View source for Magnetic reluctance

{{Short description|Resistance to magnetic flux}}

{{Infobox physical quantity
| name        = Magnetic reluctance
| image       =
| caption     =
| unit        = H<sup>−1</sup>
| otherunits  =
| symbols     = <math>\mathcal{R}</math>, <math>\mathcal{S}</math>
| baseunits   =
| dimension   = '''M'''<sup>–1</sup> '''L'''<sup>–2</sup> '''T'''<sup>2</sup> '''I'''<sup>2</sup>
| extensive   =
| intensive   =
| conserved   =
| transformsas =
| derivations  = <math>\frac{1}{\mathcal{P}}</math>, <math>\frac{\mathcal F}{\Phi}</math>, <math>\frac{l}{\mu_0 \mu_r A}</math>
}}
{{Electromagnetism|Magnetic circuit}}

'''Magnetic reluctance''', or '''magnetic resistance''', is a concept used in the analysis of [[magnetic circuit]]s. It is defined as the ratio of [[magnetomotive force]] (mmf) to [[magnetic flux]].  It represents the opposition to magnetic flux, and depends on the geometry and composition of an object.

Magnetic reluctance in a magnetic circuit is analogous to [[electrical resistance]] in an [[Electrical network|electrical circuit]] in that resistance is a measure of the opposition to the [[electric current]].  The definition of magnetic reluctance is analogous to [[Ohm's law]] in this respect.  However, magnetic flux passing through a reluctance does not give rise to dissipation of heat as it does for current through a resistance.  Thus, the analogy cannot be used for modelling energy flow in systems where energy crosses between the magnetic and electrical domains.  An alternative analogy to the reluctance model which does correctly represent energy flows is the [[gyrator–capacitor model]].

Magnetic reluctance is a [[scalar (physics)|scalar]] [[Intensive and extensive properties#Extensive properties|extensive quantity]]. The unit for magnetic reluctance is inverse [[Henry (unit)|henry]], H<sup>−1</sup>.

==History==
The term ''reluctance'' was coined in May 1888 by [[Oliver Heaviside]].<ref>Heaviside O. (1892) [https://archive.org/details/electricalpapers02heavrich Electrical Papers, Vol 2] – L.; N.Y.: Macmillan, p. 166</ref> The notion of "magnetic resistance" was first mentioned by [[James Joule]] in 1840.<ref>Joule J. (1884) [https://archive.org/details/scientificpaper05kelvgoog Scientific Papers, vol 1], p.36</ref> The idea for a [[magnetic flux]] law, similar to [[Ohm's law]] for closed [[electric circuit]]s, is attributed to [[Henry Augustus Rowland]] in an 1873 paper.<ref>{{cite journal|title=XIV. On magnetic permeability, and the maximum of magnetism of iron, steel, and nickel|author=Rowland, Henry A.|journal=Philosophical Magazine |series=Series 4|volume=46|year= 1873 |issue =304|pages =140–159|doi=10.1080/14786447308640912}}</ref> Rowland is also responsible for coining the term ''magnetomotive force'' in 1880,<ref>Rowland, Henry A, [https://www.jstor.org/stable/2369444 "On the general equations of electro-magnetic action, with application to a new theory of magnetic attractions, and to the theory of the magnetic rotation of the plane of polarization of light"] ([https://www.jstor.org/stable/2369465 part 2]), ''American Journal of Mathematics'', vol. 3, nos. 1–2, pp. 89–113, March 1880.</ref> also coined, apparently independently, a bit later in 1883 by Bosanquet.<ref>{{cite journal|doi=10.1080/14786448308628457|title=XXVIII.On magnetomotive force|journal=Philosophical Magazine |series=Series 5|volume=15|issue=93|pages=205–217|year=1883|last1=Bosanquet|first1=R.H.M.|url=https://zenodo.org/record/1431167}}</ref>

Reluctance is usually represented by a [[cursive]] capital <math> \mathcal R</math>.

== Definitions ==
In both AC and DC fields, the reluctance is the ratio of the [[magnetomotive force]] (MMF) in a [[magnetic circuit]] to the [[magnetic flux]] in this circuit.  In a pulsating DC or AC field, the reluctance also pulsates (see [[phasor (sine waves)|phasors]]).

The definition can be expressed as follows:
<math display="block">\mathcal{R} = \frac{\mathcal{F}}{\Phi}</math>
where
* <math> \mathcal{R}</math> ("R") is the reluctance in [[ampere-turn]]s per [[weber (unit)|weber]] (a unit that is equivalent to turns per [[henry (unit)|henry]]).  "[[Turn (geometry)|Turns]]" refers to the [[winding number]] of an electrical conductor comprising an inductor.
* <math> \mathcal{F}</math> ("F") is the [[magnetomotive force]] (MMF) in ampere-turns
* Φ ("Phi") is the [[magnetic flux]] in webers.

It is sometimes known as [[Hopkinson's law]] and is analogous to [[Ohm's Law]] with resistance replaced by reluctance, voltage by MMF and current by magnetic flux.

''[[Permeance]]'' is the inverse of reluctance:
<math display="block">\mathcal P = \frac{1}{\mathcal R}</math>

Its [[SI]] derived unit is the [[henry (unit)|henry]] (the same as the unit of [[inductance]], although the two concepts are distinct).

Magnetic flux always forms a closed loop, as described by [[Maxwell's equations]], but the path of the loop depends on the reluctance of the surrounding materials.  It is concentrated around the path of least reluctance.  Air and vacuum have high reluctance, while easily magnetized materials such as [[soft iron]] have low reluctance. The concentration of flux in low-reluctance materials forms strong temporary poles and causes mechanical forces that tend to move the materials towards regions of higher flux so it is always an attractive force (pull).

The reluctance of a uniform magnetic circuit can be calculated as:
<math display="block">\mathcal{R} = \frac{l}{\mu_0 \mu_r A} = \frac{l}{\mu A}</math>

where
* ''l'' is the length of the circuit in [[metre]]s
* <math> \mu_0</math> is the permeability of vacuum, equal to  <math display="inline">4 \pi \times 10^{-7} \mathrm{\frac{H}{m}} </math> (or, <math display="inline"> \mathrm{\frac{kg \cdot m}{A^2 \cdot s^2}} </math> = <math display="inline"> \mathrm{\frac{s \cdot V}{A \cdot m}} </math> = <math display="inline"> \mathrm{\frac{J}{A^2 \cdot m}} </math>)
* <math> \mu_r</math> is the relative [[magnetic permeability]] of the material (dimensionless)
* <math> \mu</math> is the permeability of the material (<math> \mu = \mu_0 \mu_r</math>)
* ''A'' is the cross-sectional area of the circuit in [[square metre]]s

== Applications ==
{{uncited section|date=January 2024}}
* Constant air gaps can be created in the core of certain transformers to reduce the effects of [[Saturation (magnetic)|saturation]]. This increases the reluctance of the magnetic circuit, and enables it to store more [[energy]] before core saturation.  This effect is also used in the [[flyback transformer]].
* Variable air gaps can be created in the cores by a movable keeper to create a flux switch that alters the amount of magnetic flux in a magnetic circuit without varying the constant [[magnetomotive force]] in that circuit.
* Variation of reluctance is the principle behind the [[reluctance motor]] (or the variable reluctance generator) and the [[Alexanderson alternator]]. Another way of saying this is that the ''reluctance forces'' strive for a maximally aligned magnetic circuit and a minimal air gap distance.
* [[Multimedia]] [[loudspeaker]]s{{clarify|date=January 2024}} are typically shielded magnetically, in order to reduce magnetic interference caused to [[television]]s and other [[Cathode ray tube|CRT]]s.  The speaker magnet is covered with a material such as [[soft iron]] to minimize the stray magnetic field.

Reluctance can also be applied to:
* [[Reluctance motor]]s
* Variable reluctance (magnetic) [[Pick up (music technology)|pickups]]
* [[Magnetic capacitance]]
* [[Magnetic circuit]]
* [[Magnetic complex reluctance]]

==References==
{{reflist}}

{{DEFAULTSORT:Magnetic Reluctance}}
[[Category:Electric and magnetic fields in matter]]
[[Category:Magnetic circuits]]