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**Magnetic scalar potential**, *ψ*, is a quantity in classical electromagnetism analogous to electric potential. It is used to specify the magnetic **H**-field in cases when there are no free currents, in a manner analogous to using the electric potential to determine the electric field in electrostatics. One important use of *ψ* is to determine the magnetic field due to permanent magnets when their magnetization is known. The potential is valid in any region with zero current density, thus if currents are confined to wires or surfaces, piecemeal solutions can be stitched together to provide a description of the magnetic field at all points in space.

The scalar potential is a useful quantity in describing the magnetic field, especially for permanent magnets.

Where there is no free current,

so if this holds in simply connected domain we can define a *magnetic scalar potential*, *ψ*, as^{ [1] }

The dimensions of *ψ* in SI base units are .

Using the definition of **H**:

it follows that

Here, ∇ ⋅ **M** acts as the source for magnetic field, much like ∇ ⋅ **P** acts as the source for electric field. So analogously to bound electric charge, the quantity

is called the *bound magnetic charge* density. Magnetic charges never occur isolated as magnetic monopoles, but only within dipoles and in magnets with a total magnetic charge sum of zero. The energy of a localized magnetic charge *q _{m}* in a magnetic scalar potential is

- ,

and of a magnetic charge density distribution *ρ _{m}* in space

- ,

where *µ _{0}* is the vacuum permeability. This is analog to the energy of an electric charge

If there is free current, one may subtract the contributions of free current per Biot–Savart law from total magnetic field and solve the remainder with the scalar potential method.

- ↑ Vanderlinde (2005 , pp. 194~199)

**Maxwell's equations**, or **Maxwell-Heaviside 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. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 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. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.

In physics, the **Navier–Stokes equations** are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

The **electric potential** is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. Furthermore, the motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used.

The **Klein–Gordon equation** is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.

A **continuity equation** or **transport equation** is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations.

In 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, **magnetic vector potential** is the vector quantity defined so that its curl is equal to the magnetic field: . Together with the electric potential *φ*, the magnetic vector potential can be used to specify the electric field **E** as well. Therefore, many equations of electromagnetism can be written either in terms of the fields **E** and **B**, or equivalently in terms of the potentials *φ* and **A**. In more advanced theories such as quantum mechanics, most equations use potentials rather than fields.

A **classical field theory** is a physical theory that predicts how one or more physical fields interact with matter through **field equations**, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically meant to describe electromagnetism and gravitation, two of the fundamental forces of nature.

**Magnetostatics** is the study of magnetic fields in systems where the currents are steady. 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. 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.

In differential geometry, the **four-gradient** is the four-vector analogue of the gradient from vector calculus.

In the physics of gauge theories, **gauge fixing** denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct configuration of the system as an equivalence class of detailed local field configurations. Any two detailed configurations in the same equivalence class are related by a gauge transformation, equivalent to a shear along unphysical axes in configuration space. Most of the quantitative physical predictions of a gauge theory can only be obtained under a coherent prescription for suppressing or ignoring these unphysical degrees of freedom.

In classical electromagnetism, **magnetization** is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Movement within this field is described by direction and is either Axial or Diametric. 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. Paramagnetic materials have a weak induced magnetization in a magnetic field, which disappears when the magnetic field is removed. Ferromagnetic and ferrimagnetic materials have strong magnetization in a magnetic field, and can be *magnetized* to have magnetization in the absence of an external field, becoming a permanent magnet. Magnetization is not necessarily uniform within a material, but may vary 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. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. It is a real vector that changes with space and time. Probability currents are analogous to mass currents in hydrodynamics and electric currents in electromagnetism. As in those fields, the probability current is related to the probability density function via a continuity equation. The probability current is invariant under gauge transformation.

In electromagnetism and applications, an **inhomogeneous electromagnetic wave equation**, or **nonhomogeneous electromagnetic wave equation**, is one of a set of wave equations describing the propagation of electromagnetic waves generated by nonzero source charges and currents. The source terms in the wave equations make the partial differential equations *inhomogeneous*, if the source terms are zero the equations reduce to the homogeneous electromagnetic wave equations. The equations follow from Maxwell's equations.

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 quantum mechanics, the **Pauli equation** or **Schrödinger–Pauli equation** is the formulation of the Schrödinger equation for spin-½ particles, which takes into account the interaction of the particle's spin with an external electromagnetic field. It is the non-relativistic limit of the Dirac equation and can be used where particles are moving at speeds much less than the speed of light, so that relativistic effects can be neglected. It was formulated by Wolfgang Pauli in 1927.

The **Madelung equations**, or the **equations of quantum hydrodynamics**, are Erwin Madelung's equivalent alternative formulation of the Schrödinger equation, written in terms of hydrodynamical variables, similar to the Navier–Stokes equations of fluid dynamics. The derivation of the Madelung equations is similar to the de Broglie–Bohm formulation, which represents the Schrödinger equation as a quantum Hamilton–Jacobi equation.

**Multipole radiation** is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

In mathematical physics, the **Gordon decomposition** of the Dirac current is a splitting of the charge or particle-number current into a part that arises from the motion of the center of mass of the particles and a part that arises from gradients of the spin density. It makes explicit use of the Dirac equation and so it applies only to "on-shell" solutions of the Dirac equation.

- Duffin, W.J. (1980).
*Electricity and Magnetism, Fourth Edition*. McGraw-Hill. ISBN 007084111X.

- Jackson, John David (1999),
*Classical Electrodynamics*(3rd ed.), John Wiley & Sons, ISBN 0-471-30932-X

- Vanderlinde, Jack (2005).
*Classical Electromagnetic Theory*. Bibcode:2005cet..book.....V. doi:10.1007/1-4020-2700-1. ISBN 1-4020-2699-4.

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