Magnetization

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In classical electromagnetism, magnetization or magnetic polarization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. [1] It is represented by a pseudovector M.

Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from one point to another point.

The density, or more precisely, the volumetric mass density, of a substance is its mass per unit volume. The symbol most often used for density is ρ, although the Latin letter D can also be used. Mathematically, density is defined as mass divided by volume:

Definition

The magnetization field or M-field can be defined according to the following equation:

${\displaystyle \mathbf {M} ={\frac {\mathrm {d} \mathbf {m} }{\mathrm {d} V}}}$

Where ${\displaystyle \mathrm {d} \mathbf {m} }$ is the elementary magnetic moment and ${\displaystyle \mathrm {d} V}$ is the volume element; in other words, the M-field is the distribution of magnetic moments in the region or manifold concerned. This is better illustrated through the following relation:

The magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current, permanent magnets, elementary particles, various molecules, and many astronomical objects.

In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.

${\displaystyle \mathbf {m} =\iiint \mathbf {M} \,\mathrm {d} V}$

where m is an ordinary magnetic moment and the triple integral denotes integration over a volume. This makes the M-field completely analogous to the electric polarisation field, or P-field, used to determine the electric dipole moment p generated by a similar region or manifold with such a polarization:

In classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.

The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, the most commonly used unit in atomic physics and chemistry is the debye (D).

${\displaystyle \mathbf {P} ={\mathrm {d} \mathbf {p} \over \mathrm {d} V},\quad \mathbf {p} =\iiint \mathbf {P} \,\mathrm {d} V}$

Where ${\displaystyle \mathrm {d} \mathbf {p} }$ is the elementary electric dipole moment.

Those definitions of P and M as a "moments per unit volume" are widely adopted, though in some cases they can lead to ambiguities and paradoxes. [1]

The M-field is measured in amperes per meter (A/m) in SI units. [2]

The ampere, often shortened to "amp", is the base unit of electric current in the International System of Units (SI). It is named after André-Marie Ampère (1775–1836), French mathematician and physicist, considered the father of electrodynamics.

Physics application

The magnetization is often not listed as a material parameter for commercially available ferromagnets. Instead the parameter that is listed is residual flux density, denoted ${\displaystyle \scriptstyle \mathbf {B} _{r}}$. Physicists often need the magnetization to calculate the moment of a ferromagnet. To calculate the dipole moment m (A⋅m2) using the formula:

${\displaystyle \mathbf {m} \;=\;\mathbf {M} V}$,

we have that

${\displaystyle \mathbf {M} ={\frac {1}{\mu _{0}}}\mathbf {B} _{\mathrm {r} }}$,

thus

${\displaystyle \mathbf {m} ={\frac {1}{\mu _{0}}}\mathbf {B} _{\mathrm {r} }V}$,

where:

• ${\displaystyle \scriptstyle \mathbf {B} _{\mathrm {r} }}$ is the residual flux density, expressed in teslas (T).
• ${\displaystyle \scriptstyle V}$ is the volume (m3) of the magnet.
• ${\displaystyle \scriptstyle \mu _{0}\;=\;4\pi \cdot 10^{-7}}$ H/m is the permeability of vacuum. [3]

In Maxwell's equations

The behavior of magnetic fields (B, H), electric fields (E, D), charge density (ρ), and current density (J) is described by Maxwell's equations. The role of the magnetization is described below.

Relations between B, H, and M

The magnetization defines the auxiliary magnetic field H as

${\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H+M} )}$ (SI units)
${\displaystyle \mathbf {B} =\mathbf {H} +4\pi \mathbf {M} }$ (Gaussian units)

which is convenient for various calculations. The vacuum permeability μ0 is, by definition, ×10−7 V·s/(A·m).

A relation between M and H exists in many materials. In diamagnets and paramagnets, the relation is usually linear:

${\displaystyle \mathbf {M} =\chi \mathbf {H} }$

where χ is called the volume magnetic susceptibility.

In ferromagnets there is no one-to-one correspondence between M and H because of magnetic hysteresis.

Magnetization current

The magnetization M makes a contribution to the current density J, known as the magnetization current. [4]

${\displaystyle \mathbf {J} _{\mathrm {m} }=\nabla \times \mathbf {M} }$

and for the bound surface current:

${\displaystyle \mathbf {K} _{\mathrm {m} }=\mathbf {M} \times \mathbf {\hat {n}} }$

so that the total current density that enters Maxwell's equations is given by

${\displaystyle \mathbf {J} =\mathbf {J} _{\mathrm {f} }+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}}$

where Jf is the electric current density of free charges (also called the free current), the second term is the contribution from the magnetization, and the last term is related to the electric polarization P.

Magnetostatics

In the absence of free electric currents and time-dependent effects, Maxwell's equations describing the magnetic quantities reduce to

{\displaystyle {\begin{aligned}\mathbf {\nabla \times H} &=0\\\mathbf {\nabla \cdot H} &=-\nabla \cdot \mathbf {M} \end{aligned}}}

These equations can be solved in analogy with electrostatic problems where

{\displaystyle {\begin{aligned}\mathbf {\nabla \times E} &=0\\\mathbf {\nabla \cdot E} &={\frac {\rho }{\epsilon _{0}}}\end{aligned}}}

In this sense −∇⋅M plays the role of a fictitious "magnetic charge density" analogous to the electric charge density ρ; (see also demagnetizing field).

Dynamics

The time-dependent behavior of magnetization becomes important when considering nanoscale and nanosecond timescale magnetization. Rather than simply aligning with an applied field, the individual magnetic moments in a material begin to precess around the applied field and come into alignment through relaxation as energy is transferred into the lattice.

Reversal

Magnetization reversal, also known as switching, refers to the process that leads to a 180° (arc) re-orientation of the magnetization vector with respect to its initial direction, from one stable orientation to the opposite one. Technologically, this is one of the most important processes in magnetism that is linked to the magnetic data storage process such as used in modern hard disk drives. [5] As it is known today, there are only a few possible ways to reverse the magnetization of a metallic magnet:

1. an applied magnetic field [5]
2. spin injection via a beam of particles with spin [5]
3. magnetization reversal by circularly polarized light; [6] i.e., incident electromagnetic radiation that is circularly polarized

Demagnetization

Demagnetization is the reduction or elimination of magnetization. [7] One way to do this is to heat the object above its Curie temperature, where thermal fluctuations have enough energy to overcome exchange interactions, the source of ferromagnetic order, and destroy that order. Another way is to pull it out of an electric coil with alternating current running through it, giving rise to fields that oppose the magnetization. [8]

One application of demagnetization is to eliminate unwanted magnetic fields. For example, magnetic fields can interfere with electronic devices such as cell phones or computers, and with machining by making cuttings cling to their parent. [8]

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