# Magnetostatics

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Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time). It is the magnetic analogue of electrostatics, where the charges are stationary. The magnetization need not be static; the equations of magnetostatics can be used to predict fast magnetic switching events that occur on time scales of nanoseconds or less. [1] Magnetostatics is even a good approximation when the currents are not static as long as the currents do not alternate rapidly. Magnetostatics is widely used in applications of micromagnetics such as models of magnetic storage devices as in computer memory. Magnetostatic focussing can be achieved either by a permanent magnet or by passing current through a coil of wire whose axis coincides with the beam axis.

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

Electrostatics is a branch of physics that studies electric charges at rest.

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: positive and negative. Like charges repel and unlike attract. An object with an absence of net charge is referred to as neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.

## Applications

### Magnetostatics as a special case of Maxwell's equations

Starting from Maxwell's equations and assuming that charges are either fixed or move as a steady current ${\displaystyle \scriptstyle \mathbf {J} }$, the equations separate into two equations for the electric field (see electrostatics) and two for the magnetic field. [2] The fields are independent of time and each other. The magnetostatic equations, in both differential and integral forms, are shown in the table below.

Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in a vacuum. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon.

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

NameForm
Partial differential Integral
Gauss's law
for magnetism
${\displaystyle \mathbf {\nabla } \cdot \mathbf {B} =0}$${\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}$
Ampère's law ${\displaystyle \mathbf {\nabla } \times \mathbf {H} =\mathbf {J} }$${\displaystyle \oint _{C}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =I_{\mathrm {enc} }}$

Where ∇ with the dot denotes divergence, and B is the magnetic flux density, the first integral is over a surface ${\displaystyle \scriptstyle S}$ with oriented surface element ${\displaystyle \scriptstyle d\mathbf {S} }$. Where ∇ with the cross denotes curl, J is the current density and H is the magnetic field intensity, the second integral is a line integral around a closed loop ${\displaystyle \scriptstyle C}$ with line element ${\displaystyle \scriptstyle \mathbf {l} }$. The current going through the loop is ${\displaystyle \scriptstyle I_{\text{enc}}}$.

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.

In vector calculus, the curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space. At every point in the field, the curl of that point is represented by a vector. The attributes of this vector characterize the rotation at that point.

In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional area at a given point in space, its direction being that of the motion of the charges at this point. In SI base units, the electric current density is measured in amperes per square metre.

The quality of this approximation may be guessed by comparing the above equations with the full version of Maxwell's equations and considering the importance of the terms that have been removed. Of particular significance is the comparison of the ${\displaystyle \scriptstyle \mathbf {J} }$ term against the ${\displaystyle \scriptstyle \partial \mathbf {D} /\partial t}$ term. If the ${\displaystyle \scriptstyle \mathbf {J} }$ term is substantially larger, then the smaller term may be ignored without significant loss of accuracy.

A common technique is to solve a series of magnetostatic problems at incremental time steps and then use these solutions to approximate the term ${\displaystyle \scriptstyle \partial \mathbf {B} /\partial t}$. Plugging this result into Faraday's Law finds a value for ${\displaystyle \scriptstyle \mathbf {E} }$ (which had previously been ignored). This method is not a true solution of Maxwell's equations but can provide a good approximation for slowly changing fields.[ citation needed ]

Faraday's law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.

## Solving for the magnetic field

### Current sources

If all currents in a system are known (i.e., if a complete description of the current density ${\displaystyle \scriptstyle \mathbf {J} (\mathbf {r} )}$ is available) then the magnetic field can be determined, at a position r, from the currents by the Biot–Savart equation: [3] :174

In physics, specifically electromagnetism, the Biot–Savart Law is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot–Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.

${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int {{\frac {\mathbf {J} (\mathbf {r} ')\times \left(\mathbf {r} -\mathbf {r} '\right)}{|\mathbf {r} -\mathbf {r} '|^{3}}}\mathrm {d} ^{3}\mathbf {r} '}}$

This technique works well for problems where the medium is a vacuum or air or some similar material with a relative permeability of 1. This includes air-core inductors and air-core transformers. One advantage of this technique is that, if a coil has a complex geometry, it can be divided into sections and the integral evaluated for each section. Since this equation is primarily used to solve linear problems, the contributions can be added. For a very difficult geometry, numerical integration may be used.

For problems where the dominant magnetic material is a highly permeable magnetic core with relatively small air gaps, a magnetic circuit approach is useful. When the air gaps are large in comparison to the magnetic circuit length, fringing becomes significant and usually requires a finite element calculation. The finite element calculation uses a modified form of the magnetostatic equations above in order to calculate magnetic potential. The value of ${\displaystyle \scriptstyle \mathbf {B} }$ can be found from the magnetic potential.

The magnetic field can be derived from the vector potential. Since the divergence of the magnetic flux density is always zero,

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} ,}$

and the relation of the vector potential to current is: [3] :176

${\displaystyle \mathbf {A} (\mathbf {r} )={\frac {\mu _{0}}{4\pi }}\int {{\frac {\mathbf {J(\mathbf {r} ')} }{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}\mathbf {r} '}.}$

### Magnetization

Strongly magnetic materials (i.e., ferromagnetic, ferrimagnetic or paramagnetic) have a magnetization that is primarily due to electron spin. In such materials the magnetization must be explicitly included using the relation

${\displaystyle \mathbf {B} =\mu _{0}(\mathbf {M} +\mathbf {H} ).}$

Except in metals, electric currents can be ignored. Then Ampère's law is simply

${\displaystyle \nabla \times \mathbf {H} =0.}$

This has the general solution

${\displaystyle \mathbf {H} =-\nabla \Phi _{M},}$

where ${\displaystyle \Phi _{M}}$ is a scalar potential. [3] :192 Substituting this in Gauss's law gives

${\displaystyle \nabla ^{2}\Phi _{M}=\nabla \cdot \mathbf {M} .}$

Thus, the divergence of the magnetization, ${\displaystyle \scriptstyle \nabla \cdot \mathbf {M} ,}$ has a role analogous to the electric charge in electrostatics [4] and is often referred to as an effective charge density ${\displaystyle \rho _{M}}$.

The vector potential method can also be employed with an effective current density

${\displaystyle \mathbf {J_{M}} =\nabla \times \mathbf {M} .}$

## Notes

1. Jackson, John David (1975). Classical electrodynamics (2d ed.). New York: Wiley. ISBN   047143132X.

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