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.

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).

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 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.