# Encyclopedia of magnetics and electromagnetics.

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# Confusion between B and H

 Stan Zurek, Confusion between B and H, Encyclopedia-Magnetica.com, {accessed on 2020-07-04} reviewed by Jeanete Leicht, 2014-12-19

Confusion between B and H - a problem recognised in the literature about the confusion between quantities and physical units of magnetic flux density B and magnetic field strength H .[(Jiles_p5_11>David C. Jiles, Introduction to Magnetism and Magnetic Materials, Second Edition, CRC Press, 1998, ISBN 9780412798603, p. 5-11)] [(Cohen>Douglas L. Cohen, Demystifying Electromagnetic Equations: A Complete Explanation of EM Unit Systems and Equation Transformations, SPIE Press, 2001, ISBN 9780819442345, p. vii-viii)][(Physics Forums, In magnetism, what is the difference between the B and H fields? {accessed 5 Feb 2014})][(https://www.researchgate.net/post/Difference_between_B_H_and_M_in_magnetics ResearchGate, Difference between B, H and M in magnetics, {accessed 5 Feb 2014})]

Both B and H are strictly defined in terms of measurement units as well as their physical meaning.[(Derived units expressed in terms of base units, SI brochure, {accessed 6 Jan 2014})][(Units with special names and symbols; units that incorporate special names and symbols, SI brochure, {accessed 6 Jan 2014})][(Jiles_p5_11)]

Electric current I generates magnetic field, in the form of magnetic field strength H, regardless the type of the surrounding medium

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

Electric current I produces around itself magnetic field strength H, whose amplitude is independent of the type of a continuous isotropic medium (regardless if it is non-magnetic, magnetic, non-linear, etc.)[(Jiles_p5_11)]

For an infinitely long straight round wire it is:

 $H = \frac{I}{2 ⋅ \pi ⋅ r}$ (A/m)

where: r - radius of a circle enclosing the current I, π - the mathematical constant.

Flux density B is a response of the medium to the applied excitation H. The relationship is defined by the magnetic permeability μ, such that:

 $B = \mu ⋅ H$ (T)

Thus, B is related to the properties of the material and its relation to the applied excitation (e.g. electric current) can be highly non-linear.

If the medium is non-continuous or anisotropic then magnetic poles or a demagnetising field could be created, which themselves become sources of local excitation and they add to the source.

## Reasons for the confusion

Some of the several possibly contributing reasons are described below.

### Systems of units

Flux density B is a different physical quantity from magnetic field strength H

$B \neq H$

There are several internationally used system of units, which include the electromagnetic units. The most popularly adopted are SI units, in the so-called “rationalised” metre-kilogram-second system (MKS). But there are at least four based on centimetre-gram-second system (CGS)[(Cohen)]

In the previously used CGS system, the permeability μ was unitless[(Goldman>Alex Goldman, Handbook of Modern Ferromagnetic Materials, Springer, 1999, ISBN 9780412146619, p. 641)] and for free space it was mathematically true that μ = 1, so it could be written that B = H.

(because for free space the magnetisation term can be omitted).[(Nicola A. Spaldin, Magnetic Materials: Fundamentals and Applications, Cambridge University Press, 2010, ISBN 9781139491556, p. 14)]

Such notation is in itself confusing, because it makes more difficult to distinguish between the two different quantities, especially that the unit of B was gauss and of H was oersted, yet they could be equal to one another (in a similar sense as 1 inch = 25.4 mm).

For instance, in the CGS system the intrinsic induction Bi (equivalent of polarisation J in the SI system) is defined as Bi = B - H (even though B is given in gauss and H in oersteds).[(Glossary of Magnet Terms, Dura Magnetics Inc. {accessed 2015-10-27})]

Such equality is no longer true in the currently used SI system.

### Advantages of the CGS system

In theoretical physics [(S. N. Ghosh, Electromagnetic Theory and Wave Propagation, CRC Press, 2002, ISBN 9780849324307, p. 51)] the CGS units are continued to be used alongside the MKS units. The approach used in the CGS system has some advantages when performing certain electromagnetic calculations (e.g. lack of the 4·π factor), even though a distinction has to be made between “electrostatic” and “electromagnetic” CGS units.[(Cohen)][(Ghosh)]

The CGS units are also used by convention, simply because of the historic reasons, especially in the area of permanent magnets, whose energies are often quoted in MGOe (mega-gauss-oersted).[(Edward P. Furlani, Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications, Academic Press, 2001, ISBN 9780122699511, p. 54)][(Alex Goldman, Modern Ferrite Technology, Springer, 2006, ISBN 9780387294131, p. 227)]

The CGS units are still used in many applications in the USA, whereas Europe and other countries rely almost exclusively on the SI units.[(Perry A. Holman, Magnetoresistance (MR) Transducers and how to use them as sensors, 1st edition, Honeywell International, 2004, p. 4, {accessed 15 Feb 2014})]

### Multiplicity of names

There are several quantities and related magnetic terms: magnetic field strength, magnetic flux density, magnetisation, polarisation, magnetic flux and also magnetic field.

Under certain conditions, the practical differences between some of these quantities are small. For instance, for magnetically soft materials under low-amplitude excitation the difference between B and J is negligible for most practical purposes.

Also for non-magnetic materials it can be assumed that B and H have a linear relationship so if one is known then the other can be easily calculated. For those not proficient in the physics of magnetism such notation could suggest that the distinction might not be significant enough to differentiate between the quantities. However, the distinction has to be made even on the basis of units.

### Scientific background

Different authors have various scientific background. From the viewpoint of their discipline some equations are more “important” than others, for instance due to frequency of application to solving various problems. Hence, either H or B can be treated as “more important” than the other.

But in theoretical physics often B is regarded as more fundamental, because magnetic forces are proportional to B not H.[(Charles F. Stevens, The Six Core Theories of Modern Physics, MIT Press, 1996, ISBN 9780262691888, p. 85)]

## Definitions of B and H

The concept of magnetic field is fundamental in physics, but it is difficult to define - in the same sense it is it difficult to define what is “time”. Therefore, the definitions of magnetic field, magnetic field strength $H$ and magnetic flux density $B$ are usually somewhat convoluted, as shown in some examples from the literature below.

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>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. 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.