Magnetic induction

In physics, and more in particular in the theory of electromagnetism, magnetic induction (commonly denoted by B) describes a magnetic force (a vector) at every point in space; it is a vector field. In non-relativistic physics, the space in question is the three-dimensional Euclidean space $\scriptstyle \mathbb {E} ^{3}$—the infinite world that we live in.

A magnetic force can act on

• a permanent magnet (which is a magnetic dipole or—approximately—two magnetic monopoles),
• magnetizable (ferromagnetic) material like iron,
• moving electric charges (through the Lorentz force).

The physical source of a magnetic field can be

• one or more permanent magnets (see Coulomb's law)
• one or more electric currents (see Biot-Savart's law),
• time-dependent electric fields (displacement currents),

or combinations of these three.

In general the strength of the magnetic field decreases as a low power of 1/R, the inverse of the distance R of the field point to the source.

Magnetic induction is also known as magnetic flux density because it is magnetic flux per unit of surface. This relationship is based on Faraday's law of magnetic induction. The SI unit measuring the strength of B is T (tesla = weber/m²), and the Gaussian unit of B is G (gauss = maxwell/cm²) . One tesla is 10 000 gauss.

To give an indication of magnitudes: the magnetic field (or better magnetic induction) of the Earth is about 0.5 G (gauss = 50 μT). A horse shoe magnet is about 100 G. A medical MRI diagnostic machine typically supports a field of up to 2 T (20 kG). The strongest magnets in laboratories are currently about 30 T (300 kG).

Note on nomenclature

Most textbooks on electricity and magnetism distinguish the magnetic field H from the magnetic induction B (both quantities are vector fields). Yet, in practice physicists and chemists almost always call B the magnetic field. It is likely that this is because the term "induction" suggests an induced magnetic moment. Because such a moment is usually not present, the term induction is confusing. In science, phrases are common 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". That is, most scientists use the term "field" with units tesla or gauss, while strictly speaking, gauss and tesla are units of the magnetic induction B.

Relation between B and H

In vacuum (also known as the microscopic case), in the absence of a 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 in Gaussian units the dimensions of H (Oer) and of B (G = gauss) are equal, 1 Oer = 1 G, although the units have an unrelated definition (Oer is based on the field of a solenoid, and G is magnetic flux/surface). Note also that for the vacuum it is needless to introduce both B and H.

In the the "macroscopic" case, the relation between B and H contains the magnetization M of the medium,

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

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 the relation between B and H is in SI and Gaussian units, respectively,

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

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

The two macroscopic Maxwell equations that contain charges and currents, are equations for H and electric displacement D. This is a consequence of the fact that current densities J and electric fields E (due to charges) are modified by the magnetization M and the polarization P of the medium. In SI units the Maxwell equation for the magnetic field is:

${\boldsymbol {\nabla }}\times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}.$

The microscopic (no medium) form of this equation is obtained by eliminating D and H via D = ε₀E and H = B/μ₀ (P = 0 and M = 0).

The two Maxwell equations that do not contain currents and charges give relations between the fundamental fields E and B, instead of between the auxiliary fields H and D. For instance, Faraday's induction law in SI units is,

${\boldsymbol {\nabla }}\times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}.$

This equation is valid microscopically (vacuum) as well as macroscopically (in presence of a ponderable medium).