# Encyclopedia Magnetica

Encyclopedia of magnetics and electromagnetics.

magnetic_field

# Magnetic field

 Stan Zurek, Magnetic field, Encyclopedia-Magnetica.com, {accessed on 2019-02-22}

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 varying electric field.1)2) The existence of magnetic field is responsible for magnetism (and electromagnetism).

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.3)

Iron filings attracted by magnetic field of a horseshoe magnet 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, E. 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.4)

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
Richard M. Bozorth
Ferromagnetism5)
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.
Soshin Chikazumi
Physics of Ferromagnetism6)
Generally a region of space in which a magnetic pole experiences an applied force is called a magnetic field.
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.
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.
Tai L. Chow
Introduction to Electromagnetic Theory: A Modern Perspective9)
The definition of the magnetic field is slightly more difficult do formulate. […] Magnetic forces only acting on moving charges, so it is natural to magnetic force F exerted on a positive charge q whose velocity is v to define B.
S.K Gupta
Electro Magnetic Field Theory10)
[…] magnetic interactions can also be described in two steps: First, the moving charge (source) produce a magnetic field. Second, this magnetic field interacts with other moving charges, called objects, to produce the magnetic force on the objects.
V.A.Bakshi, A.V.Bakshi
Electromagnetic Field Theory11)
The region around a magnet within which the influence of the magnet can be experienced is called magnetic field.

## 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 precise 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

Magnetic field is a concept of a kind of energy contained in a given volume of space. This energy can be physically and mathematically described by various means or quantities. There are several magnetic units, which refer to such measurable quantities. However, the two basic ones are magnetic field strength $H$, and magnetic flux density $B$ (also frequently referred to as magnetic induction).

Electric voltage $V$ and electric current $I$ are both required to fully quantify the effects of electricity. By analogy both magnetic flux density $B$ and magnetic field strength $H$ are required for quantifying the effects of magnetism.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)

There are also magnetisation $M$ and polarisation $J$, which are related to $B$ and $H$ through other quantities like permeability $μ$ and susceptibility $χ$.

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

### Magnetic field strength H

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

by S. Zurek, E. 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 anisotropic

by S. Zurek, E. 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 by JanDerChemiker, CC-BY-3.0
FEM simulation illustrating magnetic field lines around a bar magnet

by S. Zurek, E. 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, E. 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 linked with the demagnetising factor.

### Magnetic monopole

A magnetic monopole cannot be created by dividing a magnetic dipole

Sbyrnes321, CC-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” (more like electron holes).33) So although the behaviour of such structures resembles the monopoles, they are not monopoles in the same strict sense as the electrons and protons are a source of electric charge.

### Magnetic forces

Path of moving electrons bent into a circle by perpendicularly applied magnetic field 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.34)

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.35)

## 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: diamagnetism, paramagnetism, ferromagnetism, etc. These result from the type of chemical elements involved, atomic structure, temperature, etc.36)

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 like: geomagnetism (Earth's magnetic field37)), paleomagnetism (magnetic properties of geological structures38)), biomagnetism (magnetic phenomena in living organisms39)), cardiomagnetism and neuromagnetism (magnetic field generated by heart and brain, respectively40)), and many more.

## 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:41)

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

by S. Zurek, E. 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.46)

On sub-molecular level particles like electron, proton or neutron possess magnetic moments.47)48)

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.49)

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.50)

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).51)

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.52)