Electromagnetic field: Difference between revisions

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Latest revision as of 14:11, 12 August 2025

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{{short description|Electric and magnetic fields produced by moving charged objects}}

{{msg:electromagnetism}}

{{distinguish|Electromotive force}}

{{for|the British hacker convention|Electromagnetic Field (festival)}}

{{Electromagnetism|cTopic=Electrodynamics}}

TheAn '''electromagnetic field''' (also '''EMFEM field''') is composeda of[[physical twofield]], relatedvarying vectorialin fieldsspace and time, that represents the [[electric field]] and the [[magnetic field]].influences generated Thisby meansand thatacting theupon [[vectorelectric (spatial)|vectorcharge]]s.{{sfnp|ps=|Feynman|Leighton|Sands|1970|Lectures ('''E'''on andPhysics, '''B''')Vol. thatII|loc=[https://feynmanlectures.caltech.edu/II_01.html#Ch1-S2 characterize§1.2]}} theThe field each have a value defined at eachany point ofin space and time. Ifcan onlybe '''E''',regarded theas a combination of an [[electric field]], is nonzero and is constant in time, the field is said to be ana [[electrostaticmagnetic field]].

Because of the interrelationship between the fields, a disturbance in the electric field can create a disturbance in the magnetic field which in turn affects the electric field, leading to an oscillation that propagates through space, known as an ''[[electromagnetic wave]]''.{{sfnp|ps=|Ling|Moebs|Sanny|2023}}{{sfnp|ps=|Taylor|2012}}

The way in which charges and currents (i.e. streams of charges) interact with the electromagnetic field is described by [[Maxwell's equations]]{{sfnp|ps=|Purcell|Morin|2012|pp=436–437}} and the [[Lorentz force law]].{{sfnp|ps=|Purcell|Morin|2012|pp=277–296}} Maxwell's equations detail how the electric field converges towards or diverges away from electric charges, how the magnetic field curls around electrical currents, and how changes in the electric and magnetic fields influence each other. The Lorentz force law states that a charge subject to an electric field feels a force along the direction of the field, and a charge moving through a magnetic field feels a force that is perpendicular both to the magnetic field and to its direction of motion.

The electromagnetic field generates a [[force]] <b>F</b> on a charged particle, given by the [[Hendrik Lorentz|Lorentz]] equation

The electromagnetic field is described by [[classical electrodynamics]], an example of a [[classical field theory]]. This theory describes many macroscopic physical phenomena accurately.{{sfnp|ps=|Purcell|Morin|2012|p=2}} However, it was unable to explain the [[photoelectric effect]] and [[atomic absorption spectroscopy]], experiments at the atomic scale. That required the use of [[quantum mechanics]], specifically the [[quantization of the electromagnetic field]] and the development of [[quantum electrodynamics]].

: <math>\mathbf{F}

= q (\mathbf{E} + \mathbf{v} \times \mathbf{B}),</math>

== History ==

where <math>q</math> is the charge of the particle, <b>v</b> is its current velocity (expressed as a vector), and &times; is the [[cross product]] operator.

{{main|History of electromagnetic theory}}

[[File:Results of Michael Faraday's iron filings experiment. Wellcome M0000164.jpg|thumb|Results of Michael Faraday's iron filings experiment.]]

The empirical investigation of electromagnetism is at least as old as the ancient Greek philosopher, mathematician and scientist [[Thales of Miletus]], who around 600&nbsp;BCE described his experiments rubbing fur of animals on various materials such as amber creating static electricity.{{sfnp|ps=|ThoughtCo|2018}} By the 18th century, it was understood that objects can carry positive or negative [[electric charge]], that two objects carrying charge of the same sign repel each other, that two objects carrying charges of opposite sign attract one another, and that the strength of this force falls off as the square of the distance between them. [[Michael Faraday]] visualized this in terms of the charges interacting via the [[electric field]]. 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 are 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. In 1820, [[Hans Christian Ørsted]] showed that an electric current can deflect a nearby compass needle, establishing that electricity and magnetism are closely related phenomena.{{sfnp|ps=|Stauffer|1957}} Faraday then made the seminal observation that time-varying magnetic fields could induce electric currents in 1831.

In 1861, [[James Clerk Maxwell]] synthesized all the work to date on electrical and magnetic phenomena into a single mathematical theory, from which he then deduced that [[light]] is an electromagnetic wave. Maxwell's continuous field theory was very successful until evidence supporting the atomic model of matter emerged. Beginning in 1877, [[Hendrik Lorentz]] developed an atomic model of electromagnetism and in 1897 [[J. J. Thomson]] completed experiments that defined the [[electron]]. The Lorentz theory works for free charges in electromagnetic fields, but fails to predict the energy spectrum for bound charges in atoms and molecules. For that problem, [[quantum mechanics]] is needed, ultimately leading to the theory of [[quantum electrodynamics]].

The behaviour of electromagnetic fields can be described with [[Maxwell's equations]], and their quantum basis by [[quantum electrodynamics]].

Practical applications of the new understanding of electromagnetic fields emerged in the late 1800s. The electrical generator and motor were invented using only the empirical findings like Faraday's and Ampere's laws combined with practical experience.

==Behaviour of the electromagnetic field (a [[fluid dynamics|hydrodynamic]] interpretation)==

== Mathematical description ==

The electric and magnetic vector fields can be thought of as being the velocities of a pair of fluids which permeate space. In the absence of charges these fluids would be at rest, so that their velocity fields would be zero.

{{main|Mathematical descriptions of the electromagnetic field}}

There are different mathematical ways of representing the electromagnetic field. The first one views the electric and magnetic fields as three-dimensional [[vector field]]s. 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 {{math|'''E'''(''x'', ''y'', ''z'', ''t'')}} ([[electric field]]) and {{math|'''B'''(''x'', ''y'', ''z'', ''t'')}} ([[magnetic field]]).

If only the electric field ({{math|'''E'''}}) is non-zero, and is constant in time, the field is said to be an [[electrostatic field]]. Similarly, if only the magnetic field ({{math|'''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]].{{sfnp|ps=|Wangsness|1986|loc=Intermediate-level textbook}}

Electric charges act either as sources or sinks of the electric fluid. An electron is constantly absorbing electric fluid around it at some rate, call it &epsilon;. Protons are the reverse: they constantly pour electric "liquid" towards the surrounding space at rate &epsilon;, so liquid moves away from the proton with speed

:<math> v = {\epsilon \over 4 \pi r^2} </math>

(where ''r'' is distance of the fluid away from the proton) so that the total [[flux]] of liquid going through any (imaginary) sphere which contains that proton is the area of the sphere times the speed of the fluid flowing through it: <math> 4 \pi r^2 \cdot v = \epsilon </math>.

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

Magnetic liquid, on the other hand, has no sources or sinks: nothing can pour out or suck up magnetic fluid. Magnetic fluid is incompressible, which means that its density does not change: it is not possible to compress a lot of magnetic fluid into a smaller space, or to squash it out of a given volume. (Electric fluid is also incompressible, but it has sources and sinks.) If magnetic fluid is standing still, it can be stirred up, making it move in closed circles and closed loops (see [[vortical]] motion).

The behavior 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:

For the magnetic fluid to keep moving in the same loop, though, some force has to keep stirring it up: otherwise the energy of its circular motion will dissipate and the magnetic fluid will stop moving and will return to rest.

; [[Gauss's law]] : <math>\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}</math>

If electric fluid starts to accelerate in a certain direction, it will cause a vortex of magnetic fluid to move in circles around the direction in which the electric fluid is accelerating (according to the [[handedness|right hand rule]]). As soon as the electric fluid stops accelerating, the vortex of magnetic fluid vanishes.

; [[Gauss's law for magnetism]] : <math>\nabla \cdot \mathbf{B} = 0</math>

; [[Faraday's law of induction#Maxwell–Faraday equation|Faraday's law]] : <math>\nabla \times \mathbf{E} = -\frac {\partial \mathbf{B}}{\partial t}</math>

; [[Ampère's circuital law#Extending the original law: the Ampère–Maxwell equation|Ampère–Maxwell law]] : <math>\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}</math>

where <math>\rho</math> is the charge density, which is a function of time and position, <math>\varepsilon_0</math> is the [[vacuum permittivity]], <math>\mu_0</math> is the [[vacuum permeability]], and {{math|'''J'''}} is the current density vector, also a function of time and position. 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.

Notice that: electric fluid will not accelerate spontaneously. Something has to force it to accelerate. This same thing then causes (indirectly) the magnetic vortex to be stirred up. A magnetic vortex will not arise spontaneously.

When a field travels across to different media, the behavior of the field changes according to the properties of the media.{{sfnp|ps=|Edminister|1995|loc= Examples and practice problems}}

Finally, if magnetic fluid accelerates in a certain direction, it causes electric fluid to move in a vortex which circles around the direction of acceleration in the direction opposite to the right hand rule.

== Properties of the field ==

Summarily: an acceleration of the electric fluid causes a positive vortex of magnetic "liquid" to move around it, but an acceleration of the magnetic liquid causes a negative vortex of electric liquid to flow around it.

=== Electrostatics and magnetostatics ===

{{main|Electrostatics|Magnetostatics}}

[[File:VFPt image charge plane horizontal.svg|thumb|250px|Electric field of a positive point [[electric charge]] suspended over an infinite sheet of conducting material. The field is depicted by [[field line|electric field lines]], lines which follow the direction of the electric field in space.]]

The Maxwell equations simplify when the charge density at each point in space does not change over time and all electric currents likewise remain constant. All of the time derivatives vanish from the equations, leaving two expressions that involve the electric field,

<math display=block>\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}</math>

and

<math display=block>\nabla\times\mathbf{E} = 0,</math>

along with two formulae that involve the magnetic field:

<math display=block>\nabla \cdot \mathbf{B} = 0</math>

and

<math display=block>\nabla \times \mathbf{B} = \mu_0 \mathbf{J}.</math>

These expressions are the basic equations of [[electrostatics]], which focuses on situations where electrical charges do not move, and [[magnetostatics]], the corresponding area of magnetic phenomena.{{sfnp|ps=|Feynman|Leighton|Sands|1970|loc=[https://www.feynmanlectures.caltech.edu/II_04.html §4.1]}}

=== Transformations of electromagnetic fields ===

Why the opposite signs? The opposite signs create a negative feedback loop (see [[Lenz's law]].) An acceleration of electric fluid causes a positive magnetic vortex. This means that the magnetic fluid has been accelerated to produce this circular flow. But this causes a negative vortex of electric fluid around the magnetic vortex. This reactive vortical acceleration of electric fluid is in the direction opposite of the original acceleration of electric fluid: hence a negative feedback loop:

{{further|Classical electromagnetism and special relativity|Electromagnetic four-potential|Electromagnetic tensor}}

:<math> \Delta E \rightarrow + \Delta B </math>

:<math> - \Delta E \leftarrow + \Delta B </math>.

Whether a physical effect is attributable to an electric field or to a magnetic field is dependent upon the observer, in a way that [[special relativity]] makes mathematically precise. For example, suppose that a laboratory contains a long straight wire that carries an electrical current. In the frame of reference where the laboratory is at rest, the wire is motionless and electrically neutral: the current, composed of negatively charged electrons, moves against a background of positively charged ions, and the densities of positive and negative charges cancel each other out. A test charge near the wire would feel no electrical force from the wire. However, if the test charge is in motion parallel to the current, the situation changes. In the rest frame of the test charge, the positive and negative charges in the wire are moving at different speeds, and so the positive and negative charge distributions are [[Lorentz contraction|Lorentz-contracted]] by different amounts. Consequently, the wire has a nonzero net charge density, and the test charge must experience a nonzero electric field and thus a nonzero force. In the rest frame of the laboratory, there is no electric field to explain the test charge being pulled towards or pushed away from the wire. So, an observer in the laboratory rest frame concludes that a {{em|magnetic}} field must be present.{{sfnp|ps=|Purcell|Morin|2012|pp=259–263}}{{sfnp|ps=|Feynman|Leighton|Sands|1970|loc=[https://www.feynmanlectures.caltech.edu/II_13.html §13.6]}}

If there were a positive feedback loop, the result would be (presumably) similar to the effect produced by a microphone too close to its speaker: a deafening high pitched resonant noise. The positive feedback would cause the original acceleration of electric fluid to amplify itself continually, while at the same time the vortices around it would amplify as well: an explosive maelstrom of movement of electromagnetic fluid. Fortunately, the laws of electromagnetism being what they are, an initial disturbance (acceleration) of the electric fluid will cause feedback loop which, being negative, will tend to extinguish itself at its source but which will propagate outwards in what is called an [[electromagnetic wave]].

In general, a situation that one observer describes using only an electric field will be described by an observer in a different inertial frame using a combination of electric and magnetic fields. Analogously, a phenomenon that one observer describes using only a magnetic field will be, in a relatively moving reference frame, described by a combination of fields. The rules for relating the fields required in different reference frames are the [[Classical electromagnetism and special relativity|Lorentz transformations of the fields]].{{sfnp|ps=|Purcell|Morin|2012|p=309}}

==Flaw in the velocity field interpretation==

The fluid analogy does not work in this sense: that objects immersed in a moving fluid (e.g. a river) tend to be pushed by that fluid in such a way that the velocity of the object aligns with the velocity of the fluid. Once the velocities are aligned, the fluid's motion should vanish from the object's point of view.

Thus, electrostatics 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 a consequence of different frames of measurement. The fact that the two field variations can be reproduced just by changing the motion of the observer is further evidence that there is only a single actual field involved which is simply being observed differently.

However, the force of an electric field on a charged particle is <math> \mathbf{F} = q \mathbf{E} </math>, and this force is independent of the velocity of the particle, which means that the particle will accelerate continually in the direction of the field. If the field is the velocity field of a fluid then the fluid would be causing the object to accelerate continually in the direction of the fluid's motion, to the point that the object's speed becomes way larger than the fluid it is immersed in. This is [[paradox|paradoxical]].

=== Reciprocal behavior of electric and magnetic fields ===

From the continually accelerating object's point of view (see [[principle of relativity]]), if its speed has already surpassed the speed of the fluid, then the fluid is moving backwards, so the field should be pointing in the direction opposite to the direction in which the object keeps accelerating. This means that that the object should stop accelerating and begin decelerating, until its speed aligns with the speed of the electric fluid.

{{Main | Faraday's law of induction | Ampère's circuital law }}

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 inside a loop creates an electric voltage around the loop". This is the principle behind the [[electric generator]].

Ampere's Law roughly states that "an electrical current around a loop creates a magnetic field through the loop". Thus, this law can be applied to generate a magnetic field and run an [[electric motor]].

==The field as a stream of moving photons==

An alternative interpretation would be that the field is not actually a velocity field, but a [[flux]] density field of [[photon|photonic]] fluid, which is constantly moving at the same speed: the [[speed of light]], independent of the speed of the observer (the charged object). Photonic fluid never changes speed but can change net direction and the intensity of its net movement in that direction.

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

The velocity field interpretation is related to the hypothesis of an [[ether]] through which electromagnetic waves would propagate. The existence of the ether was disproved by the [[Michelson-Morley experiment]] and the necessity of having an ether vanished when it was replaced by [[Einstein]]'s [[theory of relativity]].

[[File:Onde electromagnetique.svg|thumb|upright=1.8|A [[linear polarization|linearly polarized]] electromagnetic [[plane wave]] propagating parallel to the z-axis is a possible solution for the [[electromagnetic wave equation]]s in [[free space]]. The [[electric field]], {{math|'''E'''}}, and the [[magnetic field]], {{math|'''B'''}}, are perpendicular to each other and the direction of propagation.|400x200px]]

[[Maxwell's equations]] can be combined to derive [[wave equation]]s. The solutions of these equations take the form of an [[electromagnetic wave]]. In a volume of space not containing charges or currents ([[free space]]) – that is, where <math>\rho</math> and {{math|'''J'''}} are zero, the electric and magnetic fields satisfy these [[electromagnetic wave equation]]s:{{sfnp|ps=|Feynman|Leighton|Sands|1970|loc=[https://www.feynmanlectures.caltech.edu/II_20.html §20.1]}}{{sfnp|ps=|Cheng|1989|loc=Intermediate-level textbook}}

: <math> \left( \nabla^2 - { 1 \over {c}^2 } {\partial^2 \over \partial t^2} \right) \mathbf{E} \ \ = \ \ 0</math>

: <math> \left( \nabla^2 - { 1 \over {c}^2 } {\partial^2 \over \partial t^2} \right) \mathbf{B} \ \ = \ \ 0</math>

[[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]]. This unified the physical understanding of electricity, magnetism, and light: visible light is but one portion of the full range of electromagnetic waves, the [[electromagnetic spectrum]].

According to [[special relativity]], the Lorentz force equation reduces to the equation

:<math> \mathbf{F} = q \mathbf{E}. </math>

The magnetic field becomes a relativistic by-product of the electric field. I.e. [[Lorentz transformation]]s cause magnetic fields to be induced from electric fields, and vice versa. So the photonic fluid describes the electric field, and relativistic effects account for the derivative magnetic field. (This can be derived by applying a Lorenz transformation to a simplified version of [[Maxwell's equations]], and it is mentioned by Einstein in his paper ''On The Electrodynamics Of Moving Bodies''.)

=== Time-varying EM fields in Maxwell's equations ===

The [[speed]] of light is invariant under a Lorenz transformation, but the [[velocity]] of light is changed. The component of the velocity of light parallel to the [[Coordinate rotation#Relativity|boost]] is left unchanged, but the transversal component is rotated: it is accelerated in a direction parallel to the boost. The addition of special relativity allows the combination of the electric and magnetic fields into a single [[tensor field]]. The tensor character of this combined electromagnetic field implies that the field is [[anisotropic]] with respect to the velocity of the charged particle on which it produces a force: the [[Lorenz force]] varies with the velocity of the charged particle.

{{main|near and far field|near field optics|virtual particle|dielectric heating|Electromagnetic induction}}

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. Such radiation can occur across a wide range of frequencies called the [[electromagnetic spectrum]], including [[radio wave]]s, [[microwave]], [[infrared]], [[visible light]], [[ultraviolet light]], [[X-rays]], and [[gamma ray]]s. 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.

==Light and Electromagnetic Waves==

Electrically charged particles are constantly emitting (or absorbing) photonic fluid, which is more commonly known as [[light]]. So how is light related to electromagnetic waves? E-M waves are undulatory movement patterns of light which can always be observed to be emitted by electric charges undergoing [[acceleration]].

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''.

If a charged particle is at rest, then it does not emit electromagnetic waves. Instead, it is surrounded by an [[electrostatic field]]. If the charged particle is in inertial motion, then the electrostatic field is joined by a [[magnetostatic field]]. These pair of static fields produce a movement of electromagnetic energy, which is similar to an electromagnetic wave, except that that the fields are not oscillating.

Changing {{em|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.

E-M waves are propagating, expanding, [[simple harmonic motion|harmonic]], [[oscillation|oscillating]], accelerations of the photonic fluid. Since the photonic fluid itself moves at the speed of light (by definition), then E-M waves can move no faster than the speed of light. E-M waves move at a speed close to the speed of light, depending on the [[medium]] through which they move (e.g. faster in air than through water, and faster through water than through a glass [[lens (optics)|lens]]).

Changing {{em|magnetic}} dipole fields (i.e., magnetic near-fields) are used commercially for many types of [[Electromagnetic induction|magnetic induction]] devices. These include motors and electrical transformers at low frequencies, and devices such as [[RFID]] tags, [[metal detector]]s, and [[MRI]] scanner coils at higher frequencies.

Water [[wave]]s are waves of water, and air [[sound]] waves are waves of air, and electromagnetic waves are [[transverse wave]]s of light.

== Health and safety ==

[[Ocean current]]s are movements of water, and [[wind]] is the movement of air, and the electromagnetic field is the movement of light.

{{main | Electromagnetic radiation and health}}

The potential effects of electromagnetic fields on human health vary widely depending on the frequency, intensity of the fields, and the length of the exposure. Low frequency, low intensity, and short duration exposure to electromagnetic radiation is generally considered safe.{{sfnp|ps=|NIOSH|1996}} On the other hand, radiation from other parts of the [[electromagnetic spectrum]], such as [[ultraviolet]] light{{sfnp|ps=|Ultraviolet Radiation Guide|1992}} and [[gamma ray]]s,{{sfnp|ps=|Valentin|2007|pp=61–79|loc=§4. Quantities used in radiological protection}} are known to cause significant harm in some circumstances.

== See also ==

==The Electromagnetic Field as a [[Feedback]] Loop==

{{div col|colwidth=16em}}

The behavior of the electromagnetic field can be resolved into four different parts of a loop: (1) the electric and magnetic fields are generated by electric charges, (2) the electric and magnetic fields interact only with each other, (3) the electric and magnetic fields produce forces on electric charges, (4) the electric charges move in space.

* [[Classification of electromagnetic fields]]

* [[Electric field]]

* [[Electromagnetism]]

* [[Electromagnetic propagation]]

* [[Electromagnetic radiation]]

* [[Electromagnetic spectrum]]

* [[EMF measurements|Electromagnetic field measurements]]

* [[Magnetic field]]

* [[Maxwell's equations]]

* [[Photoelectric effect]]

* [[Photon]]

* [[Quantization of the electromagnetic field]]

* [[Quantum electrodynamics]]

{{div col end}}

== References ==

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

=== Citations ===

* charges generate fields

{{reflist}}

** [[Gauss's Law]]: charges generate electric fields

** [[Amp&egrave;re's Law]]: currents generate magnetic fields

* the fields interact with each other

** [[Faraday's law|Faraday induction]]: changing magnetic field induces vortex of electric field

** [[displacement current]]: changing electric field acts like a current, generating vortex of magnetic field

** [[Maxwell-Hertz equations]]

** electromagnetic [[wave equation]]

* fields act upon charges

** [[Lorenz 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

* charges move

** [[continuity equation]]: current is movement of charges

=== Sources ===

'''See also:''' [[electromagnetic interaction]], [[intensity]], [[bremsstrahlung]], [[polar aurora]].

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{{refend}}

== Further reading ==

==External Link==

{{refbegin|2}}

*[http://www.fourmilab.ch/etexts/einstein/specrel/www/ ''On the Electrodynamics Of Moving Bodies''] by [[Einstein|A. Einstein]], June 30, 1905. Hosted by Fourmilab.

* {{cite book |last1=Griffiths |first1=David J. |title=Introduction to Electrodynamics |date=1999 |publisher=Prentice Hall |___location=Upper Saddle River, New Jersey |isbn=978-0138053260 |edition=3rd |url=https://archive.org/details/introductiontoel00grif_0}}

* {{cite journal |last1=Maxwell |first1=J. C.|author-link=James Clerk Maxwell |title=A Dynamical Theory of the Electromagnetic Field |journal=[[Philosophical Transactions of the Royal Society of London]] |date=1 January 1865 |volume=155 |pages=459–512 |doi=10.1098/rstl.1865.0008 |s2cid=186207827}} (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)

* {{cite book |last1=Greene |first1=Brian |author-link=Brian Greene |title=The Fabric of the Cosmos |publisher=Random House |___location=New York |at=Ch. 3, §§ "Force", "Matter", and "The Higgs Field"}}

{{refend}}

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