# Encyclopedia Magnetica

Encyclopedia of magnetics and electromagnetics.

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# Magnetic field strength

 Stan Zurek, Magnetic field strength , Encyclopedia-Magnetica.com, {accessed 2019-06-16}

Magnetic field strength $H$ - a physical quantity used as one of the basic measures of the intensity of magnetic field.1)2) The unit of magnetic field strength3) is ampere per metre or A/m.

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 values correlated with them). In vacuum, at each point the $H$ and $B$ vectors are oriented along the same direction and are directly proportional with the magnetic constant, 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.4)

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.5) Therefore, strictly speaking, other names like magnetic field intensity or magnetic field which can be encountered in everyday technical jargon6) 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 $H$
Richard M. Bozorth
Ferromagnetism7)
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.
David C. Jiles
Introduction to Magnetism and Magnetic Materials8)
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: A magnetic field strength of 1 ampere per meter is produced at the center of a single circular coil of conductor of diameter 1 meter when it carries a current of 1 ampere.
Magnetic field, Encyclopaedia Britannica9) 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 and the magnetic field B as the total magnetic field including also the contribution made by the magnetic properties of the materials in the field.
V.A.Bakshi, A.V.Bakshi
Electromagnetic Field Theory10)
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.

## 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.11)

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.12)

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 $B·H \approx 0$.13)

## 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

$H$ 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).

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 and the physical sizes involved (e.g. lenght and diameter of the conductor, etc.)

The relationship between $H$ and $I$ is often shown by employing the Biot-Savart's law.14) or Ampere's law15) However, often (but not always16)) 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 in literature there is an unspoken assumption that the derivation is carried out for vacuum and not for an arbitrary medium with a different permeability17). 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 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.18)19) 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.20)

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 mathematically identical.