Magnetic induction

In physics, and more in particular in the theory of electromagnetism, magnetic induction (commonly denoted by B) is a vector field closely related to the magnetic field H.

The SI unit measuring the strength of B is T (tesla), and the Gaussian unit is gauss. One tesla is 10 000 gauss. To indicate the order of magnitude: the magnetic field (or better magnetic induction) of the Earth is about 0.5 gauss = 50 μT. A medical MRI diagnostic machine typically supports a field of 2 T. The strongest magnets in laboratories are presently about 30 T.

Note on nomenclature

Every textbook on electricity and magnetism distinguishes clearly the magnetic field H from the magnetic induction B. Yet, in practice physicists and chemists refer to B as magnetic field, the reason being that the term "induction" implies to them an induced magnetic moment, which is usually not manifestly present; the term "induction" comes very unnatural to scientists talking about magnetic fields. It is common to hear and read phrases as: "This EPR spectrum was meausured at a magnetic field of 3400 gauss", or "Our magnet can achieve magnetic fields as high as 20 tesla". Most scientists say or write this, knowing very well that gauss and tesla are units of magnetic induction (in Gaussian and SI units, respectively), and that the corresponding units of magnetic field are oersted and ampere/meter.

Relation between B and H

In vacuum, that is, in the absence of a ponderable, continuous, and magnetizable medium, the fields B and H are related as follows,

${\begin{aligned}\mathbf {B} &=\mu _{0}\mathbf {H} \qquad {\hbox{in SI units}}\\\mathbf {B} &=k\mathbf {H} \qquad \;{\hbox{in Gaussian units}},\\\end{aligned}}$

where μ₀ is the magnetic constant (equal to 4π⋅10⁻⁷ N/A²) and k ( = 1 gauss/oersted) changes the dimension of H (Oer) into that of B (gauss).

In a continuous magnetizable medium the relation between B and H contains the magnetization M of the medium,

${\begin{aligned}\mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )\qquad &{\hbox{in SI units}}\\\mathbf {B} =\mathbf {H} +4\pi \mathbf {M} \qquad \;\;&{\hbox{in Gaussian units}},\\\end{aligned}}$

which expresses the fact that B is modified by the induction of a magnetic moment (non-zero magnetization) in the medium.

In almost all non-ferromagnetic media, the magnetization M is linear in H,

$\mathbf {M} ={\boldsymbol {\chi }}\mathbf {H} \quad \Longleftrightarrow \quad M_{\alpha }=\sum _{\beta =x,y,z}\chi _{\alpha \beta }H_{\beta }.$

For a magnetically isotropic medium the magnetic susceptibility tensor χ is a constant times the identity 3×3 matrix, χ = χ_m 1. For an isotropic medium we obtain for SI and Gaussian units, respectively, the relation between B and H,

${\begin{aligned}\mathbf {B} &=\mu _{0}(1+\chi _{m})\mathbf {H} \equiv \mu \mathbf {H} \\\mathbf {B} &=(1+4\pi \chi _{m})\mathbf {H} \equiv \mu \mathbf {H} .\\\end{aligned}}$

The material constant μ, which expresses the "ease" of magnetization of the medium, is called the magnetic permeability of the medium. In most non-ferromagnetic materials χ_m << 1 and consequently B ≈ μ₀H (SI) or B ≈ H (Gaussian).