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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.

Starting from Maxwell's equations and assuming that charges are either fixed or move as a steady current , 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.

Name | Form | |
---|---|---|

Partial differential | Integral | |

Gauss's law for magnetism | ||

Ampère's law |

Where ∇ with the dot denotes divergence, and **B** is the magnetic flux density, the first integral is over a surface with oriented surface element . 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 with line element . The current going through the loop is .

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 term against the term. If the 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 . Plugging this result into Faraday's Law finds a value for (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.

If all currents in a system are known (i.e., if a complete description of the current density 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.

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 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,

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

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

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

This has the general solution

where is a scalar potential.^{ [3] }^{:192} Substituting this in Gauss's law gives

Thus, the divergence of the magnetization, has a role analogous to the electric charge in electrostatics ^{ [4] } and is often referred to as an effective charge density .

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

- ↑ Hiebert, Ballentine & Freeman 2002
- ↑ Feynman, Leighton & Sands 2006
- 1 2 3 Jackson, John David (1975).
*Classical electrodynamics*(2d ed.). New York: Wiley. ISBN 047143132X. - ↑ Aharoni 1996

An **electromagnetic field** is a physical field produced by moving electrically charged objects. It affects the behavior of non-comoving charged objects at any distance of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction. It is one of the four fundamental forces of nature.

In physics the **Lorentz force** is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge *q* moving with a velocity **v** in an electric field **E** and a magnetic field **B** experiences a force of

**Flux** describes any effect that appears to pass or travel through a surface or substance. A flux is either a concept based in physics or used with applied mathematics. Both concepts have mathematical rigor, enabling comparison of the underlying mathematics when the terminology is unclear. For transport phenomena, flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In electromagnetism, flux is a scalar quantity, defined as the surface integral of the component of a vector field perpendicular to the surface at each point.

In classical electromagnetism, **Ampère's circuital law** relates the integrated magnetic field around a closed loop to the electric current passing through the loop. James Clerk Maxwell derived it using hydrodynamics in his 1861 paper "On Physical Lines of Force" and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.

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 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.

In electrodynamics, **Poynting's theorem** is a statement of conservation of energy for the electromagnetic field, in the form of a partial differential equation, due to the British physicist John Henry Poynting. Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution, through energy flux.

The term **magnetic potential** can be used for either of two quantities in classical electromagnetism: the *magnetic vector potential*, or simply *vector potential*, **A**; and the *magnetic scalar potential**ψ*. Both quantities can be used in certain circumstances to calculate the magnetic field **B**.

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. It is represented by a pseudovector **M**.

In quantum mechanics, the **probability current** is a mathematical quantity describing the flow of probability in terms of probability per unit time per unit area. Specifically, if one describes the probability density as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. This is analogous to mass currents in hydrodynamics and electric currents in electromagnetism. It is a real vector, like electric current density. The concept of a probability current is a useful formalism in quantum mechanics.

In electromagnetism, **Jefimenko's equations** give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay of the fields due to the finite speed of light and relativistic effects. Therefore they can be used for *moving* charges and currents. They are the general solutions to Maxwell's equations for any arbitrary distribution of charges and currents.

**Micromagnetics** is a field of physics dealing with the prediction of magnetic behaviors at sub-micrometer length scales. The length scales considered are large enough for the atomic structure of the material to be ignored, yet small enough to resolve magnetic structures such as domain walls or vortices.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

In electrodynamics, the **retarded potentials** are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light *c*, so the delay of the fields connecting cause and effect at earlier and later times is an important factor: the signal takes a finite time to propagate from a point in the charge or current distribution to another point in space, see figure below.

In physics, **Gauss's law for gravity**, also known as **Gauss's flux theorem for gravity**, is a law of physics that is essentially equivalent to Newton's law of universal gravitation. It is named after Carl Friedrich Gauss. Although Gauss's law for gravity is equivalent to Newton's law, there are many situations where Gauss's law for gravity offers a more convenient and simple way to do a calculation than Newton's law.

The **demagnetizing field**, also called the **stray field**, is the magnetic field (H-field) generated by the magnetization in a magnet. The total magnetic field in a region containing magnets is the sum of the demagnetizing fields of the magnets and the magnetic field due to any free currents or displacement currents. The term *demagnetizing field* reflects its tendency to act on the magnetization so as to reduce the total magnetic moment. It gives rise to *shape anisotropy* in ferromagnets with a single magnetic domain and to magnetic domains in larger ferromagnets.

- Aharoni, Amikam (1996).
*Introduction to the Theory of Ferromagnetism*. Clarendon Press. ISBN 0-19-851791-2. - Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (2006).
*The Feynman Lectures on Physics*.**2**. ISBN 0-8053-9045-6. - Hiebert, W; Ballentine, G; Freeman, M (2002). "Comparison of experimental and numerical micromagnetic dynamics in coherent precessional switching and modal oscillations".
*Physical Review B*.**65**(14). p. 140404. Bibcode:2002PhRvB..65n0404H. doi:10.1103/PhysRevB.65.140404.

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