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In electromagnetism, **Jefimenko's equations** (named after Oleg D. Jefimenko) 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 (retarded time) 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.^{ [1] }

**Electromagnetism** is a branch of physics involving the study of the **electromagnetic force**, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force usually exhibits electromagnetic fields such as electric fields, magnetic fields, and light, and is one of the four fundamental interactions in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force.

**Oleg Dmitrovich Jefimenko** was a physicist and Professor Emeritus at West Virginia University.

An **electric field** is a vector field surrounding an electric charge that exerts force on other charges, attracting or repelling them. Mathematically the electric field is a vector field that associates to each point in space the force, called the Coulomb force, that would be experienced per unit of charge by an infinitesimal test charge at that point. The units of the electric field in the SI system are newtons per coulomb (N/C), or volts per meter (V/m). 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.

Jefimenko's equations give the **E** field and **B** field produced by an arbitrary charge or current distribution, of charge density *ρ* and current density **J**:^{ [2] }

A **magnetic field** is a vector field that describes the magnetic influence of electrical currents and magnetized materials. In everyday life, the effects of magnetic fields are often 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. Magnetic fields 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 varies with location. As such, it is an example of a vector field.

In electromagnetism, **charge density** is the amount of electric charge per unit length, surface area, or volume. *Volume charge density* is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C•m^{−3}), at any point in a volume. *Surface charge density* (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C•m^{−2}), at any point on a surface charge distribution on a two dimensional surface. *Linear charge density* (λ) is the quantity of charge per unit length, measured in coulombs per meter (C•m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

In electromagnetism, **current density** is the electric current per unit area of 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 units, the electric current density is measured in amperes per square metre.

where **r**′ is a point in the charge distribution, **r** is a point in space, and

is the retarded time. There are similar expressions for **D** and **H**.^{ [3] }

In electromagnetism, electromagnetic waves in vacuum travel at the speed of light *c*, according to Maxwell's Equations. The **retarded time** is the time when the field began to propagate from the point where it was emitted to an observer. The term "retarded" is used in this context in the sense of propagation delays.

These equations are the time-dependent generalization of Coulomb's law and the Biot–Savart law to electrodynamics, which were originally true only for electrostatic and magnetostatic fields, and steady currents.

**Coulomb's law**, or **Coulomb's inverse-square law**, is a law of physics for quantifying Coulomb's force, or electrostatic force. Electrostatic force is the amount of force with which stationary, electrically charged particles either repel, or attract each other. This force and the law for quantifying it, represent one of the most basic forms of force used in the physical sciences, and were an essential basis to the study and development of the theory and field of classical electromagnetism. The law was first published in 1785 by French physicist Charles-Augustin de Coulomb.

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.

Jefimenko's equations can be found^{ [4] } from the retarded potentials *φ* and **A**:

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.

which are the solutions to Maxwell's equations in the potential formulation, then substituting in the definitions of the electromagnetic potentials themselves:

and using the relation

replaces the potentials *φ* and **A** by the fields **E** and **B**.

There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave^{ [5] } (electromagnetism). However, Jefimenko's equations show an alternative point of view.^{ [6] } Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents."^{ [7] }

As pointed out by McDonald,^{ [8] } Jefimenko's equations seem to appear first in 1962 in the second edition of Panofsky and Phillips's classic textbook.^{ [9] } David Griffiths, however, clarifies that "the earliest explicit statement of which I am aware was by Oleg Jefimenko, in 1966" and characterizes equations in Panofsky and Phillips's textbook as only "closely related expressions"^{ [10] }. According to Andrew Zangwill, the equations analogous to Jefimenko's but in the Fourier frequency domain were first derived by George Adolphus Schott in his treaties Electromagnetic Radiation (University Press, Cambridge, 1912)^{ [11] }.

Essential features of these equations are easily observed which is that the right hand sides involve "retarded" time which reflects the "causality" of the expressions. In other words, the left side of each equation is actually "caused" by the right side, unlike the normal differential expressions for Maxwell's equations where both sides take place simultaneously. In the typical expressions for Maxwell's equations there is no doubt that both sides are equal to each other, but as Jefimenko notes, "... since each of these equations connects quantities simultaneous in time, none of these equations can represent a causal relation."^{ [12] } The second feature is that the expression for **E** does not depend upon **B** and vice versa. Hence, it is impossible for **E** and **B** fields to be "creating" each other. Charge density and current density are creating them both.

- ↑ Oleg D. Jefimenko,
*Electricity and Magnetism: An Introduction to the Theory of Electric and Magnetic Fields*, Appleton-Century-Crofts (New-York - 1966). 2nd ed.: Electret Scientific (Star City - 1989), ISBN 978-0-917406-08-9. See also: David J. Griffiths, Mark A. Heald,*Time-dependent generalizations of the Biot–Savart and Coulomb laws*, American Journal of Physics**59 (2)**(1991), 111-117. - ↑ Introduction to Electrodynamics (3rd Edition), D. J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3.
- ↑ Oleg D. Jefimenko,
*Solutions of Maxwell's equations for electric and magnetic fields in arbitrary media*, American Journal of Physics**60 (10)**(1992), 899–902. - ↑ Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
- ↑ Kinsler, P. (2011). "How to be causal: time, spacetime, and spectra".
*Eur. J. Phys*.**32**: 1687. arXiv: 1106.1792 . Bibcode:2011EJPh...32.1687K. doi:10.1088/0143-0807/32/6/022. - ↑ Oleg D. Jefimenko,
*Causality Electromagnetic Induction and Gravitation*, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-4, page 16 ISBN 0-917406-23-0. - ↑ Oleg D. Jefimenko,
*Causality Electromagnetic Induction and Gravitation*, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-5, page 16 ISBN 0-917406-23-0. - ↑ Kirk T. McDonald,
*The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips*, American Journal of Physics**65 (11)**(1997), 1074-1076. - ↑ Wolfgang K. H. Panofsky, Melba Phillips,
*Classical Electricity And Magnetism*, Addison-Wesley (2nd. ed - 1962), Section 14.3. The electric field is written in a slightly different - but completely equivalent - form. Reprint: Dover Publications (2005), ISBN 978-0-486-43924-2. - ↑ David J. Griffiths, Introduction to Electrodynamics, Prentice Hall (New Jersey), 3rd edition (1999), pp. 427—429
- ↑ Andrew Zangwill, Modern Electrodynamics, Cambridge University Press, 1st edition (2013), pp. 726—727, 765
- ↑ Oleg D. Jefimenko,
*Causality Electromagnetic Induction and Gravitation*, 2nd ed.: Electret Scientific (Star City - 2000) Chapter 1, Sec. 1-1, page 6 ISBN 0-917406-23-0.

An **electromagnetic field** is a physical field produced by electrically charged objects. It affects the behavior of charged objects in the vicinity 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

**Maxwell's equations** are a set of 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. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at the speed of light. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum 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. He also first used the equations to propose that light is an electromagnetic phenomenon.

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.

**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.

"**A Dynamical Theory of the Electromagnetic Field**" is a paper by James Clerk Maxwell on electromagnetism, published in 1865. In the paper, Maxwell derives an electromagnetic wave equation with a velocity for light in close agreement with measurements made by experiment, and deduces that light is an electromagnetic wave.

An **electromagnetic four-potential** is a relativistic vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into a single four-vector.

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 electromagnetism, the **Lorenz gauge condition** or **Lorenz gauge** is a partial gauge fixing of the electromagnetic vector potential. The condition is that This does not completely determine the gauge: one can still make a gauge transformation where is a harmonic scalar function.

The **electromagnetic wave equation** is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. It is a three-dimensional form of the wave equation. The homogeneous form of the equation, written in terms of either the electric field **E** or the magnetic field **B**, takes the form:

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 makes 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.

The **moving magnet and conductor problem** is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity. In it, the current in a conductor moving with constant velocity, *v*, with respect to a magnet is calculated in the frame of reference of the magnet and in the frame of reference of the conductor. The observable quantity in the experiment, the current, is the same in either case, in accordance with the basic *principle of relativity*, which states: "Only *relative* motion is observable; there is no absolute standard of rest". However, according to Maxwell's equations, the charges in the conductor experience a **magnetic force** in the frame of the magnet and an **electric force** in the frame of the conductor. The same phenomenon would seem to have two different descriptions depending on the frame of reference of the observer.

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.

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. First of all, it gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another. Secondly, it sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form.

**Gravitoelectromagnetism**, abbreviated **GEM**, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. **Gravitomagnetism** is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge. The most common version of GEM is valid only far from isolated sources, and for slowly moving test particles.

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