Encyclopedia Magnetica™

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====== Magnetic field ======

|< 100% >|
| //[[user/Stan Zurek]], Magnetic field, Encyclopedia Magnetica//, \\ @PAGEL@  |

**Magnetic field** - a region in space, in which [[magnetic force|magnetic forces]] are observable. Magnetic forces are mechanical forces acting as a result of magnetic interactions. 

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Magnetic field //B// acting with a [[magnetic force]] //F<sub>m</sub>// on a positive [[electric charge]] //q// moving with [[velocity]] //v//, direction of //F<sub>m</sub>// is perpendicular to both //B// and //v//
[[file/lorentz_force_magnetic_magnetica.png|{{lorentz_force_magnetic_magnetica.png}}]]
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Magnetic field is always generated around [[electric current]], or more generally by moving [[electric charge|electric charges]] or by varying [[electric field]] **//E//**, as well as [[intrinsic]] [[magnetic dipole moment|magnetic dipole moments]] of [[sub-atomic particle|sub-atomic particles]].[(Britannica>[[http://www.britannica.com/EBchecked/topic/357048/magnetic-field|Magnetic field, Encyclopaedia Britannica, {accessed 27 Nov 2013}]])][(Jiles>[[http://google.com/books?isbn=9780412798603|David C. Jiles, Introduction to Magnetism and Magnetic Materials, 2nd edition, CRC Press, 1998, ISBN 9780412798603]])][(Spaldin>[[https://books.google.co.uk/books?isbn=9780521886697|Nicola A. Spaldin, Magnetic Materials, Fundamentals and Applications, Second edition, Cambridge University Press, 2011, ISBN 9780521886697]])] The existence of magnetic field is responsible for [[magnetism]] and [[electromagnetism]]. 

In vacuum, a force **//F//** ([[Lorentz force]]) acting on an electric charge //q// moving with velocity **//v//** is due to the electric field **//E//** and magnetic field **//B//** ([[magnetic flux density]]): $\vec{F} = q·\vec{E} + q·\vec{v} × \vec{B}$ 

The field which satisfies the action of the **//B//** vector in the Lorentz force equation is assumed to be the **definition of magnetic field**.[(Purcell>[[https://isbnsearch.org/isbn/9781107014022|E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022]])][(Griffiths>[[http://books.google.com/books?isbn=0321856562|David J. Griffiths, Introduction to electrodynamics, 4th ed., Pearson, Boston, 2013, ISBN 0321856562]])]

In physics, a [[field]] is a quantity or value assigned to each point in space or volume. Therefore, magnetic field can be mathematically described by assigning physical quantity to each point in space filled with such field. Depending on the required mathematical treatment the description can be a [[scalar field]] of [[vector field]], also being a function of position or time.[([[http://google.com/books?isbn=9780080522968|Alan Jeffrey, Advanced Engineering Mathematics, Academic Press, 2001, ISBN 9780080522968, p. 626]])]

In order to fully quantify the effects of magnetic field in a given material or medium, it is practical to use two quantities such as **magnetic flux density //B//** and **[[magnetic field strength]] //H//**.

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[[Iron filings]] attracted in magnetic field of a [[horseshoe magnet]]
[[file/magnet_with_iron_filings_magnetica_jpg|{{magnet_with_iron_filings_magnetica.jpg}}]]
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[[Iron filings]] follow [[magnetic field line|magnetic field lines]] around a single wire with current (perpendicular to the page)
[[file/Magnetic field lines around a straight conductor_png|{{Magnetic field lines around a straight conductor.png}}]]
//<sup>by Popular Science Montly, Vol. 56, 1899-1900, Public Domain</sup>//
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[[Compass needle|Compass needles]] follow the direction of [[magnetic field]] (//**B**// or //**H**//) around the wire with current //**I**// ([[right-hand rule]])
[[file/magnetic_field_around_wire_e-m_jpg|{{magnetic_field_around_wire_e-m.jpg}}]]
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===== Different approaches to magnetic fields, B and H =====

Both //**B**// and //**H**// are strictly defined in terms of measurement units as well as their physical and technical meaning.[([[http://www.bipm.org/en/si/derived_units/2-2-1.html|Derived units expressed in terms of base units, SI brochure, {accessed 6 Jan 2014}]])][([[http://www.bipm.org/en/si/si_brochure/chapter2/2-2/table3.html|Units with special names and symbols; units that incorporate special names and symbols, SI brochure, {accessed 6 Jan 2014}]])][(Jiles)] However, there are two main approaches to the analysis, usefulness and importance of //B// and //H// magnetic fields: **theoretical physics** and **engineering**.

If the same system of units is used, e.g. [[SI system of units|SI]], then both approaches are exactly equivalent, in terms of the numerical values and units. Only the names or problematic, not the units or mathematical calculations.[(Purcell)]

These two approaches are complementary, giving access to a type of equation which is more useful for a given calculation. One is not "better" than the other, simply because it is considered more "fundamental" by one group of people. The "quality" of either form of definition is only as good as its usefulness for a specific purpose, such as calculation of effects with relativistic phenomena, or designing an efficient [[transformer]] or [[electromagnet]].

However, theoretical physics puts more emphasis on //B//, whereas in engineering both //H// and //B// are treated as equally important. Naming conventions can differ significantly. Additionally, the previously used [[CGS units|CGS system of units]] (still used by some countries and branches of science) has yet another set of names and coefficients. All this creates an often confusing mix, with many books on electromagnetisms giving full list of conversion factors between the different systems.

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| {{/wiki/logo.png?20&nolink}} //See also: [[Confusion between B and H]].// |

In particular attention has to be paid to some equations, because they might be written in a given form which requires certain implicit conditions to be met - these conditions are often not stated.

For instance, the [[Biot-Savart law]] written in the physics form $B = f (μ_0 · I)$ is valid //only// in [[vacuum]], whereas the engineering form $H = f (I)$ is more general, and valid for any [[uniform]] [[isotropic]] medium.
===== Magnetic field in theoretical physics =====
In the theoretical approach more emphasis is put on the derivation of equations from first principles, and usefulness of a given expressions for deriving other equations. Microscopic (atomic and sub-atomic level) understanding is fundamental, but between and inside the atoms of any matter there is vacuum. Ferromagnetic materials receive limited attention or not at all[(Tong)], with the explanations following those of the engineering approach.[(Purcell)][(Griffiths)]

The units are used observing all the usual physical and mathematical rules. However, the relationship between the units and the names of the variables appear to be treated less rigorously, with [[tesla]] and [[A per m|A/m]] used almost interchangeably, for //B//, //H// and //M//, using the [[magnetic constant]] $μ_0$ as a conversion tool.
==== Magnetic field B ====
From theoretical physics viewpoint, the only form of magnetic field required for all analysis is the //**magnetic field B**//, which is regarded as "indisputably fundamental".[(Griffiths)][(Purcell)] The definition of //B// is a field which causes the force described by equation (1). 

Depending on type of notation in various literature, either the notation with bold fonts denoting vectors (equation (1a)) or with the arrows above the variables (equation (1b)) can be used - both are precisely equivalent.

| (1a)  |  $$  \mathbf{F} = q·\mathbf{E} + q·\mathbf{v} × \mathbf{B}  $$  |  \\ (N)  |
| (1b)  |  $$  \vec{F} = q·\vec{E} + q·\vec{v} × \vec{B}  $$  |  :::  |
| where: $q$ - electric charge (C), $E$ - electric field (V/m), $v$ - velocity (m/s) of the charge, $B$ - magnetic field (T)  |||

Force due to electric field acts in the direction of that field ([[dot product]] "·" in equations (1ab)). Force due to magnetic field acts perpendicularly to both the direction of movement of the charge and the direction of magnetic field ([[cross product]] "×" in equations (1ab)).

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[[Lorentz force]] acting on a positive charged particle //q//: a) force //F<sub>e</sub>// in [[electric field]] //E//, b) force //F<sub>m</sub>// in [[magnetic field]] //B//, c) total force //F = F<sub>e</sub>// + //F<sub>m</sub>// in both fields; green solid arrow shows velocity and direction of movement, green dotted arrow how the trajectory develops
[[/file/lorentz_force_magnetica_png|{{/lorentz_force_magnetica.png}}]]
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Other names for //B// such as "magnetic flux density" and "magnetic induction" are even regarded as "absurd choice".[(Griffiths)] The quantity //H// is treated as "auxiliary" and having "no name".[(Purcell)][(Griffiths)] Some theoreticians even regard the SI units as "arbitrary and a little daft".[(Tong>[[http://www.damtp.cam.ac.uk/user/tong/em/el1.pdf|David Tong, Electromagnetism, University of Cambridge Part IB and Part II Mathematical Tripos, Lent Term, 2015]], {accessed 2020-10-16})] Other systems of units can be used for theoretical calculations, such as [[Lorentz-Heaviside units|Lorentz-Heaviside]] or [[natural units]], which can be defined with [[permeability of vacuum]] or [[speed of light]] as unity (unitless), which can simplify certain calculations.[(Tong)]

Magnetic field //B// is generated by [[electric current]] and intrinsic [[magnetic dipole moment|magnetic dipole moments]] of subatomic particles.

Magnetic force, as defined by the [[Lorentz force]] is proportional to //B//, and such force is taken as the //definition// of flux density //B//. The full form of [[Maxwell equations|Maxwell's equation]] always shows //B//, not //H//. This is because the [[divergence]] of //B// is always zero, whereas it can be non-zero for //H//.[(Purcell)]

On a microscopic level, between and inside the atoms, there is vacuum and //B// can be used for all the calculations. 

If SI units are used then //B// has the dimension of [[tesla]].

==== Magnetisation M ====

Magnetic properties of matter are a consequence of macroscopic averaging of microscopic effects, especially due to vector summation of [[magnetic dipole moment|magnetic dipole moments]] of subatomic particles (and atoms), which can align to the magnetic field applied to the material. The material responds with magnetisation //M// when exposed to some magnetic field //B//. If complete alignment of all free magnetic moments is achieved then the material is said to be [[magnetic saturation|magnetically saturated]], and value of //M// cannot increase any more.[(Purcell)]

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[[Magnetisation]] //M// is the sum of [[magnetic dipole moment|magnetic dipole moments]] per unit of volume //V//
[[/file/volume_magnetisation_magnetica_png|{{volume_magnetisation_magnetica.png}}]]
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**//Magnetisation M//** is //defined//[(Purcell)][(Griffiths)] as the sum of magnetic moments per unit volume, within a given material, as per equation (2).

|  (2)  |   $$  \vec{M} \equiv \frac{\text{magnetic moment}}{\text{unit volume}} \equiv \frac{ \sum_i \vec{m_i} }{V}  $$   |  (A·m<sup>2</sup> / m<sup>3</sup>) \\ ≡ (A/m)  |
| where: $\vec{m}$ - magnetic dipole moment (A·m<sup>2</sup>), $i$ - counting index (unitless), $V$ - unit volume (m<sup>3</sup>)  |||

In many physics research papers the magnetisation is expressed as [[mass magnetisation]], with the response measured by a unit of mass, rather than a unit of volume, with the popular unit of [[emu per gram|emu/g]].[([[https://doi.org/10.1038/s41598-021-87786-8|Pavithra, C.L.P., Janardhana, R.K.S.K., Reddy, K.M. et al. An advancement in the synthesis of unique soft magnetic CoCuFeNiZn high entropy alloy thin films. Sci Rep 11, 8836 (2021)]])][([[https://doi.org/10.1063/1.4943242|Rajasekhar Madugundo, Ozlem Koylu-Alkan, and George C. Hadjipanayis, "Bulk Mn-Al-C permanent magnets prepared by various techniques" , AIP Advances 6, 056009 (2016)]])][([[https://doi.org/10.1063/1.4945335|T. M. Pekarek, P. S. Edwards, T. L. Olejniczak, C. Lampropoulos, I. Miotkowski, and A. K. Ramdas, "Magnetic properties of the layered III-VI diluted magnetic semiconductor Ga1−xFexTe" , AIP Advances 6, 056222 (2016)]])]

The conversion factor to SI units is: 1 emu/g = 1 A⋅m<sup>2</sup>/kg, and [[density]] of the material can be used to convert it to volume magnetisation.[(IEEE_units>[[https://www.ieeemagnetics.org/images/stories/magnetic_units.pdf|IEEE Magnetics, MAGNETIC UNITS]], {accessed 2021-05-01})]
==== Polarisation J ====
Magnetisation //M// (A/m) is sometimes also called "magnetic polarisation".[(Purcell)][(Griffiths)]

In research papers written by theoretical physicist, if magnetisation //M// is used to express a quantity in measurable in the units of [[tesla]] (rather than [[A per m|A/m]]), then the value is simply multiplied by $μ_0$ (permeability of vacuum) or stated directly in teslas (with an implicit assumption that it is multiplied by $μ_0$).[([[https://doi.org/10.1038/srep01455|Zhang, Y., Zuo, T., Cheng, Y. et al. High-entropy Alloys with High Saturation Magnetization, Electrical Resistivity and Malleability. Sci Rep 3, 1455 (2013)]])]

Even in some excellent books on theoretical [[electromagnetism]] (e.g. references [(Purcell)][(Griffiths)]) the polarisation //J// is not mentioned (and hence not defined).  

However, [[magnetic polarisation]] //J// can be defined and calculated as: $J = μ_0 · M$, with the unit of tesla.[(IEEE_units)]

==== Auxiliary field H ====
From a [[macroscopic]] viewpoint, there are additional effects because [[atom|atoms]] interact with each other magnetically, and it is recognised that an "auxiliary" field //H// can be defined, and although "illogical" it can be "useful", because it can be directly related to [[free current|free currents]], such that a direct relationship between the current and //H// can be stated.[(Purcell)]

Therefore, the **auxiliary field //H//** is //defined//[(Purcell)][(Griffiths)] as in equation (3).

|  (3)  |   $$  \vec{H} \equiv \frac{ \vec{B} }{μ_0} - \vec{M}  $$   |  (A/m)  |
| where: $B$ - magnetic field (T), $μ_0$ - [[permeability of vacuum]] (H/m), $M$ - magnetisation (A/m)  |||

Even if SI units are used, sometimes //H// is expressed in teslas.[([[https://doi.org/10.1063/1.4754582|Y. Wu, X. L. Dong, H. X. Yang, X. J. Wang, X. J. Huang, J. Q. Li, F. Zhou, X. J. Zhou, and Z. X. Zhao, "Magnetic transition in α-NaCuPO4 with Cu-O chains" , AIP Advances 2, 032172 (2012)]])][([[https://doi.org/10.1063/1.4812696|Ji Chen, Mansoor Bin Abdul Jalil, and Seng Ghee Tan, "Spin torque on the surface of graphene in the presence of spin orbit splitting" , AIP Advances 3, 062127 (2013)]])]  

==== Susceptibility and permeability ====

**[[Magnetic susceptibility]] //χ//** can be defined in more than one way.[(Purcell)] For materials which respond linearly to the applied magnetic field, with the assumption that //M// is proportional to //B// or the //H//, volume susceptibility can be defined as:[(Purcell)]

|  (4a)  |   $$  \vec{M} = χ_{(B)} · \frac{ \vec{B} }{μ_0}  $$   |  (A/m)  |
|  (4b)  |   $$  \vec{M} = χ_{(H)} · \vec{H}  $$   |  (A/m)  |

For [[diamagnets]] susceptibility is negative, for [[paramagnets]] it is positive. For vacuum it is zero.

These two definitions (4a, 4b) differ so that:

|  (5)  |   $$  χ_{(H)} = \frac{χ_{(B)}}{1 - χ_{(B)}}  $$   |  (unitless)  |

For [[non-magnetic materials]] $χ << 1$ and the difference in negligible,[(Purcell)] and $χ = χ_{(H)} \approx χ_{(B)}$.

However, for large values of $χ$, as encountered in [[ferromagnetic|ferromagnets]], the definitions (4a,b) are less useful, because (5) diverges, and [[permeability]] is used instead of susceptibility. 

**[[Magnetic permeability]] //μ//** is defined in relation to susceptibility:

|  (6a)  |  $$ μ \equiv μ_0 · (1 + χ)  $$   |  (H/m)  |
|  (6b)  |  $$ μ = μ_0 · μ_r  $$   |  (H/m)  |
| where: $μ_r$ is the [[relative permeability]] (unitless)  |||

and can be used as the proportionality factor between //B// and //H//:

|  (7)  |  $$ \vec{B} = μ · \vec{H} = μ_0 · μ_r · \vec{H} $$   |  (T)  |

However, there are limitations with this equation, because in ferromagnets $μ_r$ is a non-linear function of //B// or //H//.

===== Magnetic field in engineering =====

In the engineering approach, less emphasis is put on the fundamental physical concepts, and more on the properties directly measurable quantities, useful for practical applications and design of devices. Magnetic quantities are linked to the [[macroscopic]] electrical quantities which can be directly measured or controlled in [[electromagnetic device|electromagnetic devices]], such as electric [[voltage]] and [[current]].

This leads to a different concept of excitation-response relationship and the naming convention, as defined by the international [[SI units|SI system of units]].[(BIPM_2019>[[https://www.bipm.org/utils/common/pdf/si-brochure/SI-Brochure-9-EN.pdf|Bureau International des Poids et Mesures, The International System of Units (SI), 9th edition, 2019]], {accessed 2021-04-10})]

 
==== Magnetic field strength H ====

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| {{/wiki/logo.png?20&nolink}} //See also the main article: [[Magnetic field strength]].// |

**[[Magnetic field strength]] //H//** is measured in [[A per m|amperes per metre]], A/m (or scaled by the [[SI multipliers]]: kA/m, MA/m, etc.) 

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[[Biot-Savart law]]: $P$ - point at which the magnetic field strength $\vec H$ is calculated, $R$ - perpendicular distance from conductor, $I$ - current, $\vec J$ - [[current density]], $d \vec l$ - [[infinitesimal]] length of conductor contributing to magnetic field, $r$ - distance,  $\vec r$ - path vector, $\hat r$ - [[unit vector]] along direction of $r$, $\theta$ - angle between $d \vec l$ and $r$
[[file/biot-savart_law_magnetica_png|{{biot-savart_law_magnetica.png}}]]
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[[Electric current|Electric currents]] are the source of //H//. The proportionality is defined by the length of path under consideration ([[magnetic path length]]), and the total amount of current //I//, contained in a single conductor, or a number //N// of conductors. The calculation of //H// can be based either on the [[Biot-Savart law]] or on [[Ampere circuital law|Ampere's circuital law]], with both approaches giving identical results for the same configuration.[(Jiles)][(Spaldin)]

From Biot-Savart law:[(Tumanski>[[https://isbnsearch.org/isbn/9780367864958|Sławomir Tumański, Handbook of magnetic measurements, CRC Press / Taylor & Francis, Boca Raton, FL, 2011, ISBN 9780367864958]])]

|  (8)  | $$ d \vec H = \frac{1}{4·π}·\frac{I·(d \vec l \times \hat r)}{r^2}$$ |  (A/m)  |
| $I$ - current (A), $d \vec l$ - [[infinitesimal]] length (m) of conductor contributing to magnetic field, $\hat r$ - [[unit vector]] along direction of $r$ (unitless), $r$ - radial distance (m)  |||

In an alternative approach, an integral over a closed path of the [[tangential component]] of //H// gives the value of the enclosed free current.[(Purcell)] From Ampere's circuital law:

|  (9)  | $$ \int_C \vec{H} · d \vec{l}  = I $$ |  (A)  |
| where: //C// - closed path over which the integral is calculated |||

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Electric current //**I**// generates around itself [[magnetic field strength]] //**H**//
[[file/Electric_current_generates_magnetic_field_Magnetica_jpg|{{Electric_current_generates_magnetic_field_Magnetica.jpg}}]]
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As shown above, both equations can be formulated so that the magnetic permeability does not appear - magnetic properties of matter (or permeability of vacuum) are not relevant for generation of //H// (strictly applicable only in a [[uniform]] and [[isotropic]] [[medium]], regardless if it is non-magnetic, magnetic, non-linear, etc.)[(Jiles)]

Equation (9) can be rearranged so that the [[electric current]] //I// produces around itself well-defined [[magnetic field strength]] //H//, whose amplitude is independent of the type of magnetised material. For a uniform medium (no additional sources of magnetic field, such as [[magnetic poles]] due to [[demagnetising factor]]) there is a direct proportionality between //I// and //H// (e.g. instantaneous change of //I// produces instantaneous change of //H//, assuming changes negligibly slow with the speed of light):

|  (10)  | $$ H  = \frac{N · I}{l} $$ |  (A/m)  |
| where: $N$ - number of [[strand|strands]] in a wire or [[turn|turns]] in a [[coil]] (unitless), $I$ - current (A), $l$ - magnetic path length (m)   |||

For example, a current of 1 A in a straight wire, produces //H// circulating around the wire (according to the [[right-hand rule]]), such that at a radius of //r// = 0.1 m (from the centre of the wire), related to the circle with path length $l = 2·π·r$ = 0.628 m, which is equal to around //H// = 1.59 A/m (see also: [[calculator/magnetic_field_of_straight_conductor|Calculator of magnetic field of straight round conductor]]).

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Simple [[magnetic circuit]]: [[core]] and [[air gap]]
[[/file/Air gap diagram_png|{{/Air gap diagram.png}}]]
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If the medium is not isotropic, or there are several different materials then the field distribution is affected by [[demagnetising field]] caused by [[magnetic pole|magnetic poles]]. These poles become new sources of magnetic field which change distribution of the original applied field.[(Jiles)]

The product //N·I// is also known as **[[magnetomotive force]]** (MMF), which can be thought of as the applied excitation, and the component //H·l// represents the drop of MMF in a [[magnetic circuit]] (as an analogy to [[voltage drop]] in an electric circuit). If the [[magnetic circuit]] comprises a high-permeability [[magnetic core]] and an [[air gap]] (both of uniform [[cross-sectional area]]) then equation (10) can be applied as:

|  (11a)  |  $$ H_{core}·l_{core} + H_{gap}·l_{gap}  = N · I $$  |  (A) or \\ (//ampere-turns//)  |
|  //but $H_{core}·l_{core} \ll H_{gap}·l_{gap}$ so://  ||| 
|  (11b)  | $$ H_{gap}·l_{gap}  \approx N · I $$ |  (A) or \\ (//ampere-turns//)  |

and it is widely used for design of [[electric motor|electric motors]], [[electromagnet|electromagnets]], [[magnetic sensor|magnetic sensors]], [[flyback transformer|flyback transformers]], etc. 
 

In [[CGS units]] //H// is measured in [[oersted|oersteds]] (Oe), with the conversion factor: 1 Oe = 1000 / (4⋅π) A/m ≈ 79.58 A/m.[(IEEE_units)]


==== Magnetic flux density B ====

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| {{/wiki/logo.png?20&nolink}} //See also the main article: [[Magnetic flux density]].// |

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In [[ferromagnet|ferromagnets]] the //**B-H**// relationship is [[non-linearity|non-linear]], [[hysteresis|history-dependent]] and typically [[anisotropy|anisotropic]]
[[file/M4_B-H_loop_50Hz_png|{{M4_B-H_loop_50Hz.png}}]]
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**[[Magnetic flux density]]** //**B**// is a response of a medium when magnetised with //**H**//.[(Heaviside>[[https://macsphere.mcmaster.ca/bitstream/11375/14746/1/fulltext.pdf|Oliver Heaviside, Electromagnetic theory, Vol. I, 1893, Ernest Benn Limited]])] The amplitude of //**B**// depends both on the applied excitation as well as the material properties (and the [[magnetic history]] of the material, in case of prior magnetisation of [[ferromagnet|ferromagnets]]).

As mentioned above, //H// can be quantified with respect to the magnetising current (but direct measurement methods are also used, such as [[H-coil]]). //B// is measured by using the [[Faraday's law of induction]] (one of the [[Maxwell equations|Maxwell's equations]]), which can be written as:[(Tumanski)][(Zurek)]

|  (12a)  |  $$EMF(t) = - N · A · \frac{dB}{dt} $$  |  (V)  |
|  (12b)  |  $$V(t) = N · A · \frac{dB}{dt} $$  |  (V)  |
|  (12c) \\ <sub>(for sine)</sub>  |  $$V_{rms} =  π · \sqrt{2} · f ·B_p · N · A $$  |  (V)  |
| where: //EMF(t)// - [[electromotive force]] (V), //V(t)// - measured voltage (V), //N// - number of turns of the sensing [[coil]], //A// - active cross-sectional area (m<sup>2</sup>), //V<sub>rms</sub>// - [[RMS]] voltage (V), //B<sub>p</sub>// - peak of the sinusoidal flux density (T)  |||

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Illustration of relationship in a [[ferromagnet]] between [[magnetic field strength]] //H//, [[magnetic flux density]] //B//, [[magnetisation]] //M//, and [[magnetic polarisation]] //J// 
[[file/magnetic_field_flux_density_magnetisation_ferromagnet_e-m_png|{{magnetic_field_flux_density_magnetisation_ferromagnet_e-m.png}}]]
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Therefore, the value of //B// (or a continuous waveform) can be derived calculating an integral of the measured voltage (by an analogue circuit with an [[integrating opamp]], or by digital calculations of a sampled signal). 

The difference of sign in both equations arises, because (12a) describes [[electromotive force]] (EMF) induced internally in the given sensing coil ([[B-coil]]), but this information is not available in any way - only the external voltage (12b) can be measured. This difference is often implicitly assumed, but sometimes even misunderstood, showing the minus sign even for measured voltages.[(Zurek>[[https://isbnsearch.org/isbn/9780367891572|S. Zurek, Characterisation of Soft Magnetic Materials Under Rotational Magnetisation, CRC Press, 2019, ISBN 9780367891572]])]

Because equation (12a,b) defines the relationship by means of a derivative, it is not possible to measure a [[DC component]] in an as easy was as the [[AC component]]. Some variability is required, which can be achieved by moving the coil ([[ballistic measurement]], [[vibrating coil magnetometer]]) or moving the sample ([[vibrating sample magnetometer]]).

For energy transformation it is very useful to utilise the [[AC excitation]]. If a full [[cycle]] of excitation is applied, and both [[waveform|waveforms]] of //H// and //B// are measured then a [[B-H loop]] ([[hysteresis loop]]) can be plotted, as shown in the image.

The area of the loop is proportional to all the energy lost during magnetisation of a material, and it is a fundamental tool by which total [[magnetic loss|magnetic losses]] are measured in a material. All components of magnetic loss are automatically included ([[hysteresis]], [[eddy currents]], [[excess loss]]) which is why it is so important practically to accurately measure both: the magnetic field strength //H// and the magnetic flux density //B//. Equation (12c) is the basis for design of all power transformers operating with sinusoidal voltage.[(Zurek)]

There are several internation standards defining such loss measurement procedures for the industrially important [[soft magnetic materials]], such as [[electrical steels]].[(Tumanski)][(Zurek)]

==== Permeability and susceptibility ====

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| {{/wiki/logo.png?20&nolink}} //See also the main article: [[Magnetic permeability]].// |

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Permeabilities of magnetic materials (e.g. [[ferromagnets]] $μ_{ferro}$) are much higher than [[non-magnetic material|non-magnetic materials]] (e.g. [[permeability of free space|vacuum]] $μ_0 = μ_{vac}$, [[diamagnets]] $μ_{dia}$ and [[paramagnets]] $μ_{para}$)
[[file/permeabilities_-_magnetica_png|{{permeabilities_-_magnetica.png}}]]
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**[[Magnetic permeability]] //μ//** is the quantity expressing the proportionality factor between //H// and //B//. [[Soft magnetic materials]] are pivotal to efficient [[energy transformation]], so they receive a lot of attention, and the permeability value is used as a [[figure of merit]]. For any such material it can be written that:

|  (13a)  |  $$B = μ · H$$  |  \\ (T)  |
|  (13b)  |  $$B = μ_r · μ_0 · H$$  |  :::  |
| where: $μ = μ_r · μ_0$ (H/m), and therefore [[relative permeability]] $μ_r = μ / μ_0$ (unitless)  |||

For vacuum, the //**B-H**// relationship is fully linear, and $μ_0$ = 4·π·10<sup>-7</sup> (H/m) which is a universal physical constant,[(BIPM_2019)] and $μ_r = 1$. In practice, for most [[non-magnetic materials]] $μ_r$ is so close to unity that the difference is neglected.[(Zurek)]

Permeability of a medium different than vacuum can be defined as a product of dimensionless number relative to $μ_0$. Hence, $μ = μ_r · μ_0$. So for instance, [[relative permeability]] $μ_r$ = 100 means that the given material has permeability 100 times higher than the value of $μ_0$ (as measured in H/m). This means that for the same applied //H// the material will respond with //B// that is 100 times greater than it would be for vacuum (or non-magnetic material).

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Ratio //B// / //H// at a given point of [[magnetisation curve]] can be used for calculating permeability value
[[file/permeability_of_ferromagnet_-_magnetica_png|{{permeability_of_ferromagnet_-_magnetica.png}}]]
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For uniform and isotropic materials permeability can be expressed as [[scalar]], even though //**H**// and //**B**// are [[vector|vectors]]. For [[anisotropic material|anisotropic materials]] permeability is different in various directions and [[vector]] or [[tensor]] approach has to be used for more correct representation. However, in practice the scalar approach is often used because of its simplicity (and sufficiently accurate results for most practical purposes, such as design of [[magnetic circuit|magnetic circuits]]).

When excitation //**H**// varies in time then additional effects are introduced, like for instance [[eddy current|eddy currents]], which affect the [[apparent permeability]]. The [[effective permeability]] is also affected by [[shape anisotropy]]. In [[magnetic material|magnetic material]] like [[ferromagnet|ferromagnets]] the relationship between //**B**// and //**H**// can be [[non-linear]], [[history-dependent]] and highly [[anisotropic]].

There are many different [[types of permeability]] which can be derived from measured B-H characteristics, as needed for a specific practical application: relative, amplitude, reversal, impulse, etc. The calculation is always performed as a ratio $μ = f (ΔB/ΔH)$, with the delta Δ defining the reference point (e.g. origin of the graph, position of the previous impulse, etc.)

Soft magnetic materials saturate if sufficiently high excitation is applied. This causes the ratio of B/H and hence also the permeability to reduce. When approaching to full saturation relative permeability will approach unity (i.e. the same as for vacuum). 

For [[hard magnetic materials]] the value of permeability is of little use, and it is not discussed with the same depth, or even omitted in some books.[(Leonowicz>[[http://books.google.com/books?isbn=8320430496|Marcin Leonowicz, Jerzy J. Wysłocki, Modern magnets, Technologies, Coercivity mechanisms, Applications (in Polish: Współczesne magnesy, Technologie, Mechanizmy koercji, Zastosowania), Wydawnictwa Naukowo-Techniczne, Warsaw, 2005, ISBN 8320430496]])]

**Susceptibility** is rarely used in engineering applications.


===== Quantities and units =====

[[Electric charge|Electric charges]] (in the form of [[electron|electrons]] and [[proton|protons]]) are the most basic components responsible for all macroscopic electrical phenomena. However, it is far more useful in practical electrical engineering to operate with such values as [[voltage]] $V$ (related to concentration and distribution of electric charges) and [[electric current]] $I$ (representing movement of electric charges).  

Combination of these two values fully quantifies the effects of [[electricity]], such as dissipated [[power]] //P//, because $P = V·I$. Additional parameters such as [[resistance]] //R// define proportionality of electrical response or V-I relationship ([[Ohm law|Ohm's law]]) $I = V / R$ for a given material. [[Conductance]] //G// (reciprocal of [[resistance]], $G = 1 / R$) can be used to express the equation in a different, but equivalent form: $I = G·V$.

By analogy, in engineering, both [[magnetic flux density]] $B$ and [[magnetic field strength]] $H$ (or their equivalents) are required for quantifying the effects of [[magnetism]] in matter, for example stored or lost energy.[(White>[[http://google.com/books?isbn=9781439866511|Mary Anne White, Physical Properties of Materials, Second Edition, CRC Press, 2011, ISBN 9781439866511, p. 359]])] These two quantities are defined separately in the [[International System of Units]] (SI), with $H$ having the unit of [[ampere per metre]] (A/m), and $B$ the unit of [[tesla]] (T). [([[http://www.bipm.org/en/si/derived_units/2-2-1.html|Derived units expressed in terms of base units, SI brochure, {accessed 6 Jan 2014}]])][([[http://www.bipm.org/en/si/si_brochure/chapter2/2-2/table3.html|Units with special names and symbols; units that incorporate special names and symbols, SI brochure, {accessed 6 Jan 2014}]])] 

[[Energy density]] in a given magnetic material can be calculated as $E_{density}=B·H$. [[Permeability]] //μ// defines the proportionality of a magnetic response: $B = μ·H$. These relations are widely used in the engineering approach.

Magnetic field is a concept of energy contained in a given volume of space. This energy can be physically and mathematically described by various means or quantities. There are six most frequently used magnetic quantities, which are interrelated, as shown in the table below.

<WRAP 100% lo>
|< 100% 19% 19% 19% 19% 12% 12% >|
^  General field  ^^  Field related to magnetised matter  ^^  Material property  ^^
|  $B$ (T) \\ [[magnetic flux density]] \\ (//magnetic induction, magnetic field//)   |  $H$ (A/m) \\ [[magnetic field strength]] \\ (//magnetic field, magnetic field intensity//)  |  $J$ (T) \\ [[magnetic polarisation]] \\ (//intrinsic flux density, intrinsic induction, ferric induction//)   |  $M$ (A/m) \\ [[magnetisation]] \\ (//magnetic polarisation//)  |  $μ_r$ (unitless) \\ [[relative magnetic permeability|relative permeability]]   |  $χ$ (unitless) \\ [[magnetic susceptibility|susceptibility]]   |
|  in matter:  ||||||
|  $$B = μ_r · μ_0 · H$$  |  $$H = \frac{B}{μ_0} - M$$  |  $$J = μ_0 · M$$  |  $$M = χ · \frac{B}{μ_0}$$  |  $$μ_r = χ + 1$$  |  $$χ = \frac{μ_0·M}{B}$$  |
|  $$B = μ_0 · H + μ_0 · M$$ $$B = μ_0 · H + J$$  |    |  $$J = B - μ_0 · H$$  |  $$M = \frac{B}{μ_0} - H$$  |  $$μ_r = \frac{B}{μ_0 · H}$$  |  $$χ = μ_r - 1$$  |
|  in vacuum:  ||||||
|  $$B = μ_0 · H$$  |  $$H = \frac{B}{μ_0}$$  |  $$J = 0$$  |  $$M = 0$$  |   $$μ_r = 1$$  |   $$χ = 0$$  |
| $μ_0$ = 4·π·10<sup>-7</sup> (H/m) is the [[permeability of vacuum]], $μ = μ_r · μ_0$  (H/m) is the [[permeability]] of the given material  ||||||
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===== Energy of magnetic field =====

Magnetic field contains energy whose density per volume can be calculated with the following equation (for a medium with a constant permeability).[(Cullity>[[https://books.google.co.uk/books?isbn=9780471477419|B.D. Cullity, C.D. Graham, Introduction to Magnetic Materials, 2nd edition, Wiley, IEEE Press, 2009, ISBN 9780471477419]])]

|  (14a)  |  $$E_{vol} = \frac{μ_r · μ_0 · H^2}{2}$$  |  \\ (J/m<sup>3</sup>)  |
|  (14b)  |  $$E_{vol} = \frac{B · H}{2}$$  |  :::  |
|  (14c)  |  $$E_{vol} = \frac{B^2}{2· μ_r · μ_0}$$  |  :::  |
| units: //H// (A/m), //B// (T) |||

The energy and power dissipated in a magnetically soft material can be calculated by integrating over a full cycle of [[AC excitation]]:[(Zurek)]

|  (15a)  |  $$W_{vol} = \int_0^T \left( \frac{dB}{dt} · H \right)  dt $$  |  (J/m<sup>3</sup>)  |
|  (15b)  |  $$P_{vol} = f · \int_0^T \left( \frac{dB}{dt} · H \right)  dt $$  |  (W/m<sup>3</sup>)  |
|  (15c)  |  $$P_{mass} = \frac{f}{D} · \int_0^T \left( \frac{dB}{dt} · H \right)  dt $$  |  (W/kg)  |
| where: $T$ - period (s) of AC excitation, $f$ - frequency (Hz), $D$ - density of material (kg/m<sup>3</sup>), \\  units: //H// (A/m), //B// (T)  |||


The temporary energy storing capability can be utilised with a non-magnetic material, e.g. in the [[air gap]] of a [[choke]] or a [[flyback transformer]]. In such a magnetic circuit, if $B_{material} \approx B_{gap}$ (which is typically the case for a small air gap) then the value of $H$ will differ by the factor of a relative permeability of of the material, so the air gap can store significantly more energy than the magnetic core. 

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Optimum operating point //P// for a [[permanent magnet]] is where the product //B// and //H// reaches maximum //BH<sub>max</sub>//[(Leonowicz)]
[[/file/permanent_magnet_bhmax_curve_magnetica_png|{{permanent_magnet_bhmax_curve_magnetica.png}}]]
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In [[permanent magnet|permanent magnets]], the theoretical maximum amount of energy density which can be obtained for a given material is proportional to the square of the [[remanence]] value of $B_r = J_r$. In practice only around 60% of this value is achievable.[(Leonowicz)]

|  (16)  |  $$E_{BH_{max}} = (B·H)_{max} = \frac{{B_r}^2}{4 · μ_0} = \frac{{J_r}^2}{4 · μ_0}$$  |  \\ (J/m<sup>3</sup>)  |

The optimum operating point //P// for a [[permanent magnet]] is where the product //B// and //H// reaches a maximum //BH<sub>max</sub>//.[(Leonowicz)][(Hilzinger>[[https://isbnsearch.org/isbn/9783895783524|Rainer Hilzinger, Werner Rodewald, Magnetic Materials, Fundamentals, Products, Properties, Applications, VAC Vacuumschmelze, Publicis, 2013, ISBN 9783895783524]])]

===== Difficulty with definition of magnetic field =====
It is difficult to give a simple definition of magnetic or [[electromagnetic field]].

Progress of science and technology is based on some quantities which so far cannot be defined precisely. For example, the fundamental [[electric charge]] //**q**// or [[time]] //**t**// are referred to without defining what they are. Scientists can describe, but still cannot explain what exactly is time or electric charge. The assumption is that they exist, have some physical meaning and are measurable within the given [[system of units]].[([[http://google.com/books?isbn=9781139451925|Bhag Singh Guru, Hüseyin R. Hiziroglu, Electromagnetic Field Theory Fundamentals, Cambridge University Press, 2004, ISBN 9781139451925, p. 1]])]

Magnetic field can be produced by electric current, which itself is defined as the flow of electric charge in time. It is empirically known that magnetic field acts with mechanical force on moving charged particles in specific ways described by [[Maxwell equations|Maxwell's equations]]. Therefore, the acting of such specific mechanical forces ultimately defines the presence or absence of magnetic field. 

Or in other words, the detection of magnetic field can be reduced to analysis of such mechanical forces acting on moving electrically charged particles. This is the reason for the opening definition in this article which refers only to magnetic forces (mechanical forces caused by magnetic field). Such definition seems somewhat circular, but as stated above due to the insofar inexplicable nature of underlying physics it cannot be made more precise.

For these reasons many books and publications on magnetism either do not give definition at all or use similar definition as given above in this article. Typical examples are quoted in the table below.

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|< 100% 10% 30% 30% 30%>| 
^ Publication ^ Definition of //**[[magnetic field]]**// ^ Definition of //**[[magnetic field strength]]**// $H$ ^ Definition of //**[[magnetic flux density]]**// $B$ ^
| Richard M. Bozorth \\ **Ferromagnetism**[([[http://google.com/books?isbn=0780310322|Richard M. Bozorth, Ferromagnetism, IEEE Press, 2003, ISBN 0780310322, p. 1-3]])] | //A magnet will attract a piece of iron even though the two are not in contact, and this action-at-a-distance is said to be caused by the magnetic field, or field of force.// | //The strength of the field of force, the magnetic field strength, or magnetizing force H, may be defined in terms of magnetic poles: one centimeter from a unit pole the field strength is one oersted.// | //Faraday showed that some of the properties of magnetism may be likened to a flow and conceived endless //lines of induction// that represent the direction and, by their concentration, the flow at any point. [...] The total number of lines crossing a given area at right angles is the flux in that area. The flux per unit area is the flux density, or //magnetic induction//, and is represented by the symbol B.// |
| David C. Jiles \\ **Introduction to Magnetism and Magnetic Materials**[(Jiles)] | //One of the most fundamental ideas in magnetism is the concept of the magnetic field. When a field is generated in a volume of space it means that there is a change of energy of that volume, and furthermore that there is an energy gradient so that a force is produced which can be detected by the acceleration of an electric charge moving in the field, by the force on a current-carrying conductor, by the torque on a magnetic dipole such as a bar magnet or even by a reorientation of spins of electrons within certain types of atoms.// | //There are a number of ways in which the magnetic field strength H can be defined. In accordance with the ideas developed here we wish to emphasize the connection between the magnetic field H and the generating electric current. We shall therefore define the unit of magnetic field strength, the ampere per meter, in terms of the generating current. The simplest definition is as follows. The ampere per meter is the field strength produced by an infinitely long solenoid containing //n// turns per metre of coil and carrying a current of 1///n// amperes.// | //When a magnetic field H has been generated in a medium by a current, in accordance with Ampere's law, the response of the medium is its magnetic induction B, also sometimes called the flux density.// |
| **Magnetic field**, **Encyclopaedia Britannica**[(Britannica_Field>[[https://www.britannica.com/science/magnetic-field|Magnetic field, Encyclopædia Britannica Online, {accessed 2020-03-07}]])] | //Magnetic field, region in the neighbourhood of a magnetic, electric current, or changing electric field, in which magnetic forces are observable.// | //The magnetic field H might be thought of as the magnetic field produced by the flow of current in wires [...]//[(Britannica_H>There was no separate entry titled "magnetic field strength" at the time of writing this article {accessed 2020-03-07}. The brief definition was included in the entry [[https://www.britannica.com/science/magnetic-field|Magnetic field, Encyclopædia Britannica Online, {accessed 2020-03-07}]].)]  | //[...] the magnetic field B [might be thought of] as the total magnetic field including also the contribution made by the magnetic properties of the materials in the field.//[(Britannica_B>There was no separate entry titled "magnetic flux density" at the time of writing this article {accessed 2020-03-07}. The brief definition was included in the entry [[https://www.britannica.com/science/magnetic-field|Magnetic field, Encyclopædia Britannica Online, {accessed 2020-03-07}]].)]  |
| E.M. Purcell, D.J. Morin, **Electricity and magnetism**[(Purcell)] | //This interaction of currents and other moving charges can be described by introducing a magnetic field. [...] We propose to keep on calling $\mathbf{B}$ the magnetic field.// | //If we now //define// a vector function $\mathbf{H}(x, y, z)$ at every point in space by the relation $$ \mathbf{H} \equiv \frac{\mathbf{B}}{μ_0} - \mathbf{M} $$ [...] As for $\mathbf{H}$, although other names have been invented for it, we shall call it //the field// $\mathbf{H}$, or even //the magnetic field// $\mathbf{H}$.// | //[...] any moving charged particle that finds itself in this field, experiences a force [...] given by $$ \mathbf{F} = q·\mathbf{E} + q·\mathbf{v} × \mathbf{B} $$ [...] We shall take the equation as the definition of $\mathbf{B}$.// |
</WRAP>

===== Electromagnetic and magnetostatic field =====
The name **[[electromagnetic field]]** is used to refer to a field continuously variable in time. It is said that variable electric field (moving electric charges) produces variable magnetic field, and in turn, variable magnetic field produces variable electric field. This result in electromagnetic field, in which the electric and magnetic components are inseparably interlinked.[([[http://google.com/books?isbn=9788120342422|Mehta Neeraj, Applied Physics for Engineers, PHI Learning Pvt. Ltd., ISBN 9788120342422, p. 541]])]

However, if for instance the magnetic field is generated by a [[permanent magnet]] then the resulting field does not vary with time so the electric field component is not generated. Such field is then referred to as **[[magnetostatic field]]** (constant in time) in order to explicitly distinguish it from "electromagnetic field" (which is variable in time and coexists with the variable electric field component).

The distinction is similar as in analysis of electrical circuits with [[electrostatic]] effects.[([[http://google.com/books?isbn=9780470131534|Julius Adams Stratton, Electromagnetic Theory, John Wiley & Sons, 2007, ISBN 9780470131534, p. 225]])] If there are no changes of charge distribution (i.e. there are no currents flowing) then the analysis is said to be electrostatic, to distinguish it from other cases.

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[[Magnetic field]] around a moving [[electron]]; the distribution of magnetic field can be calculated from [[Biot-Savart law]] assuming that the velocity if much smaller than [[speed of light]][(Maxfield>[[https://books.google.co.uk/books?isbn=9780080949666|C. Maxfield et al., Electrical Engineering: Know It All, Newnes, 2011, ISBN 9780080949666]], p. 1004)]
[[file/biot-savart_electron_magnetica_png|{{biot-savart_electron_magnetica.png}}]]
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In its narrow meaning, the name "**magnetic field**" refers only to the magnetostatic field or the magnetic component of the electromagnetic field. 

However, in a wider sense "magnetic field" is also used to refer to electromagnetic field. This is because electromagnetic field contains the **magnetic field component** in which the magnetic forces are "observable" as per definition of magnetic field (see also the table above, with the quotes of various definitions). The narrower or wider meaning is typically clear from the context, if not explicitly defined.

A moving electric charge has attached to it a magnetic field, whose distribution follows the Biot-Savart law. Such magnetic is called [[velocity field]] and does not radiate away if the charge moves at a constant velocity. Electrostatic field extends away to infinity, from either static or moving charge.

However, when the charge gets accelerated (or decelerated), for the whole duration of acceleration, both the electric and magnetic field surrounding the charge gets distorted. The ripple comprises both components of field, and radiates away from the charge, carrying with it some energy.

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Diagram illustrating generation of [[electromagnetic field]] by an [[electric charge]]: a static charge (grey small circle at the centre) generates [[electrostatic field]] (blue area) which statically extends away into space. A sudden acceleration of the charge (dark blue small circle) creates an electromagnetic pulse (red ring) which [[radiation|radiates]] away into space at the [[speed of light]], and the space far away still contains the electrostatic field from the time when the charge was stationary (as indicated by grey lines). A charge moving at a constant velocity //v// generates electric and magnetic field attached with the charge (green area). The [[field lines]] (black lines) show the direction and intensity of electric field.[(Purcell)]
[[file/electromagnetic_field_magnetica_png|{{electromagnetic_field_magnetica.png}}]]
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If the charge is moved in a sinusoidal manner then the radiation also takes a sinusoidal character, providing a basis for wireless transmission of information (equivalent to some amount of energy) through the volume of space.

Reception of electromagnetic waves is the same process but in reverse - the electric and magnetic components of the field can move the charges in the antenna, and thus produce voltage and current, which can be then processed by an appropriate electronic circuitry, to recover the information ([[wireless communication]]) or energy ([[wireless charging]]). [[Resonance|Resonating LC circuits]] are used to improve efficiency of transmission-reception.

The concepts of [[near field]] and [[far field]] are introduced in analysis of electromagnetic fields. All magneto//static// effects can be treated as occurring in the near field region.[([[http://google.com/books?isbn=9780387201566|Nathan Ida, Engineering Electromagnetics, Springer Science & Business Media, 2004, ISBN 9780387201566, p. 1149]])]

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[[Electromagnetic wave]] with [[wavelength]] //λ//, propagating in direction //x//, comprising the component of [[electric field]] //E// [[in phase]] with the [[magnetic field]] //B//; all three directions (//E//, //B// and //x// are orthogonal to each other[(Purcell)]
[[file/electromagnetic_wave_magnetica_png|{{electromagnetic_wave_magnetica.png}}]]
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==== Medium of magnetic and electromagnetic field ====
Static magnetic field penetrates matter, with the exception of [[superconductor|superconductors]] which can expel the field from their inside (//B//=0, //H//>0, //M//<0), by sustaining [[surface current|surface currents]].[(Cullity)]

Mathematical equations can be used to describe electromagnetic field either as a wave ([[electromagnetic wave|electromagnetic waves]]) or as particles (e.g. [[photon|photons]]), and both representations were proved to be correct from a specific experimental point of view. If electromagnetic field is a wave then it is unknown in what medium such wave should exist, because it propagates equally well through [[vacuum]], which does not contain any matter. 

In [[quantum field]] theories such as [[quantum chromodynamics]] (QCD) the absence of [[elementary particle|elementary particles]] does not necessarily have to mean the lack of any medium. For instance, it is possible to analyse the effect of electromagnetic field on the vacuum in various energy regimes. As a result of such analysis the vacuum is no longer "free space" and theoretically particles can be continuously created and annihilated.[([[http://google.com/books?isbn=9781109199352|Elizabeth Werbos, Vacuum Properties of QCD in an Electromagnetic Field, University of Maryland, College Park. Physics, ISBN 9781109199352, p. 1-32]])]  
Such property can be therefore responsible for allowing the electromagnetic energy to propagate in a seemingly "empty" medium. 

In quantum field theory, electrostatic forces between two charges can be thought of as arising due to exchange of virtual photons, which are "fluctuations" of empty space.[(Nafousi>[[https://medium.com/encounter-in-nineveh/virtual-particles-including-virtual-photons-4111bee7430b|Mahmoud Nafousi, Virtual particles including virtual photons, https://medium.com / virtual particles]], {accessed 2021-05-15})]
==== Magnetic field lines ====

|< 100% >|
| {{/wiki/logo.png?20&nolink}} //See also the main article: [[Magnetic field lines]].// |

[[Magnetic field lines]] are a theoretical concept and there are no "lines" in reality. However, this concept is very useful in visualising the distribution of [[magnetic flux]] or [[electric field]]. The lines represent direction which is tangent to the direction of the given vector quantity, such as [[magnetic flux density]] at a given location.

If [[iron]] particles are placed in the vicinity of a magnet then they will form elongated structures which follow such lines extending between the [[magnetic pole|magnetic poles]]. In a similar way, a [[magnetic compass]] placed next to a magnet or electromagnet will have its needle following the contours created by magnetic particles. 

<WRAP clear></WRAP>

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[[FEM]] simulation illustrating imaginary [[magnetic field lines]] around a [[bar magnet]]
[[file/magnetic_field_lines_bar_magnet_magnetica_png|{{magnetic_field_lines_bar_magnet_magnetica.png}}]]
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[[Iron filings]] create elongated line-like structures around a [[permanent magnet]]
[[file/bar_magnet_with_iron_filings_e-m_jpg|{{bar_magnet_with_iron_filings_e-m.jpg}}]]
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"Magnetic field lines" around an [[electromagnet]], as indicated by small [[compass|compasses]]
[[file/compasses_and_coil_magnetica_jpg|{{compasses_and_coil_magnetica.jpg}}]]
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<WRAP clear></WRAP>

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"Magnetic field lines" extend into 3D space around a magnet
[[file/iron_powder_on_a_magnet_jpg|{{iron_powder_on_a_magnet.jpg}}]]
//<sup>by JanDerChemiker, [[https://creativecommons.org/licenses/by/3.0|CC-BY-3.0]]</sup>//
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For this reason they can be also called "magnetic force lines" and can be used to visualise the magnetic field around a given source, when plotted on a given cross-section (two-dimensional plane). The density of the lines (e.g. number of lines per unit area) represents the amplitude (more lines closer to each other means greater amplitude). The lines can never cross and each line is a closed curve (i.e. has no beginning or end). The lines are a theoretical concept and thus can extend over infinite lengths. 

From purely theoretical view point the magnetic field lines can extend to infinity, which would imply that a single moving electron could produce magnetic field extending over the whole universe - and if [[electromagnetic radiation]] takes place the [[electromagnetic field]] can indeed propagate vast distances, throughout the universe.

However, the intensity of magnetic field reduces with the distance from the source. Therefore, in practice the fields become vanishingly small at sufficiently long distances from the given source.[(Beeteson>[[http://google.com/books?isbn=9780120847310|John Stuart Beeteson, Visualising magnetic fields, Numerical equations solvers in action, Academic Press, 2001, ISBN 9780120847310, p. 7-9]])]

Similar concept is used for [[electric field lines]], but they start at positive charges and end at negative ones (electric field lines have sources, magnetic field lines do not).

==== Magnetic dipole and moment ====

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Magnetic dipole moment $\vec m$ of a [[current loop|loop of current]] $\vec I$ with the loop area $A$ and [[unit vector]] $\vec a$ (normal to $A$)
[[file/magnetic_moment_magnetica_png|{{magnetic_moment_magnetica.png}}]]
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A circular loop with current is one of the simplest circuits which generates magnetic field. At large distance from such loop the field is similar to the field produced by a **[[magnetic dipole]]** of two hypothetical [[magnetic monopole|magnetic monopoles]] with opposing charges separated by a given distance (per analogy to a dipole created by two [[electric charge|electric charges]]). 

**[[Magnetic moment]]** will act on a magnetic dipole placed in an external magnetic field. The acting torque will be proportional to the [[cross product]] of the two vectors: magnetic moment and flux density.[(Jiles)]

Magnetic moments are important in analysis of magnetic phenomena, because an electron orbit in an atom is in some sense analogous to a loop of current - hence having a magnetic moment associated with it. This analogy is the basis of explanation of [[diamagnetism]], in which the orbital "loop currents" of electrons are affected by the externally applied magnetic field.[(Cullity)]
==== Magnetic poles ====

The simplest magnetic dipole is created by a loop of current. The field lines entering the loop from one side constitute [[south magnetic pole]] and those leaving the loop on the other side create the opposing [[north magnetic pole]], so that the lines point from north to south magnetic pole. 

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[[Magnetic pole|Magnetic poles]] are created for any loop of [[electric current]] \\
[[file/magnetic_poles_convention_magnetica_jpg|{{magnetic_poles_convention_magnetica.jpg}}]] \\
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[[Bar magnet]] with marked [[magnetic poles]] N and S
[[file/bar_magnet_jpg|{{bar_magnet.jpg}}]]
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The [[right-hand rule]] defines the direction and sense of vectors and the accepted convention is that when looking at a current loop with current flowing anticlockwise then this is the north pole. At the same time, when looking from the other end, the current will appear to flow in clockwise direction and there will be the south pole.[(Jiles)]

The convention is also such that the end of the compass needle which points north is assumed to be the north pole itself. Because opposite poles attract, then at the geographic North pole of Earth there is a magnetic south pole. This is just a consequence of the asumed naming convention.[(Jiles)]

Magnetic poles can be created at different locations due to the presence of [[magnetic material]]. Due to [[magnetic induction]] the material will be [[magnetised]] and it will itself become a source of magnetic field, a simple example being a [[bar magnet]].[(Rosen>[[http://google.com/books?isbn=9781438110134|Joe Rosen, Encyclopedia of Physics, Infobase Publishing, 2009, ISBN 9781438110134, p. 198-200]])]

The location of the magnetic poles depends on the geometry of the [[magnetic circuit]], its magnetic history, distribution of sources of [[magnetomotive force]], and a [[demagnetising factor]] of the circuit.


[[Magnetic monopole|Magnetic monopoles]] were theoretically proposed by Paul Dirac in 1931 as a possible explanation of [[electric charge]] quantisation.[([[http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1062609|Trower, W., Magnetic detection of magnetic monopoles, IEEE Transactions on Magnetics, vol. 19 (5), 1983, p. 2061]])] So far, all the experiments failed at directly detecting the existence of magnetic monopoles.[(Rosen)][(Ray>[[http://www.nature.com/nature/journal/v505/n7485/full/nature12954.html|M. W. Ray, E. Ruokokoski, S. Kandel, M. Möttönen, D. S. Hall, Observation of Dirac monopoles in a synthetic magnetic field, Nature, 505, Jan 2014, p. 657]])]




===== Type of magnetisms =====
By definition, magnetic effects are caused by magnetic field and all such phenomena are referred to as [[magnetism]]. The name "[[electromagnetism]]" is used interchangeably with "magnetism" since the changes of electric field always create magnetic field.

However, the word "magnetism" also has several different meanings. All the matter built from atoms exhibits some type of magnetic behaviour, of which there are several fundamental types, resulting from the type of chemical elements involved, atomic and crystal structure, temperature, etc.[(Jiles)] They all arise because of interaction of intrinsic magnetic dipoles in atoms, due to the magnetic field:
  * [[diamagnetism]]
  * [[paramagnetism]]
  * [[ferromagnetism]]
  * [[ferrimagnetism]]
  * [[antiferromagnetism]]
  * [[superparamagnetism]]
  * and other, of less engineering importance: [[superferromagnetism]], [[superdiamagnetism]], [[helimagnetism]], [[metamagnetism]], [[mictomagnetism]], [[spin glass]], [[spin ice]]


===== Maxwell's equations =====

|< 100% >|
| {{/wiki/logo.png?20&nolink}} //See also the main article: [[Maxwell equations|Maxwell's equations]].// |


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|< 100%  40% 60% >|
^  Maxwell's equations in differential form[(Purcell)]  ^^
|  [[Gauss law for electrostatics|Gauss's law for electrostatics]]  |  $$ \text{div } \mathbf{E} = \frac {\rho_{charge}}{\epsilon_0}$$  |
|  [[Gauss law for magnetism|Gauss's law for magnetism]]  |  $$ \text{div } \mathbf{B} = 0$$  |
|  [[Faraday law of electromagnetic induction|Faraday's law of electromagnetic induction]]  |  $$ \text{curl } \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$  |
|  [[Ampere circuital law|Ampère's circuital law]]  |  $$ \text{curl } \mathbf{B} = \mu_0 ·  \mathbf{J} + \mu_0 · \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t}$$  |
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[[Maxwell's equations]] fully describe mathematically the interrelation between electric and magnetic fields.
The early version of these equations were first collated by a Scottish physicist [[James Clerk Maxwell]]. Subsequently they were unified and expressed in vector notation by Oliver Heaviside, so that today four fundamental equations are used, whose physical meaning can be summarised as follows:[([[http://www.britannica.com/EBchecked/topic/370686/Maxwells-equations|Maxwell's equations, Encyclopaedia Britannica, {accessed 18 Dec 2013}]])]
  * [[Gauss law for electrostatics|Gauss's law for electrostatics]][(Jain>[[http://google.com/books?isbn=9788120340602|Mahesh C. Jain, Textbook Of Engineering Physics, Part 2, PHI Learning Pvt. Ltd., p. 4.42-4.70, ISBN 9788120340602]])] relates distribution of electric charge to electric field,
  * [[Gauss law for magnetism|Gauss's law for magnetism]][(Jain)] states that there are no [[magnetic monopole|magnetic monopoles]],
  * [[Faraday law of electromagnetic induction|Faraday's law of electromagnetic induction]][(Jain)] states that electric fields are produced by varying magnetic fields,
  * [[Ampere circuital law|Ampère's circuital law]][([[http://google.com/books?isbn=9780471746225|Dipak L. Sengupta, Valdis V. Liepa, Applied Electromagnetics and Electromagnetic Compatibility, John Wiley & Sons, 2005, ISBN 9780471746225, p. 28]])] states that magnetic fields are produced by electric currents or changing electric fields.

The equations can be mathematically written in many ways (e.g. differential or integral form) or different units (e.g. [[CGS]] or [[MKS]]). They can also be formulated on the basis of more fundamental theory of [[quantum electrodynamics]].

In vacuum the equations simplify, because there are no charges, no currents and no material properties.
===== Generation of magnetic field =====

There are three main sources of magnetic field, all inseparably linked with the electric charges and their properties: 
  * electric current (macroscopic)
  * magnetic moments of sub-atomic particles (intrinsic magnetic dipole moments such as [[electron spin moment]])
  * emission of photons (electromagnetic radiation) due to de-excitation of electrons in atoms. 

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[[Magnetic force]] due to magnetic field, acting on [[magnetic poles]] (position of the hanging magnet is deflected accordingly), from left to right: like poles repel, opposite poles attract, nail ([[soft ferromagnet|soft ferromagnetic material]]) gets magnetised and is attracted to either pole of a magnet, the force on [[non-magnetic material|non-magnetic materials]] such as plastic and copper (without [[electric current]]) is typically negligible, [[electromagnetic coil]] with current can repel or attract the magnet (depending on the polarity of current), and the force on [[non-magnetic stainless steel]] (e.g. type 316) is negligibly small 
[[file/bar_magnets_polarity_n_magnetica_jpg|{{bar_magnets_polarity_n_magnetica.jpg}}]]
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==== By electric current ====

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[[Compass|Compasses]] detect magnetic field inside and around a [[coil]] with electric current 
[[file/right-hand_rule_compass_magnetica_jpg|{{right-hand_rule_compass_magnetica.jpg}}]]
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Macroscopically, [[generation of magnetic field]] is associated with unbalanced movement of electric charges. For example, in a conductor the positive charges remain bound to mostly stationary atoms, but electrons can freely move. 

When voltage is applied the electrons drift and generate magnetic field extending over large distances around such current. Each single electron, and each single moving electric charge (positive or negative) generates magnetic field around itself.

In practical applications, the combined amount of current can be increased by using multiple [[turn|turns]] in a [[winding]]. For example, a current at milliampere level can be made having equivalent effect to several amperes by using sufficient number of turns (this is the meaning of //N// in equation (10)).

All electric currents generate magnetic field, and even extremely weak electric signals send in nervous system and neurons generate detectable magnetic fields. Such activity can by detected by [[SQUID]] devices so that [[magnetoencephalography]] can be carried out.[([[http://google.com/books?isbn=9780123668691|Hubert Preissl, Magnetoencephalography, Gulf Professional Publishing, 2005, ISBN 9780123668691, p. xi]])]

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Current in a wire can be measured with a [[current transformer]] by detecting the magnetic field around the wire
[[file/iclamp_closed_magnetica_jpg|{{iclamp_closed_magnetica.jpg}}]]
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Lower current can be used to generate higher magnetic field, by using multiple turns of wire (as in the coil of this [[relay]])
[[file/relay_250v_8a_12v_magnetica_jpg|{{relay_250v_8a_12v_magnetica.jpg}}]] 
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[[Electromagnet]] - a source of magnetic field
[[file/Electromagnet_200mm_by_Magneto_jpg|{{Electromagnet_200mm_by_Magneto.jpg}}]]
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Change of electric field leads to generation of changing magnetic field, and vice versa. Thus electromagnetic field is created, which comprises both electric and magnetic components.[([[http://google.com/books?isbn=9781139451925|Bhag Singh Guru, Hüseyin R. Hiziroglu, Electromagnetic Field Theory Fundamentals, Cambridge University Press, 2004, ISBN 9781139451925, p. 8]])] High-frequency sinusoidal currents (with some [[modulation]]) can be used to [[electromagnetic radiation|radiate electromagnetic waves]], from and to antennas.

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[[Lightning]] is a sudden discharge of [[electric current]] which generates an impulse of magnetic field
[[file/lightning_pd_jpg|{{lightning_pd.jpg}}]]
//<sup>by Nico36, Public Domain</sup>//
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[[Magnetised]] piece of lodestone attracting and holding up a paper staple (which is made from [[iron]])
[[file/magnetite_lodestone_magnetica_jpg|{{magnetite_lodestone_magnetica.jpg}}]]
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In nature, an example of magnetic field generation is a [[lightning]], which is a sudden discharge of [[electric current]] which produces an impulse of magnetic field around it, as well as electromagnetic waves throughout a wide spectrum, including the visible light. Such magnetic pulse is capable of magnetising natural minerals, such as [[magnetite]].[(Bowles_Minerals>[[https://www.google.co.uk/search?tbo=p&tbm=bks&q=isbn:9781862393158|J. F. W. Bowles, R. A. Howie, D. J. Vaughan, J. Zussman, Rock-forming Minerals, Geological Society of London, 2011, ISBN 9781862393158, p. 403]])]

These natural effects were the first manifestations of magnetism, which allowed humans to discover, tame and utilise the magnetic and electromagnetic effects.

==== By intrinsic magnetic moments ====

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Simplified illustration of electron's [[orbital magnetic moment]] $\vec m_o$ and [[spin magnetic moment]] $\vec m_s$[(Coey)]
[[file/orbital_spin_magnetic_moment_em_png|{{orbital_spin_magnetic_moment_em.png}}]]
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On an atomic and sub-atomic level, particles such as [[electron]], [[proton]] or [[neutron]] possess [[magnetic moment|magnetic moments]].[([[http://google.com/books?isbn=9780080457055|Tapan Chatterji, Neutron Scattering from Magnetic Materials, Gulf Professional Publishing, 2005, ISBN 9780080457055, p. 3]])][([[http://google.com/books?isbn=9780306473524|Almadena Chtchelkanova, Stuart A. Wolf, Yves Idzerda, Magnetic Interactions and Spin Transport, Springer, 2003, ISBN 9780306473524, p. 316]])][(Coey>[[https://books.google.co.uk/books?isbn=0521816149|J.M.D. Coey, Magnetism and Magnetic Materials, Cambridge University Press, 2010, ISBN 9780521816144]])]

The magnetic moments can be conceptually illustrated as a movement of electron (orbital or spinning), which would be equivalent to electric current. However, these effects do not follow [[classical physics]] - they are instead quantum phenomena. 

Hence, they are also sources of magnetic field and they interact with each other accordingly. Under appropriate conditions the interactions are strong enough for the spins of electrons to align parallel to each other leading to [[ferromagnetism]], which is responsible for most "[[magnetic material|magnetic]]" effects as understood in everyday language.[(Chikazumi>[[http://google.com/books?isbn=9780199564811|Soshin Chikazumi, Physics of Ferromagnetism, 2nd edition, Oxford University Press, 2009, ISBN 9780199564811]])]


Ferromagnetic materials (so-called [[soft magnetic materials]]) have relatively high magnetic permeability so that they respond with large value of //**B**// for the same excitation of //**H**// (as compared to non-ferromagnetic materials). For this reason they are used to concentrate and conduct magnetic field so that a suitable [[magnetic circuit]] like in an [[electromagnet]], [[transformer]] or [[electric motor]] can be created. 

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Permanent magnets are used in [[electric motor|motors]], [[generator|generators]], [[electromagnetic actuator|actuators]], [[loudspeaker|loudspeakers]], holders, toys, etc.
[[file/hard_ferrites_1_magnetica_jpg|{{hard_ferrites_1_magnetica.jpg}}]] 
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If very strong magnetic fields are required then the usefulness of ferromagnets is diminished. Fields exceeding 3 T can be generated with [[Bitter electromagnet|Bitter electromagnets]], [[superconducting magnets]], [[pulse field|pulse methods]] or [[compression of magnetic flux]] techniques. These are capable of producing magnetic fields without the aid of magnetic core, but at the expense of other parameters like very high electric power levels, cryogenic cooling, very short pulse duration, etc.[(Chikazumi)]

[[Permanent magnet]] is a ferromagnetic material which is made in a way to maximise energy storage. This is typically achieved by an appropriate chemical and crystal composition, such that impedes the movement of [[domain wall|domain walls]] as to maximise the [[coercivity]]. During the [[magnetisation process]], the internal spins rearrange so their vector sum produces significant [[magnetic polarisation]].

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[[Antenna]] in a [[mobile phone]] can be just a flat strip of [[copper]] (0.2 × 10 × 30 mm)
[[file/nokia_antenna_02mm_thick_15x30mm_jpg|{{nokia_antenna_02mm_thick_15x30mm.jpg}}]]
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After removing the external field, permanent magnets retain the magnetisation, and thus become a new source of magnetic field,[(Purcell)] without the need for a power supply.

For some applications there is no need for magnetic cores, or it is impractical to use them. Most of telecommunication based on [[electromagnetic wave|electromagnetic wave]] (e.g. [[mobile phone|mobile phone]]) can use non-ferromagnetic antennas to transmit and receive electromagnetic signals. 

The electromagnetic field is then transmitted through non-conducting media (air, vacuum, building materials, glass). Conductive materials cause [[absorption]] or [[reflection]] of waves. Other wave phenomena such as [[refraction]] and [[diffraction]] also take place.[([[http://google.com/books?isbn=9780470576649|Warren L. Stutzman, Gary A. Thiele, Antenna Theory and Design, John Wiley & Sons, 2012, ISBN 9780470576649,p. 88]])]

Antennas are driven by electric currents at high frequency (typically from kHz to GHz range, depending on the application).

==== By photon emission ====

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[[Electron]] is excited to a higher energy state by absorbing a quantum of energy, and a [[photon]] is released when electron drops to a lower energy level 
[[file/photon_emission_absorption_magnetica_png|{{photon_emission_absorption_magnetica.png}}]]
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A photon is a quantum of electromagnetic energy, and as all electromagnetic waves it comprises magnetic and electric field components.

Electrons in atoms occupy quantised energy levels, and if all the electrons occupy the lowest positions it is said the the atom is in its [[ground state]]. 

Energy can be absorbed by an atom by the electrons jumping to a higher energy level. And if there is unoccupied lower energy level there is some probability that such excited electron will jump down, emitting a photon in the process. The energy of such photon is directly related to the energy difference between the two states of the electron. Very high energy photons can be generated in this way, with energies of [[hard X-rays]] and beyond.[(Holbrow)]

Using a simplistic analogy, a jump of an electron is a movement requiring acceleration, so in some sense this is akin to generation of electromagnetic wave due to electric current, as described above. However, this analogy is also incorrect, because classical physics analogy cannot fully represent the inherently quantum phenomenon of a photon emission.

Highly energetic processes, such as lightning, create [[plasma]] with highly excited electrons, which emit light when they jump to lower energy levels.[(Holbrow>[[https://books.google.co.uk/books?isbn=9780387790794|Charles H. Holbrow, James N. Lloyd, Joseph C. Amato, Enrique Galvez, M. Elizabeth Parks, Modern Introductory Physics, 2nd ed., Springer, New York, ISBN 9780387790794]])] For the same reason flames and fire produce photons, with the colour of light depending on the involved chemical components.

However, even ordinary heat is capable of exciting electrons, and all matter above [[absolute zero]] emits electromagnetic radiation. This is the basis of operation of [[thermal camera|thermal cameras]], which detect infrared radiation. If the matter is hot enough it glows in the spectrum visible to human eye.



===== Magnetic field and relativistic effects =====

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Magnetic field or its effects depend on the [[frame of reference]]: upper image show the wire (grey) with the positive charges as stationary and the negative charges moving - the magnetic force //F<sub>m</sub>// pushes the single moving charge towards the wire; in the bottom image the frame of reference travels with the moving charge (so this charge is then stationary) - the magnetic force disappears and the negative charge is pushed towards the wire due to electrostatic force //F<sub>e</sub>//[(Libre)] because the concentration of charges changes in the wire (due to the [[Lorentz contraction]])
[[file/relativistic_effects_with_current_magnetica_png|{{relativistic_effects_with_current_magnetica.png}}]]
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The existence of magnetic field depends on the [[frame of reference]]. Magnetic force depends on the velocity of the moving charge. Therefore, if the [[inertial frame of reference]] (no acceleration) is chosen such that it moves together with the charge, then the charge becomes stationary and there is no magnetic force acting on it at that instant. 

Because of the [[Lorentz contraction]] due to relativistic effects, the apparent density of the electric charges changes, so that electrostatic imbalance appears and electrostatic force can appear instead of magnetic force.[(Libre>[[https://phys.libretexts.org/@go/page/19555|Rinaldo, M., Neary, & Woodman. (2020, November 5). Electric and magnetic fields and Special Relativity, Libretexts.org]], {accessed 2021-05-10})][(Griffiths)]

The movement of electrons inside an atom cannot be described by [[classical physics]], but requires [[quantum mechanics]], which also need to take into account the relativistic effects.[(Griffiths)] 

Solutions of Maxwell's equations which include [[time dilation]] require deriving very complex mathematical equations.[(Griffiths)]

Even though the very existence of magnetic field is linked to relativistic effects, for most practical purposes and designs (especially at lower frequencies) the analysis can be reduced to simplified cases, with safely ignoring the relativity principles. This is similar to Newton's laws of motion, for which at ordinary velocities the relativistic effects can be ignored.
===== See also =====
  * [[Electromagnetism]]
  * [[Magnetic field strength]]
  * [[Flux density]]
  * [[Confusion between B and H]]
  * [[Magnetic field lines]]
  * [[Magnetic materials]]

===== External video materials =====

{{youtube>caHXwJbkbQU?large}}
===== References =====
~~REFNOTES~~

{{tag>Magnetic_field Theory_of_magnetism Counter}}