# Encyclopedia Magnetica

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 2019-11-21} 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 .1) 2)3)4)

Both B and H are strictly defined in terms of measurement units as well as their physical meaning.5)6)7)

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.)8)

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)9)

In the previously used CGS system, the permeability μ was unitless10) and for free space it was mathematically true that μ = 1, so it could be written that B = H.

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).12)

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

### Advantages of the CGS system

In theoretical physics 13) 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.14)15)

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).16)17)

The CGS units are still used in many applications in the USA, whereas Europe and other countries rely almost exclusively on the SI units.18)

### 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.

### Similarity of names

A common imprecise short-hand is to refer to H as “magnetic field” instead of the full name “magnetic field strength”19), but similarly for B, depending on the context.20)

However, in literature these terms are referred to by various names, and sometimes even with implied incorrect meaning. For instance, H can be referred to as “magnetic field” 21), “applied field”22) or “auxiliary field”.23) But also B might be called as “magnetic field” 24) or “auxiliary field”.25)

### 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, or indeed even “real”. This is because magnetic forces are proportional to B not H.26) 