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. Magnetic induction is also known as magnetic flux density.

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 the magnetic field H from the magnetic induction B. Yet, in practice physicists and chemists almost always call B a magnetic field. It is likely that this is because the term "induction" implies somehow an induced magnetic moment, which usually is not present. Hence the term "induction" is confusing. In science, phrases as: "This EPR spectrum was measured at a magnetic field of 3400 gauss", and "Our magnet can achieve magnetic fields as high as 20 tesla" are common. Most scientists use the term "field", well aware of the fact that, strictly speaking, gauss and tesla are units of magnetic induction.

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,

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

where μ₀ is the magnetic constant (equal to 4π⋅10⁻⁷ N/A²). Note that with the use of Gaussian units and in vacuum, the dimensions of H (Oer) and of B (G = gauss) are equal, 1 Oer = 1 G.

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).