Encyclopedia Magnetica™

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

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| //[[user/Stan Zurek]], Magnetic field strength, Encyclopedia Magnetica//, \\ @PAGEL@  |

**Magnetic field strength //H//** - a physical quantity used as one of the basic measures of the intensity of [[magnetic field]].[(Mansfield>[[http://google.com/books?isbn=9780470746387|Michael Mansfield, Colm O'Sullivan, Understanding Physics, John Wiley & Sons, 2010, ISBN 9780470746387, p. 407]])][(Britannica>[[https://www.britannica.com/science/magnetic-field|Magnetic field, Encyclopædia Britannica Online, {accessed 2016-07-05}]])] The unit of magnetic field strength[(Jiles>[[http://google.com/books?isbn=9781482238884|David Jiles, Introduction to Magnetism and Magnetic Materials, Third Edition, CRC Press, 2015, ISBN 9781482238884, p. 5-16]])] is //**[[ampere]] per [[metre]]**// or **A/m**.

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[[Electric current]] $I$ generates **magnetic field strength** $H$, whose magnitude is independent on the type of the uniform isotropic surrounding medium ([[magnetic]] or [[non-magnetic]])
[[file/Electric_current_generates_magnetic_field_Magnetica_jpg|{{Electric_current_generates_magnetic_field_Magnetica.jpg}}]]
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From the engineering viewpoint, **magnetic field strength** $H$ can be thought of as excitation and the **[[magnetic flux density]]** $B$ as the response of the medium.[(Zurek>[[https://isbnsearch.org/isbn/9780367891572|S. Zurek, Characterisation of Soft Magnetic Materials Under Rotational Magnetisation, CRC Press, 2019, ISBN 9780367891572]])][(Tumanski>[[https://isbnsearch.org/isbn/9780367864958|Sławomir Tumański, Handbook of magnetic measurements, CRC Press / Taylor & Francis, Boca Raton, FL, 2011, ISBN 9780367864958]])] This naming convention is defined in the [[SI system of units]].

From theoretical physics viewpoint, the field $H$ is defined as the vectorial difference between [[flux density]] $B$ and [[magnetisation]] $M$. The //H// field is sometimes referred to as "auxiliary" or simply "field H".[(Purcell>[[https://isbnsearch.org/isbn/9781107014022|E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022]])][([[https://isbnsearch.org/isbn/9352837347|R Murugeshan, Electricity and Magnetism, S Chand Publishing, 2019, ISBN 9352837347]])][(Feynman>[[https://www.feynmanlectures.caltech.edu/II_36.html|Richard Feynman, Robert Leighton, Matthew Sands, Ferromagnetism, The Feynman Lectures on Physics, Vol. II, Caltech]], {accessed 2021-05-16})]

These two approaches are identical in the sense of the physical quantities in question (with the same physical units of A/m), but are referred to by different names, and different emphasis put on their meaning and use in derivation of some equations. 

Magnetic field is a [[vector field]] in space, and is a kind of [[energy]] whose full quantification requires the knowledge of the vector fields of both magnetic field strength $H$ and [[flux density]] $B$ (or other values correlated with them, such as [[magnetisation]] //M// or [[polarisation]] //J//). In vacuum, at each point the $H$ and $B$ vectors are oriented along the same direction and are directly proportional through [[permeability]] of free space, but in other media they can be misaligned (especially in [[uniform material|non-uniform]] or [[anisotropy|anisotropic]] materials).

The requirement of two quantities is analogous for example to [[electricity]]. Both [[electric voltage]] $V$ and [[electric current]] $I$ are required to fully quantify the effects of electricity, e.g. the amount of transferred energy.[(White>[[http://google.com/books?isbn=9781439866511|Mary Anne White, Physical Properties of Materials, Second Edition, CRC Press, 2011, ISBN 9781439866511, p. 359]])]

The name //**magnetic field strength**// and the symbol $H$ are defined by //[[International Bureau of Weights and Measures]] (BIPM)// as a one of the [[coherent derived unit|coherent derived physical units]].[(BIPM>[[http://www.bipm.org/en/publications/si-brochure/section2-2.html|SI Brochure: The International System of Units (SI) [8th edition, 2006; updated in 2014], Section 2.2: SI derived units, {accessed 2016-06-15}]])] Therefore, strictly speaking, other names like //**magnetic field intensity**// or //**magnetic field**// (or even just //**field**//) which can be encountered in everyday technical jargon[(Mansfield)] are incorrect if used when referring to a specific value of //H// in A/m.

There are many other names which are used in the literature, all denoting the same quantity:
  * //magnetic field intensity H// [(Britannica>[[https://www.britannica.com/science/magnetic-field|Encyclopaedia Britannica, Magnetism, {accessed 2021-05-02}]])]
  * //magnetic field H// [(Britannica)][(Purcell)][([[http://books.google.com/books?isbn=9780124912465|Ronald T. Merrill, M. W. McElhinny, Phillip L. McFadden (ed.), The Magnetic Field of the Earth: Paleomagnetism, the Core, and the Deep Mantle, Volume 63 of International geophysics series, ISBN 9780124912465]])]
  * //field H// [(Purcell)]
  * //field H'// [(Feynman)]
  * //H-field strength// [(Poljak>[[https://books.google.co.uk/books?id=gWOcDwAAQBAJ|D. Poljak, M. Cvetković, Human interaction with electromagnetic fields, Computational models in dosimetry, Academic Press, Elsevier, 2019, ISBN 9780128164433]])]
  * //H-field// [([[https://www.antenna-theory.com/definitions/hfield.php|Magnetic Field (H-Field), Antenna-Theory.com]], {accessed 2021-05-05})]
  * //magnetization field strength H// [(Hilzinger>[[https://isbnsearch.org/isbn/9783895783524|Rainer Hilzinger, Werner Rodewald, Magnetic Materials, Fundamentals, Products, Properties, Applications, VAC Vacuumschmelze, Publicis, 2013, ISBN 9783895783524]])]
  * //magnetizing field strength H// [(Hilzinger)]
  * //magnetising force H// [(Thompson_1890>[[https://books.google.co.uk/books?id=kQwAAAAAMAAJ|Silvanus P. Thompson, The electro-magnet, The Telegraphic Journal and Electrical Review, Vol. XXVII, July 4 - December 26, 1890, pp. 372-377]])]
  * //magnetic force H// [(Thompson_1890)]
  * //intensity of magnetic force H// [(Thompson_1890)]
  * //auxiliary field H// [([[https://isbnsearch.org/isbn/9352837347|R Murugeshan, Electricity and Magnetism, S Chand Publishing, 2019, ISBN 9352837347]])]
  * and probably several others.

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

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

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

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Illustration of [[magnetic field strength]] //H//, [[flux density]] //B//, [[magnetisation]] //M//, and [[magnetic polarisation|polarisation]] //J// in a **[[ferromagnet]]**
[[file/magnetic_field_flux_density_magnetisation_ferromagnet_e-m_png|{{magnetic_field_flux_density_magnetisation_ferromagnet_e-m.png}}]]
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The [[magnetic flux density]] //B// is a separate physical quantity, with different physical units in the [[SI system]]. The //H// and //B// are interlinked such that:[(Purcell)]


|  $$\vec{B} = \vec{J} + μ_0 · \vec{H} = μ_0 · (\vec{H} + \vec{M}) $$  |  (T)  |
| where: $μ_0$ - absolute [[permeability of vacuum]] (H/m), $μ_r$ - [[relative permeability]] of material (unitless), $μ = μ_0 · μ_r$ - absolute permeability of material (H/m), $J$ - [[magnetic polarisation]] (T), $M$ - [[magnetisation]] (A/m)  ||

[[Magnetisation]] //M// represents orientation of subatomic [[magnetic dipole moment|magnetic dipole moments]] per unit volume, and [[magnetic polarisation]] //J// is //M// scaled by the [[permeability of vacuum]].

In a general case, all the three vectors //B//, //H// and //J// (or //B//, //H// and //M//) can point in different directions (as shown in the illustration for an [[anisotropic material]]), but always such that the vector sum in the equation above is fulfilled. 

For uniaxial magnetisation the equation can be simplified to the scalar form, which is used widely in engineering applications:

|  $$B = μ_0 · μ_r · H = μ · H $$  |  (T)  |

[[Relative permeability]] $μ_r$ is a figure of merit of [[soft magnetic materials]] and has values significantly greater than unity. 

For [[hard magnetic materials]] $μ_r \approx$ 1, and it is a much less important parameter.

For non-magnetic materials also $μ_r \approx$ 1, but such that [[paramagnet|paramagnets]] are weakly attracted to any polarity of magnetic field ($μ_r$ slightly greater than unity), and [[diamagnet|diamagnets]] are always weakly repelled it ($μ_r$ slightly less than unity). Depending on the viewpoint, [[superconductor|superconductors]] can be classified as ideal diamagnets for which $μ_r$ = 0, and thus they are quite strongly repelled from magnetic field, sufficiently for [[magnetic levitation]].[(Tumanski)]
===== Difficulty with definition =====
It is difficult to give a concise definition of such a basic quantity like [[magnetic field]], but various authors give at least a descriptive version. The same applies to **magnetic field strength**, as well as the other basic quantity - **[[magnetic flux density]]**.

The table below shows some examples of definitions of $H$ given in the literature (exact quotations are shown).

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^ Publication ^ Definition of //**[[magnetic field]]**// ^ Definition of //**[[magnetic field strength]]**// $H$ ^ Definition of //**[[magnetic flux density]]**// $B$ ^
| R. Feynman, R. Leighton, M. Sands \\ **The Feynman Lectures on Physics**[(Feynman)]  | //First, we must extend, somewhat, our ideas of the electric and magnetic vectors, **E** and **B**. We have defined them in terms of the forces that are felt by a charge. We wish now to speak of electric and magnetic fields at a point even when there is no charge present. We are saying, in effect, that since there are forces “acting on” the charge, there is still “something” there when the charge is removed. //   | //We choose to define a new vector field **H** by $$\mathbf{H} = \mathbf{B} − \frac{\mathbf{M}}{ε_0 c^2} $$ [...] Most people who use the mks units have chosen to use a different definition of **H**. Calling //their// field **H'** (of course, they still call it **H** without the prime), it is defined by $$\mathbf{H'} = ε_0 c^2\mathbf{B} − \mathbf{M}$$ Also, they usually write $ε_0 c^2$ as a new number 1/μ<sub>0</sub>//   | //We can write the force F on a charge q moving with a velocity v as  $$\mathbf{F} = q(\mathbf{E} + \mathbf{v} × \mathbf{B})$$  We call **E** the electric field and **B** the magnetic field at the location of the charge.//  |
| 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>[[http://google.com/books?isbn=9780412386404|David Jiles, Introduction to Magnetism and Magnetic Materials, Chapman and Hall, 1991, ISBN 9780412386404, p. 1-13]])]  | //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. [...] 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 $\mathbf{B}$.// |
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===== Analogy to electric circuits =====

==== Electric circuit ====
At the fundamental level, all the electricity is linked to the presence and movement of electric charges, so knowing their positions would be sufficient to fully quantify all electric effects, including [[electric field]]. However, in practice, it is much simpler to operate with directly measurable quantities such as [[current]] $I$ and [[voltage]] $V$. 

From a macroscopic viewpoint, values of $I$ and $V$ are both required to fully quantify the effects of [[electricity]] in electric circuits. In [[direct current]] circuits the proportionality between $V$ and $I$ is dictated by [[electrical resistance]] $R$ of a given medium (according to [[Ohm's law]]), such that $V = R·I$.

The product of $V$ and $I$ is proportional to [[power]] $P$ and [[/energy]] $E$ in a given electric circuit.


==== Magnetic circuit ====
By analogy both [[magnetic field strength]] $H$ and [[magnetic flux density]] $B$ (or their representations by other related variables) are required for quantifying the effects of [[magnetism]] in [[magnetic circuit|magnetic circuits]]. The proportionality between $H$ and $B$ is dictated by [[magnetic permeability]] $μ$ of a given medium.[(White)]

All magnetic field effects are also linked to the movement and [[intrinsic properties]] of electric charges. Knowing these properties (such as [[spin magnetic moment]]) and the details of movement of the charges (taking into account relativistic effects) it would be possible to completely describe the magnetic field. However, in practice it is much simpler, especially from engineering viewpoint, to utilise the directly measurable quantities such as $H$ and $B$ to quantify power and energy in a given magnetic circuit.

Under [[steady state]] conditions, the product of $H$ and $B$ is a measure of [[specific energy]] in J/m<sup>3</sup>, stored in the magnetic field contained in the given medium. The $B·H$ product (the amount of stored energy) is used for example for classification of [[permanent magnet|permanent magnets]].[([[http://www.ndfeb-info.com/neodymium_grades.aspx|Grades of Neodymium, NdFeB-Info.com, {accessed 2016-06-28}]])]


===== H due to electric current  =====

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Amplitude of **magnetic field strength** $H$ reduces with the distance from a conductor with electric current $I$
[[file/h_around_i_-_magnetica_png|{{h_around_i_-_magnetica.png}}]]
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Orientation of **magnetic field strength** $H$ vector with respect to the current $I$ follows the [[right-hand rule]]
[[file/right-hand_rule_magnetica_png|{{right-hand_rule_magnetica.png}}]]
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From a [[macroscopic]] viewpoint, the fields can be treated as averaged over some volume of material, and their magnitudes can be linked to measurable signals such as [[current]] or [[voltage]]. Therefore, this approach is used extensively in engineering.[(Tumanski)][(Zurek)][(Jiles)]

$H$ is always generated around [[electrical current]] $I$, which can be a solid [[conductor]] with current or just a moving [[electric charge]] (also in [[free space]]). The direction of the $H$ vector is perpendicular to the direction of the current $I$ generating it, and the senses of the vectors are assumed to follow the [[right-hand rule]].[(Jiles)] It can be said that //H// "circulates" around the current //I//.

Without other sources of magnetic field and in a [[uniformity|uniform]] and [[isotropic]] medium the generated magnetic field strength $H$ depends only on the magnitude and direction of the electric current $I$ and the physical sizes involved (e.g. length and diameter of the conductor, etc.) so according to the [[Ampere's law]] the proportionality is dictated by the [[magnetic path length]] $l$: 

|  $$ \int_C \vec{H} · d \vec{l}  = I $$  |  (A)  |
| where: //C// - closed path over which the integral is calculated, $dl$ - [[infinitesimal]] fragment of [[magnetic path length]] (m), $I$ - current (A)  ||



In a linear isotropic medium the values from various sources combine and can be calculated from the superposition of the sources. For simple geometrical cases the value of $H$ can be calculated analytically, but for very complex systems it is possible to perform computation for example with [[finite-element modelling]].

The relationship between $H$ and $I$ is often shown by employing the [[Biot-Savart law|Biot-Savart's law]][(Jiles)] or [[Ampere law|Ampere's law]].[(MIT>[[http://ocw.mit.edu/high-school/physics/exam-prep/magnetic-fields/biot-savarts-law-amperes-law/8_02_spring_2007_ch9sourc_b_field.pdf|MIT OpenCourseWare, Biot-Savart's Law & Ampere's Law, Chapter 9]])] Often (but not always[(Britannica_Maxwell_equations>[[http://www.britannica.com/science/Maxwells-equations|Maxwell’s equations, Encyclopædia Britannica. Encyclopædia Britannica Online, {accessed 2016-06-28}]])]) both of these are stated with the variable of [[flux density]] $B$ so that the [[permeability]] of the medium is automatically taken into account.

In many examples given in the literature there is an implicit assumption (typically not stated) that the derivation is carried out for vacuum and not for an arbitrary medium with a different permeability[(MIT)]. When the $μ_0$ permeability is reduced in the equations on both sides then $H$ is proportional only to $I$ and this is true for any uniform isotropic medium with any permeability, even non-linear.[(Tumanski)]

The situation is slightly different for [[anisotropic]] or discontinuous medium. They can give rise to additional sources of magnetic field because new [[magnetic pole|magnetic poles]] can be generated by the excited medium, and these poles must be taken into account in order to accurately describe distribution of $H$. For instance, pole pieces in an [[electromagnet]] affect $H$, whose distribution is no longer dictated by just the coils with electric current. 

===== H due to M and B =====

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//H//, //B//, //M//, and //J// in a **[[paramagnet]]**
[[file/magnetic_field_flux_density_magnetisation_paramagnet_e-m_png|{{magnetic_field_flux_density_magnetisation_paramagnet_e-m.png}}]]
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The [[microscopic]] viewpoint is used often in theoretical physics.[(Purcell)][(Griffiths>[[http://books.google.com/books?isbn=0321856562|David J. Griffiths, Introduction to electrodynamics, 4th ed., Pearson, Boston, 2013, ISBN 0321856562]])]

Each atom responds to the externally applied magnetic field //B// with some [[magnetisation]] //M//, which is defined as the vector sum of magnetic moments per given volume. The "auxiliary" magnetic field //H// is then //defined// as the vector difference between the applied magnetic field //B// and the magnetisation //M//:[(Purcell)]

|  $$\vec{H} = \frac{\vec{B}}{μ_0} - \vec{M}$$  |  (A/m)  |
| where: $μ_0$ - absolute [[permeability of vacuum]] (H/m)  ||

For DC excitation, in non-magnetic or magnetic but isotropic materials //B// and //H// vectors are parallel. For [[ferromagnetic]] (and other ordered structures) the [[crystal anisotropy|crystal]] or [[shape anisotropy]] can introduce significant angle between the two vectors.

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

[[Maxwell's equations]] are typically given with respect to [[magnetic flux density]] //B// because in that form they are valid under more general conditions.[(Griffiths)]

However, under certain conditions it is also possible to express them with respect to //H//. This approach is extensively used in numerical calculations such as [[finite-element modelling]] (FEM), where the direct link between the electric current (expressed by [[current density]] //J//) and //H// is exploited, through the Ampere's law, both for solutions and formulations of [[boundary condition|boundary conditions]].[(COMSOL_Cyclopedia>[[https://www.comsol.com/multiphysics/electromagnetics|Introduction to Field Electromagnetics, Multiphysics Cyclopedia, COMSOL]], {accessed 2021-06-25})][(COMSOL_Guide>[[https://doc.comsol.com/5.4/doc/com.comsol.help.acdc/ACDCModuleUsersGuide.pdf|AC/DC ModuleUser’s Guide, COMSOL 5.4]], {accessed 2021-06-25})][(FEMM>[[https://www.femm.info/wiki/Documentation/|David Meeker, Finite Element Method Magnetics: Documentation]], {accessed 2021-06-25})][(Multiphysics>[[http://www.multiphysics.us/electromagnetics.html|Electromagnetics, Multiphysics Learning & Networking, http://www.multiphysics.us]], {accessed 2021-06-25})][(ANSYS>[[https://www.ozeninc.com/wp-content/uploads/2021/01/ANSYS-Maxwell-Magnetic-Field-Formulation-Application-Brief.pdf|ANSYS Maxwell Magnetic Field Formulation, Application Brief, ANSYS, 2013]], {accessed 2021-06-25})]

^  Example of notation used in FEM documentation (after reference [(COMSOL_Cyclopedia)] )  ^^
|  $$ \nabla · \mathbf{D} = ρ$$  |  $$ \nabla · \mathbf{B} = 0$$ |  
|  $$ \nabla \times \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$  |  $$ \nabla \times \mathbf{H} = \mathbf{J} +  \frac {\partial \mathbf{D}}{\partial t} $$  |

==== H in electromagnetic waves ====

In vacuum, in the absence of charges and currents, the Maxwell's equations simplify, and they can be written either with respect to [[magnetic flux density]] //B// (as shown in the table below), or [[magnetic field strength]] //H//. This can be done because of the linearity of vacuum (or other non-magnetic medium), which has no free charges, so there are no additional electric currents which have to be taken into account.[(Kwok>[[https://www.sjsu.edu/people/raymond.kwok/docs/ee140/Maxwell_Eqns.pdf|Ray Kwok, Maxwell Equations, San José State University, CA, USA, www.sjsu.edu]], {accessed 2021-06-22})] The format with //B// is valid under more general conditions.[(Griffiths)]

|< 100% >|
^  Maxwell's equations in vacuum (in a differential form)[(Purcell)][(Kwok)]  ^^^^
^  magnetic field represented by //H//  ^^  magnetic field represented by //B//  ^^
|  $$ \text{div } \mathbf{E} = 0$$  |  $$ \text{div } \mathbf{H} = 0$$  |  $$ \text{div } \mathbf{E} = 0$$  |  $$ \text{div } \mathbf{B} = 0$$  |
|  $$ \text{curl } \mathbf{E} = - \mu_0 · \frac {\partial \mathbf{H}}{\partial t}$$  |  $$ \text{curl } \mathbf{H} =  \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t} $$  |  $$ \text{curl } \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$  |  $$ \text{curl } \mathbf{B} = \mu_0 · \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t} $$  |


In vacuum the two notations, with //B// or //H// are exactly equivalent, with the latter quite popular for analysing radiation from [[antenna|antennas]].[(Suriano>[[https://interferencetechnology.com/antenna-fundamentals/|Candace Suriano, et al., Antenna Fundamentals, Interference Technology, May 3, 2007]], {accessed 2021-04-22})] For example, using the [[Poynting vector]] which represents [[power]], as a product of electric field //E// in V/m and magnetic field //H// in A/m, the result is V·A/m<sup>2</sup> or W/m<sup>2</sup> ([[power density]]).

===== Defining H with force =====

It is shown in the literature that magnetic field strength at a given point in space can be defined as the mechanical [[force]] acting on [[unit pole]] at the given point.[(Mansfield)] However, calculation of force requires $B$, which depends on the properties of medium. Indeed, the original experiment performed by Biot and Savart involved physical forces acting on wires.[(Biot>[[http://www.ampere.cnrs.fr/ice/ice_book_detail-fr-text-koyre_ampere-ampere_text-7-2.html|Jean-Baptiste Biot and Felix Savart. Note sur le magnetisme de la pile de Volta. Ann. Chim.Phys., 15:222–223, 1820]])]

The forces acting on two magnetised bodies will be different if they are placed in oxygen (which is [[paramagnetic]]) or in water (which is [[diamagnetic]]). This difference will be directly proportional to the relative permeabilities of the involved media. However, the $H$ produced around the wire will be the same (as long as the medium is uniform and isotropic).

The magnitude of [[magnetic force]] (Lorentz force) is always proportional flux density $B$.[(Purcell)]



===== Generation of H =====

Known values of //H// are generated by utilising the Ampere or Biot-Savart laws mentioned above. If relativistic effects can be ignored, then the proportionality is exactly direct such that instantaneous values of magnetic field strength $H$ correspond to instantaneous values of the applied current $I$:

|  $$ H(t) = c · I(t) $$  |  (A/m)  | 
| where: $c$ - proportionality constant of a given circuit (1/m)  ||

Under certain conditions the generated magnetic field can be calculated so precisely that it can be used for calibration of other sensors or definition of values, as recommended by [[BIPM]].[(SI_Appendix2>[[https://www.bipm.org/en/publications/mises-en-pratique|SI Brochure – 9th edition (2019) – Appendix 2, 20 May 2019]], {accessed 2021-06-27})] 

Two typical devices which can be used for generating known values of //H// are  the [[solenoid]] and the [[Helmholtz coil]].[(SI_Appendix2)][(Tumanski)] They can be even used in a combined setup, in which the external Helmholtz coils compensate for [[Earth's magnetic field]] (or other unwanted sources) and the internal solenoid for generation of the precisely known magnetic field.[(Fiorillo>[[https://isbnsearch.org/isbn/9780122572517|Fausto Fiorillo, Measurement and Characterization of Magnetic Materials, Academic Press, 2005, ISBN 9780122572517]])]

==== Solenoid ==== 

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[[Solenoid]] is often used a source of known value of magnetic field strength //H//, which can be calculated for its geometrical centre (black dot)
[[file/solenoid_magnetica_png|{{solenoid_magnetica.png}}]]
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| {{/wiki/logo.png?20&nolink}} //See also the main article:// [[Solenoid]] |

In an infinitely long uniform **[[solenoid]]**, the value of //H// at its axis depends only on the value of current in the coil and the number of turns per unit length. 

For a solenoid with a finite length, the magnetic field at its geometrical centre can be calculated as in the equation below. For "thin" (wire diameter much smaller than coil diameter) and "long" (coil diameter much smaller than its length $d \ll l$) solenoid, the equation simplifies:[(Jiles)]

|  $$ H_{centre} = \frac{N·I}{\sqrt{l^2 + d^2}} \approx \frac{N·I}{l} $$  |  (A/m)  |
| where: $N$ - total [[number of turns]] in the [[solenoid]] (unitless), $I$ - current (A), $l$ - solenoid length (m), $d$ - solenoid diameter (m)  ||

If the thickness of the wire in the solenoid is significant, or there are many layers of the coil, some additional correction terms are required in the equation.[(Jiles)]


==== Helmholtz coil ====

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

Another widely used source of //H// is the **[[Helmholtz coil]]**. The device comprises two identical coils resembling circular [[current loop|current loops]], positioned parallel on the same axis, and separated precisely by the radius of the circle.

For two coils, each with radius //r// and each comprising number of turns //N<sub>each</sub>//, the value of magnetic field at the geometrical centre can be calculated as:[(Jiles)]

|  (turns per coil)  |  $$ H_{centre} = \frac{N_{each}·I·\sqrt{0.8^3}}{r} \approx \frac{0.71554·N_{each}·I}{r} $$  |  (A/m)  |
|  (turns total)  |  $$ H_{centre} = \frac{N_{total}·I·\sqrt{0.8^3}}{2·r} \approx \frac{0.35777·N_{total}·I}{r} $$  |  (A/m)  |
| where: $N_{each}$ - [[number of turns]] of each [[coil]] (unitless), $N_{total}$ - total number of turns of both coils (unitless), such that $N_{total}=2·N_{each}$, $I$ - current (A), $r$ - radius of each coil and spacing between them (m)  |||

Shapes other than circular are also used (e.g. square) but at the expense of uniformity of the obtained field distribution.

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[[Helmholtz coil]] is a precise source of magnetic field at its geometrical centre (black dot)
[[file/helmholtz_coils_magnetica_png|{{helmholtz_coils_magnetica.png}}]]
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Path of moving [[electron|electrons]] bent into a circle by [[magnetic field]] generated by external [[Helmholtz coil]] (red circular shape in the background)
[[file/Cyclotron motion wider view_jpg|{{Cyclotron motion wider view.jpg}}]] 
//<sup>by M. Białek, Wikimedia Commons, CC-BY-SA-3.0</sup>// 
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Set of three large orthogonal Helmholtz coils used for compensation of [[Earth magnetic field|Earth's magnetic field]] in 3D
[[file/3d_helmholtz_coils_magnetica_jpg|{{3d_helmholtz_coils_magnetica.jpg}}]] 
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==== Magnetic circuit with a small gap ====

The Ampere's law relates the integral around a closed path, to the current enclosed by such path. 

This relationship is used extensively in engineering by employing the concept of [[magnetomotive force]] (product of current and turns of the coil, expressed in [[ampere-turn|ampere-turns]]). For a simple magnetic circuit with one air gap it can be written that:

|  $$ N·I = H_{core}·l_{core} + H_{gap}·l_{gap} $$  |  (A-turns) ≡ (A)  |
| where: $N$ - [[number of turns]] of the [[winding]] (unitless), $I$ - current (A), $H_{core}$ - //H// in the core (A/m), $l_{core}$ - length of the core (m), $H_{gap}$ - //H// in the [[air gap]] (A/m), $l_{gap}$ - length of the air gap (m)  ||

In a magnetic circuit with a relatively small air gap the value of [[magnetic flux density]] is such that $B_{gap} \approx B_{core}$. However, the value of //H// required to support some value of //B// is scaled by the inverse of [[relative permeability]]. Hence, for a magnetic material with large permeability, leads to the condition of $H_{core} \ll H_{gap}$ and also to $H_{core}·l_{core} \ll H_{gap}·l_{gap}$, and therefore only the terms related to the air gap are significant. This allows simplifying the equation as:[(Kozlowski>[[https://doi.org/10.1109/TMAG.2013.2288322|A. Kozłowski, R. Rygał and S. Zurek, Large DC Electromagnet for Semi-Industrial Thermomagnetic Processing of Nanocrystalline Ribbon, IEEE Transactions on Magnetics, 50 (4), pp. 1-4, 2014, Art no. 8000404, DOI 10.1109/TMAG.2013.2288322]])]


|  $$ H_{gap} \approx \frac{N·I}{l_{gap}} $$  |  (A/m)  |
| where: $N$ - [[number of turns]] of the [[winding]] (unitless), $I$ - current (A), $l_{gap}$ - length of the [[air gap]] (m)  ||

However, for more complex magnetic circuits, effects such as [[flux fringing]] or [[magnetic energy]] stored in the material must be taken into account, and this can be done by numerical methods such as [[finite-element modelling]].[(Kozlowski)]

Addition of air gap allows storing energy in it. The B-H loop is "sheared", extending the operation to higher //H//. These effects are widely used for example in [[flyback transformer|flyback transformers]].

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[[Magnetic circuit]] with [[magnetic path]] and [[air gap]]
[[file/magnetic_circuit_with_air_gap_magnetica_png|{{magnetic_circuit_with_air_gap_magnetica.png}}]]
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Large [[electromagnet]] with an [[air gap]]; designed with the simplified equation[(Kozlowski)]
[[file/electromagnet_200mm_by_magneto_jpg|{{electromagnet_200mm_by_magneto.jpg}}]]
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Addition of [[air gap]] cases [[B-H loop shearing]], linearising the loop and extending the saturation to higher //H//
[[file/air_gap_in_bh_loop_png|{{air_gap_in_bh_loop.png}}]]
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==== Demagnetising field ====

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Simplified illustration of [[demagnetising field]] $H_d$ in a body magnetised with [[magnetisation]] $M$
[[/file/demagnetising_field_h_magnetica.png|{{demagnetising_field_h_magnetica.png}}]]
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Magnetic field is generated not only by the electric currents, but also by the magnetic moments which can store magnetic energy, for example by alignment due to previously applied [[process of magnetisation]]. The collections of such intrinsic magnetic moments amounts to [[magnetisation]] $M$ and becomes a source of magnetic field, as it is the case in [[permanent magnet|permanent magnets]]. If magnetic poles are created then [[magnetic field lines]] (of $H$) are by convention assumed to point from the [[north pole|N]] to the [[south pole|S pole]]. 

The [[magnetic field lines]] will close through the medium surrounding the magnet, but also through the magnet itself, in the direction opposite to the magnetisation $M$, thus lowering the effective magnetisation of the body, which is the reason why this effect is called the [[demagnetising field]] $H_d$. 

The effect can be quantified with  a unitless [[demagnetising factor]] $N_d$, which is proportional to the magnetisation and it is a function of dimensions of the body, such that for for very long structures or for magnetically closed circuits $N_d=0$ and for thin flat structures of infinite dimensions magnetised perpendicularly to the surface $N_d=1$. 

The value of demagnetising factor can be calculated analytically for ellipsoids and other very simple geometric shapes, and for a sphere $N_d=1/3$.[(Jiles)]

|  $$H_d = - N_d·M $$  |  (A/m)  |
| where: $N_d$ - demagnetising factor (unitless), $M$ - magnetisation (A/m)  ||

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[[Demagnetising field]] $H_d$ in a body magnetised with [[uniform]] magnetisation $M$[(Fiorillo)]
[[/file/demagnetising_field_m_j_h_hd_b_magnetica_png|{{demagnetising_field_m_j_h_hd_b_magnetica.png}}]]
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The illustration shows a [[permanent magnet]] magnetised with uniform //M// (or //J//), surrounded by [[vacuum]]. The magnetic field lines are shown separately for each field: //M//, //H// and //B//, which is the vector sum of //M// and //H//. 

The demagnetising field //H<sub>d</sub>// points in the opposite direction but it is non-uniform because some [[magnetic flux]] closes also through the outside of the body (where //M// = 0 and //J// = 0). 

As a result, inside the body //B// is also non-uniform and in terms of magnitude //B// < //J//. 

Outside the body, the field lines of //B// and //H// have the same shape, because in vacuum the two [[vector]] quantities differ only by a [[scalar]] constant of the [[vacuum permeability]] μ<sub>0</sub> (both //M// and //J// are zero).

This illustration also shows that at the boundary between the two media with different [[permeability]] values, for //H// the [[tangential component]] //H<sub>t</sub>// is preserved, and for //B// the [[normal component]] //B<sub>t</sub>// is preserved.

===== Measurement of H =====

The value of //H// cannot be measured directly, but it is derived by other means. 

In some [[magnetic measurement system|magnetic measurement systems]] the proportionality to current is used explicitly, as for example in such devices as [[Epstein frame]], [[single-sheet tester]] or [[toroidal sample]]. The measured quantity is current (e.g. by means of a [[shunt resistor]]), and //H// is calculated from it.[(Zurek)][(Tumanski)][(Fiorillo)]

In some other applications, it is possible to utilise the principle that the tangential component of //H// does not change at the interface between two materials. Therefore, by measuring the tangential component of magnetic field right at the surface of the sample it is possible to learn about the field immediately below the surface. However, such measurement relies on the assumption that the B-H relationship inside the sensor is linear, because the sensing coil operation is based on the [[Faraday's law of induction]], in which the measured value is the [[magnetic flux density]] //B//. This is typically achieved by using a [[non-magnetic material]] as the former on which the H-coil is wound. The signal in the coil is induced proportionally to //B// in the H-coil, but due to linearity it can be recalculated to extract the information about //H//.[(Zurek)] Other detectors such us [[Rogowski-Chattock potentiometer]] or [[Hall-effect sensor]] can be also used to detect the tangential component of //H//, but they too measure the quantity of //B// which can be then re-calculated and expressed as //H//.[(Zurek)] 

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At the interface between two materials (with different permeabilities $μ_1$ and $μ_2$) the [[tangential component]] of //H// does not change,[(Zurek)] so: $H_{t1} = H_{t2}$
[[file/tangential_h_magnetica_png|{{tangential_h_magnetica.png}}]]
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Flat [[H-coil]] made with [[PCB]] tracks[(Zurek)]
[[file/pcb_h-coil_overview_jpg|{{2dmch6/pcb_h-coil_overview.jpg}}]]
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Simple wire-wound H-coil[(Zurek)]
[[file/simple_h-coil_magnetica_jpg|{{simple_h-coil_magnetica.jpg}}]]
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===== Energy and energy density =====


[[Energy density]] $E_d$ of the energy stored in magnetic field, in a given material, can be calculated as:[(Dobbs>[[https://isbnsearch.org/isbn/9401121125|E.R. Dobbs, Basic Electromagnetism, Springer Science & Business Media, 2013, ISBN: 9401121125]])]

|  $$E_d = \int H · dB $$ |  (J/m<sup>3</sup>)  |

which for a material with linear characteristics, including high-energy permanent magnets, can be simplified to:

|  $$E_d = \frac{H·B}{2} $$ |  (J/m<sup>3</sup>)  |

It should be noted that the last equation above encompasses both the field which is applied as well as the response of the material to being magnetised (regardless which quantity is assumed to be "fundamental", //B// or //H//).


However, in non-magnetic materials for which $μ_r$ ≈ 1 it can be written that: 

|  $$B = μ_0 · H $$ |  (T)  |  and  |  $$\frac{B}{μ_0} = H $$ |  (A/m)  |

Therefore, substitution can be made such that eliminates one of the variables, making the energy density proportional to the square of either just //B// or just //H//. Depending on the publication, both forms are used,[([[https://isbnsearch.org/isbn/4431545263|Teruo Matsushita, Electricity and Magnetism, Springer Nature, 2013, ISBN 4431545263]])][([[http://dx.doi.org/10.4236/msa.2013.412109|Ji-Yeon Shim, Bong-Yong Kang, Distribution of Electromagnetic Force of Square Working Coil for High-Speed Magnetic Pulse Welding Using FEM, Materials Sciences and Applications, Vol. 4 (12), 2013, pp. 856-862. doi: 10.4236/msa.2013.412109]])] often not stating the implicit assumption of $μ_r$ ≈ 1. These two forms are equivalent, although expression with //B// appears to be more popular. If the assumption $μ_r$ ≈ 1 cannot be made then energy is proportional to the product of $B·H$, or the equation has to include also the relative permeability $μ_r$.[(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]])]

|  for $μ_r \approx$ 1  |  $$E_d = \frac{B^2}{2·μ_0} = \frac{μ_0·H^2}{2} $$ |  (J/m<sup>3</sup>)  |
|  for $μ_r \neq 1$  |  $$E_d = \frac{B^2}{2·μ_r·μ_0} = \frac{μ_r·μ_0·H^2}{2} $$ |  (J/m<sup>3</sup>)  |



===== Hysteresis loop and power loss =====

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In [[ferromagnet|ferromagnets]] the power or energy loss is proportional to the area of the [[B-H loop]]
[[file/M4 B-H loop 50Hz_png|{{M4 B-H loop 50Hz.png}}]]
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[[Soft magnetic materials]] are used for energy transformation under alternating or pulsed magnetisation regimes. Energy efficiency of a [[magnetic circuit]] depends on the power lost in the given magnetic material. 

For one cycle of magnetisation (for time from 0 to //T//), the total energy lost in the material is proportional to the area of the traced [[B-H loop]] (hysteresis loop). The numerical value of loss can be calculated as:[(Zurek)]

|  $$P = \frac{f}{D}·\int_0^T \left(\frac{dB}{dt} · H \right) dt   $$  |  (W/kg)  |
| where: $f$ - frequency of magnetisation (Hz), $D$ - density of material (kg/m<sup>3</sup>)  ||

The specific power loss is an important [[figure of merit]] for soft magnetic materials, and for example it is the basis of categorisation of [[electrical steel|electrical steels]].[(Tumanski)]

Because of the operating conditions such B-H loops are measured under conditions of sinusoidal voltage, which also enforces sinusoidal //B//. The waveform of //H// can become severely distorted especially when material operates close to saturation. This is effect is responsible for example for the [[inrush current]] in [[transformer|transformers]].

==== Models of B-H loop ====
There are several analytical, statistical and numerical models which are used for mathematical description of the B-H loop trajectories, for the purpose of "prediction" of material behaviour under pre-defined or arbitrary magnetisation conditions. Total magnetic losses can be calculated because real B-H loops represent such total losses, and the models attempt to represent the non-linear trajectories of such loops.

[[Hysteresis model|Models]] such as [[Jiles-Atherton model|Jiles-Atherton]] or [[Preisach model|Preisach]] use //H// as the independent variable representing the applied excitation, as dictated by the [[Ampere's circuital law]].[(Tumanski)]

===== Coercivity =====

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In high-energy [[magnet|magnets]] there are two values of coercivity: //<sub>J</sub>H<sub>c</sub>// and //<sub>B</sub>H<sub>c</sub>//[(Arnold)]
[[file/permanent_magnet_bhmax_curve_magnetica_png|{{permanent_magnet_bhmax_curve_magnetica.png}}]]
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Coercivity //H<sub>c</sub>// is defined as the point at which the [[hysteresis loop]] crosses the horizontal axis (//B// or //J// or //M// = 0), as measured for a given material. The values of coercivity are used for the broad classification of magnetic materials into: [[soft magnetic materials|soft]] (//H<sub>c</sub>// < 1 kA/m), [[hard magnetic materials|hard]] (//H<sub>c</sub>// > 100 kA/m) and [[semi-hard magnetic materials|semi-hard]] (1 kA < //H<sub>c</sub>// < 100 kA/m).[(Zurek)]

The of coercivity is linked to the amount of energy which is required to magnetise (and demagnetise) a given magnetic material. Soft magnetic materials have narrow B-H loop, they are easy to magnetise and therefore they have low values of coercivity. 

In high-energy permanent magnets the coercivity values are very high (wide hysteresis loop), and because of the significant differences between the values of [[magnetic flux density]] //B// and [[magnetic polarisation]] //J// two values of coercivity can be distinguished: //<sub>J</sub>H<sub>c</sub>// and //<sub>B</sub>H<sub>c</sub>//. For such magnets the almost straight line extending from the point of coercivity //<sub>B</sub>H<sub>c</sub>// to the point of remanence //B<sub>r</sub>// denotes the operating conditions of a magnetic in a given magnetic circuit.[(Arnold>[[https://www.arnoldmagnetics.com/wp-content/uploads/2017/10/Vocabulary-of-Magnetism.pdf|Arnold Magnetic Technologies, TECHNotes, The Vocabulary of Magnetism, 2016]], {accessed 2021-05-02})] Application of magnetic field greater than coercivity can permanently demagnetise even a high-energy magnet. For lower energy magnets the demagnetisation can happen even at fields lower than coercivity.

In soft magnetic materials under normal operating conditions (significantly below [[saturation]]) $B \approx J$ and therefore just single value of coercivity //H<sub>c</sub>// is measured (because //<sub>J</sub>H<sub>c</sub>// ≈ //<sub>B</sub>H<sub>c</sub>//).


===== See also =====
  *[[Magnetic field]]
  *[[Magnetic flux density]] //B//
  *[[Confusion between B and H]]

===== References =====
~~REFNOTES~~

{{tag> Magnetic_field Counter}}