# Magnetic induction

In physics, and more in particular in the theory of electromagnetism, **magnetic induction** (also known as **magnetic flux density**) describes a magnetic field (a vector) at every point in space. The magnetic induction is commonly denoted by **B**(**r**,*t*) and is a vector field, that is, it depends on position **r** and time *t*. In non-relativistic physics, the space on which **B** is defined is the three-dimensional Euclidean space <math>\scriptstyle \mathbb{E}^3</math>—the infinite world that we live in. The field **B** is closely related to the magnetic field **H**, often called the *magnetic field intensity*, and sometimes just the *H*-field. In fact, some authors refer to **B** as the magnetic field and to **H** as an auxiliary field.

The physical source of the field **B** can be

- one or more permanent magnets (see Coulomb's magnetic law); more microscopically, the fundamental spins of elementary particles like electrons, and their orbital angular momentum.
- one or more electric currents (see Biot-Savart's law),
- time-dependent electric fields (see displacement current),

or combinations of these three. A magnetic field exists in the neighborhood of these sources. In general the strength of the magnetic field decreases as a low power of 1/*R*, the inverse of the distance *R* to the source.

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)
- elementary particles through their intrinsic spin, which is related to their intrinsic magnetic properties through their gyromagnetic ratios.

The term *magnetic flux density* refers to the fact that **B** is magnetic flux
per unit surface. This relationship is based on Faraday's law of magnetic induction.

The SI unit measuring the strength of **B** is T (tesla = weber/m^{2}), and the Gaussian unit of **B** is G (gauss = maxwell/cm^{2}) . 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 (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** and the magnetic induction **B**. Yet, in practice physicists and chemists almost always call **B** the *magnetic field*, which is because the term "induction" suggests an induced magnetic moment. Since an induced moment is usually not in evidence, the term induction is felt to be confusing. Among scientists 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 **B**. Some authors go one step further and reserve the name "magnetic field" for **B** and refer to **H** as the "auxiliary magnetic field".

## Relation between **B** and **H**

In vacuum (also known as the microscopic case, see Maxwell equations), in the absence of a magnetizable medium, the fields **B** and **H** are related as follows,

- <math>

\begin{align}
\mathbf{B} &= \mu_0 \mathbf{H} \qquad \hbox{in SI units}\\
\mathbf{B} &= \mathbf{H} \qquad\quad \hbox{in Gaussian units},\\
\end{align}
</math>
where μ_{0} is the magnetic constant (equal to 4π⋅10^{−7} N/A^{2}). 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). In the absence of a magnetizable medium it is unnecessary to introduce both **B** and **H**, because they differ by an exact and constant factor (unity for Gaussian units and μ_{0} for SI units).

At a microscopic level, the magnetic flux **B** and the electric field **E** determine the behavior of charges. For example, a single moving charge is subject to the Lorentz force law, which in SI units is:

- <math>\mathbf{F} = q \left( \mathbf{E + v \times B }\right) \ . </math>

However, treating all the charges in a system at a microscopic level is impractical, and approximations are introduced. Some of the system is treated microscopically, and some is treated as "materials", in particular, dielectrics and magnetic materials. The response of a magnetic material to magnetic flux is introduced through the *magnetization* of the material, another vector field **M**(**r**, *t*).

In the presence of a magnetizable medium the relation between **B** and **H** involves the magnetization **M** of the medium,

- <math>

\begin{align} \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{align} </math>

To actually determine the system behavior, the magnetization **M** must be determined in terms of either **B** or **H** so that the system response depends only upon one field variable. This determination of **M** can be very complicated. For example, it may involve introduction of quantum mechanics and statistical mechanics as studied in the field of condensed matter physics. However, in many non-ferromagnetic media, the magnetization **M** is linear in **H**,

- <math>

\mathbf{M} = \boldsymbol{\chi} \mathbf{H} \quad \Longleftrightarrow \quad
M_\alpha = \sum_{\beta = x,y,z} \chi_{\alpha\beta} H_\beta.
</math>
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,

- <math>

\begin{align}
\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{align}
</math>
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** ≈ μ_{0}**H** (SI) or **B** ≈ **H** (Gaussian). For ferromagnetic materials the magnetic permeability μ can be sizeable (χ_{m} >> 1). In that case the magnetization of the medium greatly enhances the magnetic field.

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:

- <math>

\boldsymbol{\nabla} \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}}{\partial t}.
</math>
The microscopic (no medium) form of this equation is obtained by eliminating **D** and **H** via
**D** = ε_{0}**E** and **H** = **B**/μ_{0} (**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,

- <math>

\boldsymbol{\nabla} \times \mathbf{E} = - \frac{\partial \mathbf{B}}{\partial t} .
</math>
This equation is valid microscopically (vacuum) as well as macroscopically (in presence of a medium). But, of course, in the microscopic case the detailed microscopic currents and charges due to the elementary sources appear, while in the macroscopic case some of these microscopic currents and charges are subsumed in the material properties, the various permittivities and permeabilities, for example. Thus the * E*- and

*-fields in the two situations differ, with the macroscopic fields being averaged to remove some of the microscopic detail.*

**B**