# Electromagnetic field

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An electromagnetic field (also EMF or EM field) is a physical field produced by electrically charged objects. [1] 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 (the others are gravitation, weak interaction and strong interaction).

In physics, a field is a physical quantity, represented by a number or tensor, that has a value for each point in space-time. For example, on a weather map, the surface temperature is described by assigning a real number to each point on a map; the temperature can be considered at a fixed point in time or over some time interval, if one wants to study the dynamics of temperature change. A surface wind map, assigning a vector to each point on a map that describes the wind velocity at that point, would be an example of a 1 dimensional tensor field. Field theories, mathematical descriptions of how field values change in space and time, are ubiquitous in physics. For instance, the electric field is another rank 1 tensor field, and the full description of electrodynamics can be formulated in terms of two interacting vector fields at each point in space-time, or as a single rank 2 tensor field theory.

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 charges; 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.

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.

## Contents

The field can be viewed as the combination of an electric field and a magnetic field. The electric field is produced by stationary charges, and the magnetic field by moving charges (currents); these two are often described as the sources of the field. The way in which charges and currents interact with the electromagnetic field is described by Maxwell's equations and the Lorentz force law. [2] The force created by the electric field is much stronger than the force created by the magnetic field. [3]

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. Mathematically the electric field is 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 strengh. 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.

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.

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.

From a classical perspective in the history of electromagnetism, the electromagnetic field can be regarded as a smooth, continuous field, propagated in a wavelike manner; whereas from the perspective of quantum field theory, the field is seen as quantized, being composed of individual particles.

Classical physics refers to theories of physics that predate modern, more complete, or more widely applicable theories. If a currently accepted theory is considered to be modern, and its introduction represented a major paradigm shift, then the previous theories, or new theories based on the older paradigm, will often be referred to as belonging to the realm of "classical physics".

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics and is used to construct physical models of subatomic particles and quasiparticles.

In the physical sciences, subatomic particles are particles much smaller than atoms. The two types of subatomic particles are: elementary particles, which according to current theories are not made of other particles; and composite particles. Particle physics and nuclear physics study these particles and how they interact. The idea of a particle underwent serious rethinking when experiments showed that light could behave like a stream of particles as well as exhibiting wave-like properties. This led to the new concept of wave–particle duality to reflect that quantum-scale "particles" behave like both particles and waves. Another new concept, the uncertainty principle, states that some of their properties taken together, such as their simultaneous position and momentum, cannot be measured exactly. In more recent times, wave–particle duality has been shown to apply not only to photons but to increasingly massive particles as well.

## Structure

The electromagnetic field may be viewed in two distinct ways: a continuous structure or a discrete structure.

### Continuous structure

Classically, electric and magnetic fields are thought of as being produced by smooth motions of charged objects. For example, oscillating charges produce variations in electric and magnetic fields that may be viewed in a 'smooth', continuous, wavelike fashion. In this case, energy is viewed as being transferred continuously through the electromagnetic field between any two locations. For instance, the metal atoms in a radio transmitter appear to transfer energy continuously. This view is useful to a certain extent (radiation of low frequency), but problems are found at high frequencies (see ultraviolet catastrophe). [4]

Radio is the technology of using radio waves to carry information, such as sound and images, by systematically modulating properties of electromagnetic energy waves transmitted through space, such as their amplitude, frequency, phase, or pulse width. When radio waves strike an electrical conductor, the oscillating fields induce an alternating current in the conductor. The information in the waves can be extracted and transformed back into its original form.

In electronics and telecommunications, a transmitter or radio transmitter is an electronic device which produces radio waves with an antenna. The transmitter itself generates a radio frequency alternating current, which is applied to the antenna. When excited by this alternating current, the antenna radiates radio waves.

The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation in all frequency ranges, emitting more energy as the frequency increases. By calculating the total amount of radiated energy, it can be shown that a blackbody is likely to release an arbitrarily high amount of energy. This would cause all matter to instantaneously radiate all of its energy until it is near absolute zero - indicating that a new model for the behaviour of blackbodies was needed.

### Discrete structure

The electromagnetic field may be thought of in a more 'coarse' way. Experiments reveal that in some circumstances electromagnetic energy transfer is better described as being carried in the form of packets called quanta (in this case, photons) with a fixed frequency. Planck's relation links the photon energy E of a photon to its frequency ν through the equation: [5]

The photon is a type of elementary particle, the quantum of the electromagnetic field including electromagnetic radiation such as light, and the force carrier for the electromagnetic force. The photon has zero rest mass and always moves at the speed of light within a vacuum.

Frequency is the number of occurrences of a repeating event per unit of time. It is also referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light.

Photon energy is the energy carried by a single photon. The amount of energy is directly proportional to the photon's electromagnetic frequency and inversely proportional to the wavelength. The higher the photon's frequency, the higher its energy. Equivalently, the longer the photon's wavelength, the lower its energy.

${\displaystyle E=\,h\,\nu }$

where h is Planck's constant, and ν is the frequency of the photon . Although modern quantum optics tells us that there also is a semi-classical explanation of the photoelectric effect—the emission of electrons from metallic surfaces subjected to electromagnetic radiation—the photon was historically (although not strictly necessarily) used to explain certain observations. It is found that increasing the intensity of the incident radiation (so long as one remains in the linear regime) increases only the number of electrons ejected, and has almost no effect on the energy distribution of their ejection. Only the frequency of the radiation is relevant to the energy of the ejected electrons.

The photoelectric effect is the emission of electrons or other free carriers when light falls on a material. Electrons emitted in this manner can be called photoelectrons. This phenomenon is commonly studied in electronic physics, as well as in fields of chemistry, such as quantum chemistry or electrochemistry.

In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

This quantum picture of the electromagnetic field (which treats it as analogous to harmonic oscillators) has proven very successful, giving rise to quantum electrodynamics, a quantum field theory describing the interaction of electromagnetic radiation with charged matter. It also gives rise to quantum optics, which is different from quantum electrodynamics in that the matter itself is modelled using quantum mechanics rather than quantum field theory.

## Dynamics

In the past, electrically charged objects were thought to produce two different, unrelated types of field associated with their charge property. An electric field is produced when the charge is stationary with respect to an observer measuring the properties of the charge, and a magnetic field as well as an electric field is produced when the charge moves, creating an electric current with respect to this observer. Over time, it was realized that the electric and magnetic fields are better thought of as two parts of a greater whole — the electromagnetic field. Until 1820, when the Danish physicist H. C. Ørsted showed the effect of electric current on a compass needle, electricity and magnetism had been viewed as unrelated phenomena [6] . In 1831, Michael Faraday, one of the great thinkers of his time, made the seminal observation that time-varying magnetic fields could induce electric currents and then, in 1864, James Clerk Maxwell published his famous paper A Dynamical Theory of the Electromagnetic Field . [7]

Once this electromagnetic field has been produced from a given charge distribution, other charged objects in this field will experience a force in a similar way that planets experience a force in the gravitational field of the sun. If these other charges and currents are comparable in size to the sources producing the above electromagnetic field, then a new net electromagnetic field will be produced. Thus, the electromagnetic field may be viewed as a dynamic entity that causes other charges and currents to move, and which is also affected by them. These interactions are described by Maxwell's equations and the Lorentz force law. This discussion ignores the radiation reaction force.

## Feedback loop

The behavior of the electromagnetic field can be divided into four different parts of a loop: [8]

• the electric and magnetic fields are generated by electric charges,
• the electric and magnetic fields interact with each other,
• the electric and magnetic fields produce forces on electric charges,
• the electric charges move in space.

A common misunderstanding is that (a) the quanta of the fields act in the same manner as (b) the charged particles that generate the fields. In our everyday world, charged particles, such as electrons, move slowly through matter with a drift velocity of a fraction of a centimeter (or inch) per second, but fields propagate at the speed of light - approximately 300 thousand kilometers (or 186 thousand miles) a second. The mundane speed difference between charged particles and field quanta is on the order of one to a million, more or less. Maxwell's equations relate (a) the presence and movement of charged particles with (b) the generation of fields. Those fields can then affect the force on, and can then move other slowly moving charged particles. Charged particles can move at relativistic speeds nearing field propagation speeds, but, as Einstein showed[ citation needed ], this requires enormous field energies, which are not present in our everyday experiences with electricity, magnetism, matter, and time and space.

The feedback loop can be summarized in a list, including phenomena belonging to each part of the loop:[ citation needed ]

• charged particles generate electric and magnetic fields
• the fields interact with each other
• changing electric field acts like a current, generating 'vortex' of magnetic field
• Faraday induction: changing magnetic field induces (negative) vortex of electric field
• Lenz's law: negative feedback loop between electric and magnetic fields
• fields act upon particles
• Lorentz force: force due to electromagnetic field
• electric force: same direction as electric field
• magnetic force: perpendicular both to magnetic field and to velocity of charge
• particles move
• current is movement of particles
• particles generate more electric and magnetic fields; cycle repeats

## Mathematical description

There are different mathematical ways of representing the electromagnetic field. The first one views the electric and magnetic fields as three-dimensional vector fields. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as E(x, y, z, t) (electric field) and B(x, y, z, t) (magnetic field).

If only the electric field (E) is non-zero, and is constant in time, the field is said to be an electrostatic field. Similarly, if only the magnetic field (B) is non-zero and is constant in time, the field is said to be a magnetostatic field. However, if either the electric or magnetic field has a time-dependence, then both fields must be considered together as a coupled electromagnetic field using Maxwell's equations. [9]

With the advent of special relativity, physical laws became susceptible to the formalism of tensors. Maxwell's equations can be written in tensor form, generally viewed by physicists as a more elegant means of expressing physical laws.

The behaviour of electric and magnetic fields, whether in cases of electrostatics, magnetostatics, or electrodynamics (electromagnetic fields), is governed by Maxwell's equations. In the vector field formalism, these are:

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}$ (Gauss's law)
${\displaystyle \nabla \cdot \mathbf {B} =0}$ (Gauss's law for magnetism)
${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$ (Faraday's law)
${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}$ (Maxwell–Ampère law)

where ${\displaystyle \rho }$ is the charge density, which can (and often does) depend on time and position, ${\displaystyle \epsilon _{0}}$ is the permittivity of free space, ${\displaystyle \mu _{0}}$ is the permeability of free space, and J is the current density vector, also a function of time and position. The units used above are the standard SI units. Inside a linear material, Maxwell's equations change by switching the permeability and permittivity of free space with the permeability and permittivity of the linear material in question. Inside other materials which possess more complex responses to electromagnetic fields, these terms are often represented by complex numbers, or tensors.

The Lorentz force law governs the interaction of the electromagnetic field with charged matter.

When a field travels across to different media, the properties of the field change according to the various boundary conditions. These equations are derived from Maxwell's equations. The tangential components of the electric and magnetic fields as they relate on the boundary of two media are as follows: [10]

${\displaystyle \mathbf {E} _{1}=\mathbf {E} _{2}}$
${\displaystyle \mathbf {H} _{1}=\mathbf {H} _{2}}$ (current-free)
${\displaystyle \mathbf {D} _{1}=\mathbf {D} _{2}}$ (charge-free)
${\displaystyle \mathbf {B} _{1}=\mathbf {B} _{2}}$

The angle of refraction of an electric field between media is related to the permittivity ${\displaystyle (\varepsilon )}$ of each medium:

${\displaystyle {\frac {\tan \theta _{1}}{\tan \theta _{2}}}={\frac {\varepsilon _{r2}}{\varepsilon _{r1}}}}$

The angle of refraction of a magnetic field between media is related to the permeability ${\displaystyle (\mu )}$ of each medium:

${\displaystyle {\frac {\tan \theta _{1}}{\tan \theta _{2}}}={\frac {\mu _{r2}}{\mu _{r1}}}}$

## Properties of the field

### Reciprocal behavior of electric and magnetic fields

The two Maxwell equations, Faraday's Law and the Ampère-Maxwell Law, illustrate a very practical feature of the electromagnetic field. Faraday's Law may be stated roughly as 'a changing magnetic field creates an electric field'. This is the principle behind the electric generator.

Ampere's Law roughly states that 'a changing electric field creates a magnetic field'. Thus, this law can be applied to generate a magnetic field and run an electric motor.

### Behavior of the fields in the absence of charges or currents

Maxwell's equations take the form of an electromagnetic wave in a volume of space not containing charges or currents (free space) – that is, where ${\displaystyle \rho }$ and J are zero. Under these conditions, the electric and magnetic fields satisfy the electromagnetic wave equation: [11]

${\displaystyle \left(\nabla ^{2}-{1 \over {c}^{2}}{\partial ^{2} \over \partial t^{2}}\right)\mathbf {E} \ \ =\ \ 0}$
${\displaystyle \left(\nabla ^{2}-{1 \over {c}^{2}}{\partial ^{2} \over \partial t^{2}}\right)\mathbf {B} \ \ =\ \ 0}$

James Clerk Maxwell was the first to obtain this relationship by his completion of Maxwell's equations with the addition of a displacement current term to Ampere's circuital law.

## Relation to and comparison with other physical fields

Being one of the four fundamental forces of nature, it is useful to compare the electromagnetic field with the gravitational, strong and weak fields. The word 'force' is sometimes replaced by 'interaction' because modern particle physics models electromagnetism as an exchange of particles known as gauge bosons.

### Electromagnetic and gravitational fields

Sources of electromagnetic fields consist of two types of charge – positive and negative. This contrasts with the sources of the gravitational field, which are masses. Masses are sometimes described as gravitational charges, the important feature of them being that there are only positive masses and no negative masses. Further, gravity differs from electromagnetism in that positive masses attract other positive masses whereas same charges in electromagnetism repel each other.

The relative strengths and ranges of the four interactions and other information are tabulated below:

TheoryInteractionmediatorRelative MagnitudeBehaviorRange
Chromodynamics Strong interaction gluon 1038110−15 m
Electrodynamics Electromagnetic interaction photon 10361/r2infinite
Flavordynamics Weak interaction W and Z bosons 10251/r5 to 1/r710−16 m
Geometrodynamics Gravitation graviton (hypothesised)1001/r2infinite

## Applications

### Static E and M fields and static EM fields

When an EM field (see electromagnetic tensor) is not varying in time, it may be seen as a purely electrical field or a purely magnetic field, or a mixture of both. However the general case of a static EM field with both electric and magnetic components present, is the case that appears to most observers. Observers who see only an electric or magnetic field component of a static EM field, have the other (electric or magnetic) component suppressed, due to the special case of the immobile state of the charges that produce the EM field in that case. In such cases the other component becomes manifest in other observer frames.

A consequence of this, is that any case that seems to consist of a "pure" static electric or magnetic field, can be converted to an EM field, with both E and M components present, by simply moving the observer into a frame of reference which is moving with regard to the frame in which only the “pure” electric or magnetic field appears. That is, a pure static electric field will show the familiar magnetic field associated with a current, in any frame of reference where the charge moves. Likewise, any new motion of a charge in a region that seemed previously to contain only a magnetic field, will show that the space now contains an electric field as well, which will be found to produces an additional Lorentz force upon the moving charge.

Thus, electrostatics, as well as magnetism and magnetostatics, are now seen as studies of the static EM field when a particular frame has been selected to suppress the other type of field, and since an EM field with both electric and magnetic will appear in any other frame, these "simpler" effects are merely the observer's. The "applications" of all such non-time varying (static) fields are discussed in the main articles linked in this section.

### Time-varying EM fields in Maxwell’s equations

An EM field that varies in time has two “causes” in Maxwell’s equations. One is charges and currents (so-called “sources”), and the other cause for an E or M field is a change in the other type of field (this last cause also appears in “free space” very far from currents and charges).

An electromagnetic field very far from currents and charges (sources) is called electromagnetic radiation (EMR) since it radiates from the charges and currents in the source, and has no "feedback" effect on them, and is also not affected directly by them in the present time (rather, it is indirectly produced by a sequences of changes in fields radiating out from them in the past). EMR consists of the radiations in the electromagnetic spectrum, including radio waves, microwave, infrared, visible light, ultraviolet light, X-rays, and gamma rays. The many commercial applications of these radiations are discussed in the named and linked articles.

A notable application of visible light is that this type of energy from the Sun powers all life on Earth that either makes or uses oxygen.

A changing electromagnetic field which is physically close to currents and charges (see near and far field for a definition of “close”) will have a dipole characteristic that is dominated by either a changing electric dipole, or a changing magnetic dipole. This type of dipole field near sources is called an electromagnetic near-field.

Changing electric dipole fields, as such, are used commercially as near-fields mainly as a source of dielectric heating. Otherwise, they appear parasitically around conductors which absorb EMR, and around antennas which have the purpose of generating EMR at greater distances.

Changing magnetic dipole fields (i.e., magnetic near-fields) are used commercially for many types of magnetic induction devices. These include motors and electrical transformers at low frequencies, and devices such as metal detectors and MRI scanner coils at higher frequencies. Sometimes these high-frequency magnetic fields change at radio frequencies without being far-field waves and thus radio waves; see RFID tags. See also near-field communication. Further uses of near-field EM effects commercially, may be found in the article on virtual photons, since at the quantum level, these fields are represented by these particles. Far-field effects (EMR) in the quantum picture of radiation, are represented by ordinary photons.

### Other

• Electromagnetic field can be used to record data on static electricity.
• Old televisions can be traced with Electromagnetic fields.

## Health and safety

The potential effects of electromagnetic fields on human health vary widely depending on the frequency and intensity of the fields.

The potential health effects of the very low frequency EMFs surrounding power lines and electrical devices are the subject of on-going research and a significant amount of public debate. The US National Institute for Occupational Safety and Health (NIOSH) and other US government agencies do not consider EMFs a proven health hazard. NIOSH has issued some cautionary advisories but stresses that the data are currently too limited to draw good conclusions. [12]

On the other hand, radiation from other parts of the electromagnetic spectrum, such as ultraviolet light and gamma rays, are known to cause significant harm in some circumstances. For more information on the health effects due to specific electromagnetic phenomena and parts of the electromagnetic spectrum, see the following articles:

## Related Research Articles

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

An electric potential is the amount of work needed to move a unit of positive charge from a reference point to a specific point inside the field without producing an acceleration. Typically, the reference point is the Earth or a point at infinity, although any point beyond the influence of the electric field charge can be used.

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.

Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first mathematically demonstrated by British mathematician Samuel Earnshaw in 1842. It is usually referenced to magnetic fields, but was first applied to electrostatic fields.

"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.

The Abraham–Minkowski controversy is a physics debate concerning electromagnetic momentum within dielectric media. Traditionally, it is argued that in the presence of matter the electromagnetic stress-energy tensor by itself is not conserved (divergenceless). Only the total stress-energy tensor carries unambiguous physical significance, and how one apportions it between an "electromagnetic" part and a "matter" part depends on context and convenience. In other words, the electromagnetic part and the matter part in the total momentum can be arbitrarily distributed as long as the total momentum is kept the same. There are two incompatible equations to describe momentum transfer between matter and electromagnetic fields. These two equations were first suggested by Hermann Minkowski (1908) and Max Abraham (1909), from which the controversy's name derives. Both were claimed to be supported by experimental data. Theoretically, it is usually argued that Abraham's version of momentum "does indeed represent the true momentum density of electromagnetic fields" for electromagnetic waves, while Minkowski's version of momentum is "pseudomomentum" or "wave momentum".

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 potentialψ. 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, 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.

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 magnetic radiation reaction force is a force on an electromagnet when its magnetic moment changes. One can derive an electric radiation reaction force for an accelerating charged particle caused by the particle emitting electromagnetic radiation. Likewise, a magnetic radiation reaction force can be derived for an accelerating magnetic moment emitting electromagnetic radiation.

In theoretical physics, the dual photon is a hypothetical elementary particle that is a dual of the photon under electric-magnetic duality which is predicted by some theoretical models and some results of M-theory in eleven dimensions.

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2. Purcell. p5-11;p61;p277-296
3. Purcell, p235: We then calculate the electric field due to a charge moving with constant velocity; it does not equal the spherically symmetric Coulomb field.
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