# Encyclopedia of magnetics and electromagnetics.

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# Magnetic field strength

 Stan Zurek, Magnetic field strength , Encyclopedia-Magnetica.com, {accessed 2020-07-04}

Magnetic field strength $H$ - a physical quantity used as one of the basic measures of the intensity of magnetic field.[(Mansfield>Michael Mansfield, Colm O'Sullivan, Understanding Physics, John Wiley & Sons, 2010, ISBN 9780470746387, p. 407)][(Britannica>Magnetic field, Encyclopædia Britannica Online, {accessed 2016-07-05})] The unit of magnetic field strength[(Jiles>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.

Magnetic field strength $H$ can be thought of as excitation and the magnetic flux density $B$ as the response of the medium.

Fig. 1. 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)

by S. Zurek, E. Magnetica, CC-BY-3.0

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, like magnetisation or polarisation). 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 non-uniform or 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>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 physical units.[(BIPM>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.

## 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).

Publication Definition of magnetic field Definition of magnetic field strength $H$ Definition of magnetic flux density $B$
Richard M. Bozorth
Ferromagnetism[(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 ara 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>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 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 Magnetic field, Encyclopædia Britannica Online, {accessed 2020-03-07}.)]
V.A.Bakshi, A.V.Bakshi
Electromagnetic Field Theory[(Bakshi>V.A.Bakshi, A.V.Bakshi, Electromagnetic Field Theory, Technical Publications, 2009, ISBN 9788184316803, p. 7-1..7-4)]
The region around a magnet within which the influence of the magnet can be experienced is called magnetic field. The quantitative measure of strongness or weakness of the magnetic field is given by magnetic field intensity or magnetic field strength. The magnetic field intensity at any point in the magnetic field is defined as the force experienced by a unit north pole of one weber strength, placed at that point. The total magnetic lines of force i.e. magnetic flux crossing a unit area in a plane at right angles to the direction of flux is called magnetic flux density. It is denoted as B and it is a vector quantity.

## Analogy to electric circuits

### Electric circuit

Electric voltage $V$ and electric current $I$ 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 conductance $G$ (or resistance $R$) of a given medium.

The product of $V$ and $I$ is a measure of power and can be expressed in watts (W), which over time $t$ gives energy $E$ in joules dissipated or transformed in the circuit.

If a fixed value of $V$ is applied to a uniform electric circuit then the resulting amplitude of $I$ is dictated by the conductivity of the circuit. For the same voltage, higher values of conductivity will result with higher current.

If a fixed amplitude of current is forced through a conductor then the voltage drop will be proportional to the resistance of the conductor. For a superconductor the resistance is zero, there is no voltage drop and therefore $V · I = 0$. Conversely, if voltage is applied to electric insulator then very little current flows and thus only small amount of energy is dissipated, so that $V · I \approx 0$.

### 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 circuits. The proportionality between $H$ and $B$ is dictated by magnetic permeability $μ$ of a given medium.[(White)]

Under steady state conditions, the product of $H$ and $B$ is a measure of specific energy in J/m3, 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 magnets.[(Grades of Neodymium, NdFeB-Info.com, {accessed 2016-06-28})]

If a fixed value of $H$ is applied to a magnetic circuit then the resulting amplitude of $B$ is dictated by permeability $μ$ of the circuit. For the same magnetic field strength, higher values of permeability will result with higher flux density.

Soft ferromagnets have large values of permeability and thus application of small $H$ results with large values of $B$ without storing much energy in the magnetic field, so that $B·H \approx 0$ (e.g. as compared to permanent magnets which can store a lot of energy).

However, superconductors can completely expel magnetic field from their volume so that they behave as “magnetic insulators”, and hence also $B·H \approx 0$.[(Karl-Heinz Bennemann, John B. Ketterson (editors), Superconductivity: Volume 1: Conventional and Unconventional Superconductors Volume 2: Novel Superconductors, Springer Science & Business Media, 2008, ISBN 9783540732532, p. 34)]

## H around electric current

Fig. 2. Amplitude of magnetic field strength $H$ reduces with the distance from a conductor with electric current $I$

by S. Zurek, E. Magnetica, CC-BY-3.0

Fig. 3. Orientation of magnetic field strength $H$ vector with respect to the current $I$ follows the right-hand rule

by S. Zurek, E. Magnetica, CC-BY-3.0

$H$ is always generated around electrical current $I$, which can be a solid conductor with current or just a moving electrical 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)]

Without other sources of magnetic field and in a 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 accordingly to the Ampere's law the proportionality is dictated by the magnetic path length $l$:

 (1) $H = \frac{I}{l}$ (A/m)

For the simplest case of a round, straight and infinitely long conductor with current (Fig. 2) the value of $H$ for a given circle with a radius $r$ can be calculated from the magnetic path length of the circle:

 (2) $H = \frac{I}{2⋅π⋅r}$ (A/m)

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's law.[(Jiles)] or Ampere's law[(MIT>MIT OpenCourseWare, Biot-Savart's Law & Ampere's Law, Chapter 9)] Often (but not always[(Britannica_Maxwell_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 (and if other sources of magnetic field are absent).

The situation is slightly different for anisotropic or discontinuous medium. They can give rise to additional sources of magnetic field because new 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.

## 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)][(Bakshi)] 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>Jean-Baptiste Biot and Felix Savart. Note sur le magnetisme de la pile de Volta. Ann. Chim.Phys., 15:222–223, 1820)]

Therefore, 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 (Fig. 2) will be the same (as long as the medium is uniform and isotropic).