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Magnetic field

Stan Zurek, Magnetic field,, {accessed on 2020-10-22}


Magnetic field - a region in space, in which magnetic forces are observable. Magnetic forces are mechanical forces acting as a result of magnetic interactions. Magnetic field is always generated around electric current, or more generally by moving electric charges or by varying electric field E, as well as magnetic dipole moment of sub-atomic particles.1)2) The existence of magnetic field is responsible for magnetism and electromagnetism.

In vacuum, a force F acting on an electric charge q moving with velocity v is due to the electric field E and magnetic field B (magnetic flux density), such that the B vector which satisfies the following equation is assumed to be the definition of magnetic field:3)

$$ \mathbf{F} = q·\mathbf{E} + q·\mathbf{v} × \mathbf{B} $$

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.4)

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Iron filings attracted by magnetic field of a horseshoe magnet horseshoe_magnet_and_iron_filings.jpg by Oguraclutch, CC-BY-SA-3.0
Magnetic field lines are used for graphic visualisation of magnetic field (the image shows a cross-section - the wire is perpendicular to the page)

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

Iron filings follow magnetic field lines around a single wire with current (perpendicular to the page) by Popular Science Montly, Vol. 56, 1899-1900, Public Domain

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.5)

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'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.

Publication Definition of magnetic field Definition of magnetic field strength $H$ Definition of magnetic flux density $B$
Richard M. Bozorth
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 Materials7)
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 Britannica8) 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 […]9) […] 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.10)
E.M. Purcell, D.J. Morin, Electricity and magnetism11) 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 $\mathbf{B}$.

Electromagnetic and magnetostatic field

The name electromagnetic field is used to refer to a field continuously variable in time. Variable electric field (moving electric charges) produces variable magnetic field. In turn, variable magnetic field produces variable electric field. This result in electromagnetic field, in which the electric and magnetic components are inseparably interlinked.12)

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).

The distinction is similar as in analysis of electrical circuits where electrostatic is used.13) 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.

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 table above).

The concepts of near field and far field are introduced in analysis of electromagnetic fields. All magnetostatic effects can be treated as occurring in the near field region.14)

Medium in which magnetic and electromagnetic field propagates

Mathematical equations can be used to describe electromagnetic field either as a wave (electromagnetic waves) or as particles (e.g. photons), and both representations were proved to be correct from 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 free space or vacuum.

In all quantum field theories like quantum chromodynamics (QCD) the absence of 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.15) Such property can be therefore responsible for allowing the electromagnetic energy to propagate in a seemingly “empty” medium. However, this is only a theoretical treatment and with the current state of science and technology cannot be proved or disproved.

Quantities and units

See also the main article: Confusion between B and H.

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:

Electric charge (in the form of electron and proton) are the most basic components responsible for all electrical phenomena. However, it is more useful in electrical engineering to operate with electric voltage $V$ and electric current $I$, combination of which fully quantifies the effects of electricity (such us transmitted power or energy). Additional parameters such as resistivity or conductivity are also very relevant.

By analogy, in engineering, both magnetic flux density $B$ and magnetic field strength $H$ are required for quantifying the effects of magnetism in matter, for example stored or lost energy.16) 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). 17)18)

Magnetic field strength H

See also the main article: Magnetic field strength.
Electric current I generates around itself magnetic field strength H electric_current_generates_magnetic_field_magnetica.jpg

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

A current I produces around itself magnetic field strength H, whose amplitude is independent of the type of a uniform isotropic medium (regardless if it is non-magnetic, magnetic, non-linear, etc.)19)

As an example, H = 1 A/m could be produced at the centre of a single circular loop with diameter of 1 m with a current of 1 A.20) Alternatively, an infinitely long, cylindrical conductor with a current of 1 A generates H = 1 A/m around itself in a circle whose circumference is 1 m (which translates to a radius of around 15.9 cm).

The direction of generated magnetic field is such that it follows the right-hand rule, so that if the thumb points towards the direction of current flow then the fingers show direction of the generated magnetic field strength (see the image on the right).

If the medium is not isotropic, or there are several different materials then the field distribution is affected by demagnetising field caused by magnetic poles. These poles become new sources of magnetic field which change distribution of the original applied field.21)

Magnetic flux density B

See also the main article: Magnetic flux density.
In ferromagnets the B-H relationship is non-linear, history-dependent and typically anisotropic

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

Application of H to a medium (e.g. a given material) causes it to respond with magnetic flux density B. The amplitude of B depends both on the applied excitation as well as the material properties. Constant H (e.g. generated by a DC current) produces constant B, whose value depends on the magnetic permeability μ of the material.

For free space (vacuum) the B-H relationship is fully linear, with the proportionality factor μ0 = 4·π·10-7 (H/m) which is a universal physical constant: 22)

$B = \mu_0 · H$ (T)

Permeability of a medium different than free space can be defined as a product of dimensionless number relative to μ0 and the value of μ0 itself. Hence, μ = μr·μ0. So for instance, relative permeability μr 2 means that the given material has permeability twice the value of μ0.23)

For uniform and isotropic materials permeability can be expressed as scalar, even though H and B are vectors. For anisotropic materials permeability is different in various directions and tensor approach has to be used for more correct representation.

When excitation H varies in time then additional effects are introduced, like for instance eddy currents, which affect the apparent permeability which defines the B and H relationship. In magnetic material like ferromagnets the relationship between B and H can be non-linear, history-dependent and highly anisotropic.

Magnetic field lines

See also the main article: Magnetic field lines.

Magnetic field lines are a theoretical concept and there are no “lines” in reality. However, the lines represent direction which is tangent to the direction of the vector of magnetic flux density at a given location. If iron particles are placed in vicinity of a magnet then they will form elongated structures which seem to follow lines extending between magnetic poles. In a similar way, a magnetic compass placed next to a magnet will have its needle following the contours created by magnetic particles.

Iron filings create elongated line-like structures around a permanent magnet File source: by JanDerChemiker, CC-BY-3.0
FEM simulation illustrating magnetic field lines around a bar magnet

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

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 means greater amplitude. The lines can never cross and each line is a closed curve (i.e. has no beginning or end). In theory, the lines 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. 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.24)

Magnetic dipole and moment

Animation of field lines around a magnetic dipole and an electric dipole

by Geek3, CC-BY-SA-3.0

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 monopoles with opposing charges separated by a given distance (per analogy to a dipole created by two 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.25)

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.

Magnetic poles are created for any loop of electric current

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

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 south pole. 26)

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, an simple example being a bar magnet.27)

The location of the magnetic poles depends on the geometry of the magnetic circuit and is also linked with the demagnetising factor.

Magnetic monopole

A magnetic monopole cannot be created by dividing a magnetic dipole

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

Magnetic monopoles were theoretically proposed by Paul Dirac in 1931 as a possible explanation of electric charge quantisation.28)

A monopole is a hypothetical elementary particle which exhibits properties of a single magnetic pole. This cannot happen simply by dividing an existing magnetic dipole, because every division will lead to creation of smaller dipoles.29)

So far, all the experiments failed at directly detecting the existence of magnetic monopoles.30)31)

The results of some experiments were claimed to prove existence of magnetic monopoles, but these experiments had to be carried out under special conditions. For instance, the “monopole-like structures” were observed not in magnetic field, but in the so-called “syntetic magnetic field”.32)

In a different experiment the “monopoles” were not particles in the same sense as for example the electron, but rather “quasi-particles” or “topological monopoles”.33)34) So although the behaviour of such structures resembles monopoles in certain aspects, they are not magnetic monopoles in the same strict sense as the electrons and protons are electric monopoles (the most fundamental sources of electric charge).

Magnetic forces

Path of moving electrons bent into a circle by perpendicularly applied magnetic field ASCII by M. Białek, CC-BY-SA-3.0

Electric charges at rest act with electrostatic force, which is described by Coulomb's law. When these charges are in motion there is an additional force, which is commonly called the magnetic force. By using relativistic Lorentz transformation it can be theoretically shown that such force arises as an extension of the electrostatic force between electrical charges.35)

A path of moving electric charge is deflected in magnetic field. This basic behaviour can exhibit itself as attraction or repulsion between any sources of magnetic field (moving electric charges, current carrying conductors, permanent magnets, electromagnets, etc.)

In a lossless medium the magnetic field can bend a path of the moving charges into circles or spirals, with a dimension defined by the Larmor radius. This phenomenon is used for instance in a type of particle accelerator called cyclotron.36)

Type of magnetisms

By definition, magnetic effects are caused by magnetic field and the 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 build from atoms exhibits some type of magnetic behaviour, of which there are several fundamental types, resulting from the type of chemical elements involved, atomic structure, temperature, etc.37)

There are several ways that the magnetic field can interact with other branches of science and technology. Such cross-related disciplines often have their own names, such as (to give just a few examples):

Maxwell's equations

See also the main article: Maxwell's equations.

Maxwell's equations fully describe mathematically the interrelation between electric and magnetic fields. The early version of these equations were first collated by Scottish physicist James Clerk Maxwell. Subsequently they were simplified and unified so that today four fundamental equations are used, whose physical meaning can be summarised as follows:42)

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.

Generation of magnetic field

Electromagnet - a source of magnetic field electromagnet_200mm_by_magneto.jpg

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

Generation of magnetic field is associated with movement of electric charges. On macroscopic level the magnetic field is generated always if there is electric current flowing. However, also changing 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.47)

Lightning is a sudden discharge of electric current which generates an impulse of magnetic field by Nico36, Public Domain

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 wide spectrum, including the visible light. Such magnetic pulse is capable of magnetising natural minerals, such as magnetite.48)

There are many other mechanisms in which magnetic field can be generated, for example with the core of a planet, on a global level (see: Earth's magnetic field).

On sub-molecular level particles like electron, proton or neutron possess magnetic moments.49)50)

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” effects as understood in everyday language.51)

Ferromagnetic materials have 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 an electromagnet, transformer or electric motor can be created. The excitation is still applied either as an electric current in a winding or as a permanent magnet, but the generation of magnetic field is aided greatly by the presence of magnetic core made of a suitable ferromagnetic material.

If very strong magnetic fields are required then the usefulness of ferromagnets is diminished. Fields exceeding 3 teslas can be generated with Bitter electromagnets, superconducting magnets, pulse methods or compression of magnetic flux can be used. 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.52)

Weak magnetic fields do not need magnetic cores. Most of telecommunication based on electromagnetic wave (like mobile phone and television) can use non-ferromagnetic antennas to transmit and receive electromagnetic signals (and hence by definition the magnetic component within the electromagnetic field).53)

All electric currents generate magnetic field, and even 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.54)

See also


3), 11) E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022, p. 239-562
9) 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}.
10) 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}.
magnetic_field.txt · Last modified: 2020/10/18 22:01 by stan_zurek

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