and very high current electromagnets, cannot use cores.

When the current in the coil is turned off, most of the domains in the core material lose alignment and return to a random state, and the electromagnetic field disappears. However, some of the alignment persists because the domains resist turning their direction of magnetization, which leaves the core magnetized as a weak permanent magnet. This phenomenon is called [[hysteresis]] and the remaining magnetic field is called [[remanence|remanent magnetism]]. The residual magnetization of the core can be removed by [[degaussing]]. In alternating current electromagnets, such as those used in motors, the core's magnetization is constantly reversed, and the remanence contributes to the motor's losses.

===Ampere's law===
The magnetic field of electromagnets in the general case is given by [[Ampere's Law]]:

:<math>\int \mathbf{J}\cdot d\mathbf{A} = \oint \mathbf{H}\cdot d\boldsymbol{\ell}</math>

which says that the integral of the magnetizing field <math>\mathbf{H}</math> around any closed loop is equal to the sum of the current flowing through the loop. A related equation is the [[Biot–Savart law]], which gives the magnetic field due to each small segment of current.

=== Force exerted by magnetic field ===
Likewise, on the solenoid, the force exerted by an electromagnet on a conductor located at a section of core material is:
{{NumBlk|:|<math>F = \frac{B^2 A}{2 \mu_0} </math>|{{EquationRef|2}}}}

This equation can be derived from the [[Magnetic energy|energy stored in a magnetic field]]. [[Energy]] is force times distance. Rearranging terms yields the equation above.

The 1.6&nbsp;T limit on the field<ref name="Pauley" /><ref name="Short" /> previously mentioned sets a limit on the maximum force per unit core area, or [[magnetic pressure]], an iron-core electromagnet can exert; roughly:

:<math>\frac{F}{A} = \frac {(B_\text{sat})^2}{2 \mu_0}  \approx 1000\ \mathrm{kPa} = 10^6 \mathrm{N/m^2} = 145\ \mathrm{{lbf/in^2}}</math>
<!--(1.6**2)/(2*4*pi*10**-7)-->
for the core's saturation limit, <math>B_{sat}</math>. In more intuitive units, it is useful to remember that at 1&nbsp;T the magnetic pressure is approximately {{Convert|4|atm|kg/cm2}}.

Given a core geometry, the magnetic field needed for a given force can be calculated from ({{EquationNote|Eq. 2}}); if the result is much more than 1.6&nbsp;T, a larger core must be used. 

However, computing the magnetic field and force exerted by ferromagnetic materials in general is difficult for two reasons. First, the strength of the field varies from point to point in a complicated way, particularly outside the core and in air gaps, where ''fringing fields'' and ''[[leakage flux]]'' must be considered. Second, the magnetic field and force are [[Nonlinear system|nonlinear]] functions of the current, depending on the nonlinear relation between <math>B</math> and <math>\mathbf{H}</math> for the particular core material used. For precise calculations, computer programs that can produce a model of the magnetic field using the [[finite element method]] are employed.

=== Magnetic circuit ===
[[Image:Electromagnet with gap.svg|thumb|upright=1.7|Figure 1. Magnetic field (<span style="color:green;">green</span>) of a typical electromagnet, with the iron core ''C'' forming a closed loop with two air gaps ''G'' in it.<br /> 
''B'' – magnetic field in the core<br />
''B<sub>F</sub>'' – fringing fields; in the gaps ''G'', the magnetic field lines bulge out, so the field strength is less than in the core: ''B<sub>F</sub>''&nbsp;<&nbsp;''B''<br />
''B<sub>L</sub>'' – [[leakage flux]]; magnetic field lines which do not follow complete magnetic circuit<br />
''L'' – average length of the magnetic circuit (used in {{EquationNote|3|Eq. 3}}). It is the sum of the length ''L<sub>core</sub>'' in the iron core pieces and the length ''L<sub>gap</sub>'' in the air gaps ''G''.<br />
Both the leakage flux and the fringing fields get larger as the gaps are increased, reducing the force exerted by the magnet. ]]

In many practical applications of electromagnets, such as motors, generators, transformers, lifting magnets, and loudspeakers, the iron core is in the form of a loop or [[magnetic circuit]], possibly broken by a few narrow air gaps. Iron presents much less "resistance" ([[reluctance]]) to the magnetic field than air, so a stronger field can be obtained if most of the magnetic field's path is within the core.<ref name="Merzouki" /> Since the magnetic field lines are closed loops, the core is usually made in the form of a loop.

Since most of the magnetic field is confined within the outlines of the core loop, this allows a simplification of the mathematical analysis.<ref name="Merzouki" /> A common simplifying assumption satisfied by many electromagnets, which will be used in this section, is that the magnetic field strength ''<math>B</math>'' is constant around the magnetic circuit (within the core and air gaps) and zero outside it. Most of the magnetic field will be concentrated in the core material (''C'') (see Fig. 1). Within the core, the magnetic field (''B'') will be approximately uniform across any cross-section; if the core also has roughly constant area throughout its length, the field in the core will be constant.<ref name="Merzouki" />

At any air gaps (''G'') between core sections, the magnetic field lines are no longer confined by the core. Here, they bulge out beyond the core geometry over the length of the gap, reducing the field strength in the gap.<ref name="Merzouki" /> The "bulges" (''B<sub>F</sub>'') are called fringing fields.<ref name="Merzouki" /> However, as long as the length of the gap is smaller than the cross-section dimensions of the core, the field in the gap will be approximately the same as in the core.

In addition, some of the magnetic field lines (''B<sub>L</sub>'') will take "short cuts" and not pass through the entire core circuit, and thus will not contribute to the force exerted by the magnet. This also includes field lines that encircle the wire windings but do not enter the core. This is called [[leakage flux]].

The equations in this section are valid for electromagnets for which:
# the magnetic circuit is a single loop of core material, possibly broken by a few air gaps;
# the core has roughly the same cross-sectional area throughout its length;
# any air gaps between sections of core material are not large compared with the cross-sectional dimensions of the core;
# there is negligible leakage flux.

==== Magnetic field in magnetic circuit ====
The magnetic field created by an electromagnet is proportional to both the number of turns of wire <math>N</math> and the current <math>I</math>; their product, <math>NI</math>, is called [[magnetomotive force]]. For an electromagnet with a single [[magnetic circuit]], Ampere's Law reduces to:<ref name="Merzouki" /><ref>{{cite book
  | last = Feynman
  | first = Richard P.
  | title = Lectures on Physics, Vol. 2
  | publisher = Addison-Wesley
  | year = 1963
  | location = New York
  | pages = 36–9 to 36–11, eq. 36–26
  | isbn =  978-8185015842
  | url = https://www.feynmanlectures.caltech.edu/II_36.html#Ch36-S5
}}</ref><ref name="Fitzgerald">{{cite book
  | last1 = Fitzgerald
  | first1 = A.
  | last2 = Kingsley |first2=Charles |last3=Kusko |first3=Alexander
  | title = Electric Machinery, 3rd Ed
  | publisher = McGraw-Hill
  | year = 1971
  | location = USA
  | pages = 3–5}}</ref>

:<math>NI = H_{\mathrm{core}} L_{\mathrm{core}} + H_{\mathrm{gap}} L_{\mathrm{gap}}</math>
{{NumBlk|:|<math>NI = B \left(\frac{L_{\mathrm{core}}}{\mu} + \frac{L_{\mathrm{gap}}}{\mu_0} \right) </math>|{{EquationRef|3}}}}

This is a [[nonlinear equation]], because the permeability of the core <math>\mu</math> varies with <math>B</math>. For an exact solution, <math>\mu(B)</math> must be obtained from the core material [[Hysteresis loop|hysteresis curve]].<ref name="Merzouki" /> If <math>B</math> is unknown, the equation must be solved by [[Numerical analysis|numerical methods]].  

However, if the magnetomotive force is well above saturation (so the core material is in saturation), the magnetic field will be approximately the material's saturation value <math>B_{sat}</math>, and will not vary much with changes in <math>NI</math>. For a closed magnetic circuit (no air gap), most core materials saturate at a magnetomotive force of roughly 800&nbsp;ampere-turns per meter of flux path.

For most core materials, the relative permeability <math>\mu_r \approx 2000 \text{–} 6000\,</math>.<ref name="Fitzgerald" /> So in ({{EquationNote|3|Eq. 3}}), the second term dominates. Therefore, in magnetic circuits with an air gap, <math>B</math> depends strongly on the length of the air gap, and the length of the flux path in the core does not matter much. Given an air gap of 1mm, a magnetomotive force of about 796&nbsp;ampere-turns is required to produce a magnetic field of 1&nbsp;T.

==== Closed magnetic circuit ====
[[File: Lifting electromagnet cross section.png|thumb|Cross section of a lifting electromagnet, showing the cylindrical construction. The windings (''C'') are flat copper strips to withstand the [[Lorentz force]] of the magnetic field. The core is formed by the thick iron housing (''D'') that wraps around the windings.]]

For a closed magnetic circuit (no air gap), such as would be found in an electromagnet lifting a piece of iron bridged across its poles, equation ({{EquationNote|3|Eq. 3}}) becomes:

{{NumBlk|:|<math>B = \frac{NI\mu}{L} </math>|{{EquationRef|4}}}}

Substituting into ({{EquationNote|2|Eq. 2}}), the force is:

{{NumBlk|:|<math>F = \frac{\mu^2 N^2 I^2 A}{2\mu_0 L^2} </math>|{{EquationRef|5}}}}

To maximize the force, a core with a short flux path ''<math>L</math>'' and a wide cross-sectional area ''<math>A</math>'' is preferred (this also applies to magnets with an air gap). To achieve this, in applications like lifting magnets and [[loudspeaker]]s, a flat cylindrical design is often used. The winding is wrapped around a short wide cylindrical core that forms one pole, and a thick metal housing that wraps around the outside of the windings forms the other part of the magnetic circuit, bringing the magnetic field to the front to form the other pole.

=== Force between electromagnets ===
The previous methods are applicable to electromagnets with a [[magnetic circuit]]; however, they do not apply when a large part of the magnetic field path is outside the core. (A non-circuit example would be a magnet with a straight cylindrical core.) To determine the force between two electromagnets (or permanent magnets) in these cases, a special analogy called a ''magnetic-charge model'' can be used. In this model, it is assumed that the magnets have well-defined "poles" where the field lines emerge from the core, and that the magnetic field is produced by fictitious "magnetic charges" on the surface of the poles. This model assumes point-like poles (instead of surfaces), and thus it only yields a good approximation when the distance between the magnets is much larger than their diameter; thus, it is useful just for determining a force between them.

The magnetic pole strength ''<math>m</math>'' of an electromagnet is given by
<math display="block">m = \frac{NIA}{L}</math>
and thus the force between two poles is
<math display="block">F = \frac{\mu_0 m_1 m_2}{4\pi r^2}.</math>
Each electromagnet has two poles, so the total force on magnet 1 from magnet 2 is equal to the [[Vector (mathematics and physics)|vector]] sum of the forces of magnet 2's poles acting on each pole of magnet 1.