TheInfoList

Maxwell's equations are a set of coupled
partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s that, together with the
Lorentz force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ... law, form the foundation of
classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and ...
, classical
optics Optics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other wo ... , and
electric circuit An electrical network is an interconnection of electrical component An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic componen ... s. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors,
wireless Wireless communication (or just wireless, when the context allows) is the transfer of information between two or more points that do not use an electrical conductor In physics Physics (from grc, φυσική (ἐπιστήμη), ph ... communication, lenses, radar etc. They describe how
electric Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnet ... and
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ... s are generated by charges,
currents Currents or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stream), c ...
, and changes of the fields.''Electric'' and ''magnetic'' fields, according to the
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
, are the components of a single electromagnetic field.
The equations are named after the physicist and mathematician
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classica ... , 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 Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English autodidactic Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education without the guidance of masters (such as teach ...
. An important consequence of Maxwell's equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed ('' c'') in a vacuum. Known as
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ... , these waves may occur at various wavelengths to produce a
spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a Continuum (theory), continuum. The word was first used scientifically in optics to describe the ... radio wave Radio waves are a type of electromagnetic radiation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies ma ...
s to
gamma ray A gamma ray, also known as gamma radiation (symbol γ or \gamma), is a penetrating form of electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, it ...
s. The equations have two major variants. The ''microscopic'' equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the
atomic scale Atomic spacing refers to the distance between the Atomic nucleus, nuclei of atoms in a material. This space is extremely large compared to the size of the atomic nucleus, and is related to the chemical bonds which bind atoms together. In solid mate ...
. The ''macroscopic'' equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials. The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the
electric Electricity is the set of physics, physical Phenomenon, phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnet ... and
magnetic scalar potential Magnetic scalar potential, ''ψ'', is a quantity in classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charge ...
s are preferred for explicitly solving the equations as a
boundary value problem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
,
analytical mechanics Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic can also have the following meanings: Natural sciences Chemistry * ...
, or for use in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
. The covariant formulation (on
spacetime In , spacetime is any which fuses the and the one of into a single . can be used to visualize effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three ...
rather than space and time separately) makes the compatibility of Maxwell's equations with
special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
manifest Manifest may refer to: Computing * Manifest file, a metadata file that enumerates files in a program or package * Manifest (CLI), a metadata text file for CLI assemblies Events * Manifest (convention), a defunct anime festival in Melbourne, Austr ...
.
Maxwell's equations in curved spacetime In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
, commonly used in high energy and
gravitational physics Gravity (), or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxy, galaxies, and even light—are brought toward (or ''gravitate'' toward) one another. On Earth, gravity gives we ...
, are compatible with
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
.In general relativity, however, they must enter, through its
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the Cauchy str ...
, into
Einstein field equations In the General relativity, general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of Matter#In general relativity and cosmology, matter within it. ...
that include the spacetime curvature.
In fact,
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ... developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences. The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light and associated radiation. Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a
classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B.C.E. to the 5th century C.E. centered on the Mediterranean Sea *Classical architecture, architecture derived from Greek and ...
limit of the more precise theory of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativity theory, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum m ...
.

# Conceptual descriptions

## Gauss's law

Gauss's law In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
describes the relationship between a static
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ... and
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
s: a static electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a
closed surface with ''x''-, ''y''-, and ''z''-contours shown. In the part of mathematics referred to as topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Gree ... is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced as "epsilon nought" or "epsilon zero") is the value of the absolute dielectric permittivity of classical vacuum. Alternatively it may be referred to as the permittivity of free space, the elec ...
.

## Gauss's law for magnetism

Gauss's law for magnetism In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
states that electric charges have no magnetic analogues, called
magnetic monopole In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is th ...
s. Instead, the magnetic field of a material is attributed to a
dipole In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... , and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite 'magnetic charges'. Precisely, the total
magnetic flux In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ... through a Gaussian surface is zero, and the magnetic field is a
solenoidal vector field In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes use ...
.The absence of sinks/sources of the field does not imply that the field lines must be closed or escape to infinity. They can also wrap around indefinitely, without self-intersections. Moreover, around points where the field is zero (that cannot be intersected by field lines, because their direction would not be defined), there can be the simultaneous begin of some lines and end of other lines. This happens, for instance, in the middle between two identical cylindrical magnets, whose north poles face each other. In the middle between those magnets, the field is zero and the axial field lines coming from the magnets end. At the same time, an infinite number of divergent lines emanate radially from this point. The simultaneous presence of lines which end and begin around the point preserves the divergence-free character of the field. For a detailed discussion of non-closed field lines, see L. Zilberti "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017 (available at https://zenodo.org/record/4518772#.YCJU_WhKjIU) The Maxwell–Faraday version of
Faraday's law of induction Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric c ...
describes how a time varying
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ... creates ("induces") an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ... . In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface. The
electromagnetic induction Electromagnetic or magnetic induction is the production of an electromotive force In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs b ... is the operating principle behind many
electric generator In electricity generation Electricity generation is the process of generating electric power from sources of primary energy. For electric utility, utilities in the electric power industry, it is the stage prior to its Electricity delivery, deliv ...
s: for example, a rotating
bar magnet Magnetic field lines of a solenoid electromagnet, which are similar to a bar magnet as illustrated below with the iron filings">electromagnet.html" ;"title="solenoid electromagnet">solenoid electromagnet, which are similar to a bar magnet as ... creates a changing magnetic field, which in turn generates an electric field in a nearby wire.

## Ampère's law with Maxwell's addition

Ampère's law with Maxwell's addition states that magnetic fields can be generated in two ways: by
electric current An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, ...
(this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's addition", which he called
displacement current In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... ). In integral form, the magnetic field induced around any closed loop is proportional to the electric current plus displacement current (proportional to the rate of change of electric flux) through the enclosed surface. Maxwell's addition to Ampère's law is particularly important: it makes the set of equations mathematically consistent for non static fields, without changing the laws of Ampere and Gauss for static fields. However, as a consequence, it predicts that a changing magnetic field induces an electric field and vice versa. Therefore, these equations allow self-sustaining "
electromagnetic waves In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ... " to travel through empty space (see
electromagnetic wave equation The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a Medium (optics), medium or in a vacuum. It is a Wave equation#Scalar wave equation in three space d ...
). The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents,The quantity we would now call , with units of velocity, was directly measured before Maxwell's equations, in an 1855 experiment by
Wilhelm Eduard Weber Wilhelm Eduard Weber (; ; 24 October 1804 – 23 June 1891) was a German physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area ... and
Rudolf Kohlrausch Rudolf Hermann Arndt Kohlrausch (November 6, 1809 in Göttingen Göttingen (, also , ; nds, Chöttingen) is a college town, university city in Lower Saxony, Germany, the Capital (political), capital of Göttingen (district), the eponymous dist ... . They charged a
leyden jar A Leyden jar (or Leiden jar, or archaically, sometimes Kleistian jar) is an electrical component An electronic component is any basic discrete device or physical entity in an electronic system Electronic may refer to: *Electronics, the scien ... (a kind of
capacitor A capacitor is a device that stores electric charge in an electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπ� ... ), and measured the
electrostatic force ''F'' between two point charges ''q''1 and ''q''2 is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Like charges repel each other, and opposite charges mut ...
associated with the potential; then, they discharged it while measuring the
magnetic force In physics (specifically in electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The elec ...
from the current in the discharge wire. Their result was , remarkably close to the speed of light. See Joseph F. Keithley
''The story of electrical and magnetic measurements: from 500 B.C. to the 1940s'', p. 115
/ref> matches the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
; indeed,
light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nan ... ''is'' one form of
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ... (as are
X-ray An X-ray, or, much less commonly, X-radiation, is a penetrating form of high-energy electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Moti ... s,
radio wave Radio waves are a type of electromagnetic radiation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies ma ...
s, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of
electromagnetism Electromagnetism is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in ... and
optics Optics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other wo ... .

# Formulation in terms of electric and magnetic fields (microscopic or in vacuum version)

In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate law of nature, the
Lorentz force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ... law, describes how, conversely, the electric and magnetic fields act on charged particles and currents. A version of this law was included in the original equations by Maxwell but, by convention, is included no longer. The
vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product differentiation, in ...
formalism below, the work of
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English autodidactic Autodidacticism (also autodidactism) or self-education (also self-learning and self-teaching) is education without the guidance of masters (such as teach ...
, has become standard. It is manifestly rotation invariant, and therefore mathematically much more transparent than Maxwell's original 20 equations in x,y,z components. The relativistic formulations are even more symmetric and manifestly Lorentz invariant. For the same equations expressed using tensor calculus or differential forms, see alternative formulations. The differential and integral formulations are mathematically equivalent and are both useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely ''local'' and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using
finite element analysis The finite element method (FEM) is a widely used method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, ...
.

## Key to the notation

Symbols in bold represent
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
quantities, and symbols in ''italics'' represent
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
quantities, unless otherwise indicated. The equations introduce the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ... , , a
vector field In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Product ... , and the
magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with ... , , a
pseudovector In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ... field, each generally having a time and location dependence. The sources are *the total electric
charge density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... (total charge per unit volume), , and *the total electric
current density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... (total current per unit area), . The universal constants appearing in the equations (the first two ones explicitly only in the SI units formulation) are: *the
permittivity of free space Vacuum permittivity, commonly denoted (pronounced as "epsilon nought" or "epsilon zero") is the value of the absolute dielectric permittivity of classical vacuum. Alternatively it may be referred to as the permittivity of free space, the elec ...
, , and *the
permeability of free space Vacuum permeability is the magnetic permeability in a classical vacuum. ''Vacuum permeability'' is derived from production of a magnetic field by an electric current or by a moving electric charge and in all other formulas for magnetic-field pro ...
, , and *the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
, $c = \frac$

### Differential equations

In the differential equations, *the
nabla symbol ∇ The nabla symbol The nabla is a triangular symbol resembling an inverted Greek delta Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, a landform at the mouth of a river * D (NATO phonetic ...
, , denotes the three-dimensional
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ... operator,
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional ... , *the symbol (pronounced "del dot") denotes the
divergence In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes ... operator, *the symbol (pronounced "del cross") denotes the
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimension ...
operator.

### Integral equations

In the integral equations, * is any fixed volume with closed
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
surface , and * is any fixed surface with closed boundary curve , Here a ''fixed'' volume or surface means that it does not change over time. The equations are correct, complete, and a little easier to interpret with time-independent surfaces. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law: $\frac \iint_ \mathbf \cdot \mathrm\mathbf = \iint_ \frac \cdot \mathrm\mathbf\,,$ Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss and Stokes formula appropriately. * is a
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... over the boundary surface , with the loop indicating the surface is closed *$\iiint_\Omega$ is a
volume integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
over the volume , *$\oint_$ is a
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
around the boundary curve , with the loop indicating the curve is closed. *$\iint_\Sigma$ is a
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... over the surface , * The ''total''
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
enclosed in is the
volume integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
over of the
charge density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ... (see the "macroscopic formulation" section below): $Q = \iiint_\Omega \rho \ \mathrmV,$ where is the
volume elementIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. * The ''net''
electric current An electric current is a stream of charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, ...
is the
surface integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ... of the
electric current density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is car ...
passing through a fixed surface, : $I = \iint_ \mathbf \cdot \mathrm \mathbf,$ where denotes the differential
vector element of surface area , normal to surface . (Vector area is sometimes denoted by rather than , but this conflicts with the notation for
magnetic vector potential Magnetic vector potential, A, is the vector quantity in classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charg ...
).

## Formulation in Gaussian units convention

The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of and into the units of calculation, by convention. With a corresponding change in convention for the
Lorentz force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ... law this yields the same physics, i.e. trajectories of charged particles, or
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking * Work (physics), the product of ... done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the
electromagnetic tensor In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ...
: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension. Such modified definitions are conventionally used with the Gaussian ( CGS) units. Using these definitions and conventions, colloquially "in Gaussian units", the Maxwell equations become: The equations are particularly readable when length and time are measured in compatible units like seconds and lightseconds i.e. in units such that c = 1 unit of length/unit of time. Ever since 1983 (see
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
), metres and seconds are compatible except for historical legacy since ''by definition'' c = 299 792 458 m/s (≈ 1.0 feet/nanosecond). Further cosmetic changes, called rationalisations, are possible by absorbing factors of depending on whether we want
Coulomb's law Coulomb's law, or Coulomb's inverse-square law, is an experimental physical law, law of physics that quantifies the amount of force between two stationary, electric charge, electrically charged particles. The electric force between charged bodi ...
or
Gauss's law In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
to come out nicely, see
Lorentz–Heaviside units Lorentz–Heaviside units (or Heaviside–Lorentz units) constitute a system of units (particularly electromagnetic units) within Centimetre–gram–second system of units, CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share wit ...
(used mainly in
particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which rel ...
).

# Relationship between differential and integral formulations

The equivalence of the differential and integral formulations are a consequence of the
Gauss divergence theorem and the
Kelvin–Stokes theorem Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, B ...
.

## Flux and divergence According to the (purely mathematical)
Gauss divergence theorem , the
electric flux In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carr ...
through the boundary surface can be rewritten as : The integral version of Gauss's equation can thus be rewritten as $\iiint_ \left(\nabla \cdot \mathbf - \frac\right) \, \mathrmV = 0$ Since is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
the integrand is zero everywhere. This is the differential equations formulation of Gauss equation up to a trivial rearrangement. Similarly rewriting the
magnetic flux In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ... in Gauss's law for magnetism in integral form gives : which is satisfied for all if and only if $\nabla \cdot \mathbf = 0$ everywhere.

## Circulation and curl By the
Kelvin–Stokes theorem Stokes' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, B ...
we can rewrite the
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s of the fields around the closed boundary curve to an integral of the "circulation of the fields" (i.e. their
curl Curl or CURL may refer to: Science and technology * Curl (mathematics) In vector calculus Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimension ...
s) over a surface it bounds, i.e. $\oint_ \mathbf \cdot \mathrm\boldsymbol = \iint_\Sigma (\nabla \times \mathbf) \cdot \mathrm\mathbf,$ Hence the modified Ampere law in integral form can be rewritten as $\iint_\Sigma \left(\nabla \times \mathbf - \mu_0 \left(\mathbf + \varepsilon_0 \frac\right)\right)\cdot \mathrm\mathbf = 0.$ Since can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that the integrand is zero
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
Ampere's modified law in differential equations form is satisfied. The equivalence of Faraday's law in differential and integral form follows likewise. The line integrals and curls are analogous to quantities in classical
fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) and ...
: the
circulation Circulation may refer to: Science and technology * Atmospheric circulation, the large-scale movement of air * Circulation (physics), the path integral of the fluid velocity around a closed curve in a fluid flow field * Circulatory system, a biolo ...
of a fluid is the line integral of the fluid's
flow velocityIn continuum mechanics the flow velocity in fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including aerodyna ...
field around a closed loop, and the
vorticity In continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as point particle, discrete particles. The French mathematician Augustin-Louis C ... of the fluid is the curl of the velocity field.

# Charge conservation

The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the modified Ampere's Law has zero divergence by the div–curl identity. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives: $0 = \nabla\cdot (\nabla\times \mathbf) = \nabla \cdot \left(\mu_0 \left(\mathbf + \varepsilon_0 \frac \right) \right) = \mu_0\left(\nabla\cdot \mathbf + \varepsilon_0\frac\nabla\cdot \mathbf\right) = \mu_0\left(\nabla\cdot \mathbf +\frac\right)$ i.e., $\frac + \nabla \cdot \mathbf = 0.$ By the Gauss Divergence Theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary: : In particular, in an isolated system the total charge is conserved.

# Vacuum equations, electromagnetic waves and speed of light In a region with no charges () and no currents (), such as in a vacuum, Maxwell's equations reduce to: $\begin \nabla \cdot \mathbf &= 0 \quad & \nabla \times \mathbf &= -\frac, \\ \nabla \cdot \mathbf &= 0 \quad & \nabla \times \mathbf &= \mu_0\varepsilon_0 \frac. \end$ Taking the curl of the curl equations, and using the Vector calculus identities#Curl of curl, curl of the curl identity we obtain $\begin \mu_0\varepsilon_0 \frac - \nabla^2 \mathbf = 0 \\ \mu_0\varepsilon_0 \frac - \nabla^2 \mathbf = 0 \end$ The quantity $\mu_0\varepsilon_0$ has the dimension of (time/length)2. Defining $c = \left(\mu_0 \varepsilon_0\right)^$, the equations above have the form of the standard wave equations $\begin \frac \frac - \nabla^2 \mathbf = 0 \\ \frac \frac - \nabla^2 \mathbf = 0 \end$ Already during Maxwell's lifetime, it was found that the known values for $\varepsilon_0$ and $\mu_0$ give $c \approx 2.998 \times 10^8 \, \text$, then already known to be the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In the SI system, old SI system of units, the values of $\mu_0 = 4\pi\times 10^$ and $c = 299 792 458 \,\text$ are defined constants, (which means that by definition $\varepsilon_0 = 8.854... \times 10^ \,\text$) that define the ampere and the metre. In the new SI system, only ''c'' keeps its defined value, and the electron charge gets a defined value. In materials with relative permittivity, , and Permeability (electromagnetism)#Relative permeability and magnetic susceptibility, relative permeability, , the phase velocity of light becomes $v_\text = \frac\sqrt$ which is usuallyThere are cases (anomalous dispersion) where the phase velocity can exceed , but the "signal velocity" will still be less than . In addition, and are perpendicular to each other and to the direction of wave propagation, and are in phase (waves), phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law of induction, Faraday's law. In turn, that electric field creates a changing magnetic field through Ampère's circuital law, Maxwell's addition to Ampère's law. This perpetual cycle allows these waves, now known as
electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ... , to move through space at velocity .

# Macroscopic formulation

The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping. The microscopic version is sometimes called "Maxwell's equations in a vacuum": this refers to the fact that the material medium is not built into the structure of the equations, but appears only in the charge and current terms. The microscopic version was introduced by Lorentz, who tried to use it to derive the macroscopic properties of bulk matter from its microscopic constituents. "Maxwell's macroscopic equations", also known as Maxwell's equations in matter, are more similar to those that Maxwell introduced himself. In the macroscopic equations, the influence of bound charge and bound current is incorporated into the electric displacement field, displacement field and the magnetizing field , while the equations depend only on the free charges and free currents . This reflects a splitting of the total electric charge ''Q'' and current ''I'' (and their densities and J) into free and bound parts: $\begin Q &= Q_\text + Q_\text = \iiint_\Omega \left(\rho_\text + \rho_\text \right) \, \mathrmV = \iiint_\Omega \rho \,\mathrmV \\ I &= I_\text + I_\text = \iint_\Sigma \left(\mathbf_\text + \mathbf_\text \right) \cdot \mathrm\mathbf = \iint_\Sigma \mathbf \cdot \mathrm\mathbf \end$ The cost of this splitting is that the additional fields and need to be determined through phenomenological constituent equations relating these fields to the electric field and the magnetic field , together with the bound charge and current. See below for a detailed description of the differences between the microscopic equations, dealing with ''total'' charge and current including material contributions, useful in air/vacuum;In some books—e.g., in U. Krey and A. Owen's Basic Theoretical Physics (Springer 2007)—the term ''effective charge'' is used instead of ''total charge'', while ''free charge'' is simply called ''charge''. and the macroscopic equations, dealing with ''free'' charge and current, practical to use within materials.

## Bound charge and current When an electric field is applied to a dielectric, dielectric material its molecules respond by forming microscopic electric dipoles – their atomic nucleus, atomic nuclei move a tiny distance in the direction of the field, while their electrons move a tiny distance in the opposite direction. This produces a ''macroscopic'' ''bound charge'' in the material even though all of the charges involved are bound to individual molecules. For example, if every molecule responds the same, similar to that shown in the figure, these tiny movements of charge combine to produce a layer of positive Bound charge#Bound charge, bound charge on one side of the material and a layer of negative charge on the other side. The bound charge is most conveniently described in terms of the polarization density, polarization of the material, its dipole moment per unit volume. If is uniform, a macroscopic separation of charge is produced only at the surfaces where enters and leaves the material. For non-uniform , a charge is also produced in the bulk. Somewhat similarly, in all materials the constituent atoms exhibit magnetic moment#Examples of magnetic moments, magnetic moments that are intrinsically linked to the gyromagnetic ratio, angular momentum of the components of the atoms, most notably their electrons. The magnetic field#Magnetic dipoles, connection to angular momentum suggests the picture of an assembly of microscopic current loops. Outside the material, an assembly of such microscopic current loops is not different from a macroscopic current circulating around the material's surface, despite the fact that no individual charge is traveling a large distance. These ''Bound current#Magnetization current, bound currents'' can be described using the magnetization . The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of and , which average these charges and currents on a sufficiently large scale so as not to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material. As such, ''Maxwell's macroscopic equations'' ignore many details on a fine scale that can be unimportant to understanding matters on a gross scale by calculating fields that are averaged over some suitable volume.

## Auxiliary fields, polarization and magnetization

The ''List of electromagnetism equations#Definitions, definitions'' of the auxiliary fields are: $\begin \mathbf(\mathbf, t) &= \varepsilon_0 \mathbf(\mathbf, t) + \mathbf(\mathbf, t) \\ \mathbf(\mathbf, t) &= \frac \mathbf(\mathbf, t) - \mathbf(\mathbf, t) \end$ where is the polarization density, polarization field and is the magnetization field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density and bound current density in terms of polarization density, polarization and magnetization are then defined as $\begin \rho_\text &= -\nabla\cdot\mathbf \\ \mathbf_\text &= \nabla\times\mathbf + \frac \end$ If we define the total, bound, and free charge and current density by $\begin \rho &= \rho_\text + \rho_\text, \\ \mathbf &= \mathbf_\text + \mathbf_\text, \end$ and use the defining relations above to eliminate , and , the "macroscopic" Maxwell's equations reproduce the "microscopic" equations.

## Constitutive relations

In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between Electric displacement field, displacement field and the electric field , as well as the Magnetic field#H-field and magnetic materials, magnetizing field and the magnetic field . Equivalently, we have to specify the dependence of the polarization (hence the bound charge) and the magnetization (hence the bound current) on the applied electric and magnetic field. The equations specifying this response are called constitutive relations. For real-world materials, the constitutive relations are rarely simple, except approximately, and usually determined by experiment. See the main article on constitutive relations for a fuller description. For materials without polarization and magnetization, the constitutive relations are (by definition) $\mathbf = \varepsilon_0\mathbf, \quad \mathbf = \frac\mathbf$ where is the permittivity of free space and the permeability (electromagnetism), permeability of free space. Since there is no bound charge, the total and the free charge and current are equal. An alternative viewpoint on the microscopic equations is that they are the macroscopic equations ''together'' with the statement that vacuum behaves like a perfect linear "material" without additional polarization and magnetization. More generally, for linear materials the constitutive relations are $\mathbf = \varepsilon\mathbf\,,\quad \mathbf = \frac\mathbf$ where is the permittivity and the permeability (electromagnetism), permeability of the material. For the displacement field the linear approximation is usually excellent because for all but the most extreme electric fields or temperatures obtainable in the laboratory (high power pulsed lasers) the interatomic electric fields of materials of the order of 1011 V/m are much higher than the external field. For the magnetizing field $\mathbf$, however, the linear approximation can break down in common materials like iron leading to phenomena like hysteresis. Even the linear case can have various complications, however. *For homogeneous materials, and are constant throughout the material, while for inhomogeneous materials they depend on position vector, location within the material (and perhaps time). *For isotropic materials, and are scalars, while for anisotropic materials (e.g. due to crystal structure) they are tensors. *Materials are generally dispersion (optics), dispersive, so and depend on the frequency of any incident EM waves. Even more generally, in the case of non-linear materials (see for example nonlinear optics), and are not necessarily proportional to , similarly or is not necessarily proportional to . In general and depend on both and , on location and time, and possibly other physical quantities. In applications one also has to describe how the free currents and charge density behave in terms of and possibly coupled to other physical quantities like pressure, and the mass, number density, and velocity of charge-carrying particles. E.g., the original equations given by Maxwell (see History of Maxwell's equations) included Ohm's law in the form $\mathbf_\text = \sigma \mathbf\,.$

# Alternative formulations

Following is a summary of some of the numerous other mathematical formalisms to write the microscopic Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones involving charge and current. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the electrical potential and the vector potential . Potentials were introduced as a convenient way to solve the homogeneous equations, but it was thought that all observable physics was contained in the electric and magnetic fields (or relativistically, the Faraday tensor). The potentials play a central role in quantum mechanics, however, and act quantum mechanically with observable consequences even when the electric and magnetic fields vanish (Aharonov–Bohm effect). Each table describes one formalism. See the Mathematical descriptions of the electromagnetic field, main article for details of each formulation. SI units are used throughout.

# Relativistic formulations

The Maxwell equations can also be formulated on a spacetime-like Minkowski space where space and time are treated on equal footing. The direct spacetime formulations make manifest that the Maxwell equations are relativistically invariant. Because of this symmetry electric and magnetic field are treated on equal footing and are recognised as components of the Faraday tensor. This reduces the four Maxwell equations to two, which simplifies the equations, although we can no longer use the familiar vector formulation. In fact the Maxwell equations in the space + time formulation are not Galilean transformation, Galileo invariant and have Lorentz invariance as a hidden symmetry. This was a major source of inspiration for the development of relativity theory. Indeed, even the formulation that treats space and time separately is not a non-relativistic approximation and describes the same physics by simply renaming variables. For this reason the relativistic invariant equations are usually called the Maxwell equations as well. Each table describes one formalism. *In the tensor calculus formulation, the
electromagnetic tensor In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ...
is an antisymmetric covariant order 2 tensor; the four-potential, , is a covariant vector; the current, , is a vector; the square brackets, , denote Ricci calculus#Symmetric and antisymmetric parts, antisymmetrization of indices; is the derivative with respect to the coordinate, . In Minkowski space coordinates are chosen with respect to an inertial frame; , so that the metric tensor used to raise and lower indices is . The d'Alembert operator on Minkowski space is as in the vector formulation. In general spacetimes, the coordinate system is arbitrary, the covariant derivative , the Ricci tensor, and raising and lowering of indices are defined by the Lorentzian metric, and the d'Alembert operator is defined as . The topological restriction is that the second real cohomology group of the space vanishes (see the differential form formulation for an explanation). This is violated for Minkowski space with a line removed, which can model a (flat) spacetime with a point-like monopole on the complement of the line. *In the differential form formulation on arbitrary space times, is the electromagnetic tensor considered as a 2-form, is the potential 1-form, $J = - J_\alpha \mathrmx^\alpha$ is the current 3-form, is the exterior derivative, and  is the Hodge star on forms defined (up to its orientation, i.e. its sign) by the Lorentzian metric of spacetime. In the special case of 2-forms such as ''F'', the Hodge star  depends on the metric tensor only for its local scale. This means that, as formulated, the differential form field equations are conformal geometry, conformally invariant, but the Lorenz gauge condition breaks conformal invariance. The operator $\Box = \left(- \mathrm \mathrm - \mathrm \mathrm \right)$ is the Laplace–Beltrami operator, d'Alembert–Laplace–Beltrami operator on 1-forms on an arbitrary pseudo-Riemannian manifold#Lorentzian manifold, Lorentzian spacetime. The topological condition is again that the second real cohomology group is 'trivial' (meaning that its form follows from a definition). By the isomorphism with the second de Rham cohomology this condition means that every closed 2-form is exact. Other formalisms include the Geometric algebra#Spacetime Model, geometric algebra formulation and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation was used.

# Solutions

Maxwell's equations are partial differential equations that relate the electric and magnetic fields to each other and to the electric charges and currents. Often, the charges and currents are themselves dependent on the electric and magnetic fields via the Lorentz force, Lorentz force equation and the #Constitutive relations, constitutive relations. These all form a set of coupled partial differential equations which are often very difficult to solve: the solutions encompass all the diverse phenomena of
classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and ...
. Some general remarks follow. As for any differential equation, boundary conditions and initial conditions are necessary for a Electromagnetism uniqueness theorem, unique solution. For example, even with no charges and no currents anywhere in spacetime, there are the obvious solutions for which E and B are zero or constant, but there are also non-trivial solutions corresponding to electromagnetic waves. In some cases, Maxwell's equations are solved over the whole of space, and boundary conditions are given as asymptotic limits at infinity. In other cases, Maxwell's equations are solved in a finite region of space, with appropriate conditions on the boundary of that region, for example an Perfectly matched layer, artificial absorbing boundary representing the rest of the universe, or periodic boundary conditions, or walls that isolate a small region from the outside world (as with a waveguide or cavity resonator). Jefimenko's equations (or the closely related Liénard–Wiechert potentials) are the explicit solution to Maxwell's equations for the electric and magnetic fields created by any given distribution of charges and currents. It assumes specific initial conditions to obtain the so-called "retarded solution", where the only fields present are the ones created by the charges. However, Jefimenko's equations are unhelpful in situations when the charges and currents are themselves affected by the fields they create. Numerical partial differential equations, Numerical methods for differential equations can be used to compute approximate solutions of Maxwell's equations when exact solutions are impossible. These include the finite element method and finite-difference time-domain method. For more details, see Computational electromagnetics.

# Overdetermination of Maxwell's equations

Maxwell's equations ''seem'' Overdetermined system, overdetermined, in that they involve six unknowns (the three components of and ) but eight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampere's laws). (The currents and charges are not unknowns, being freely specifiable subject to charge conservation.) This is related to a certain limited kind of redundancy in Maxwell's equations: It can be proven that any system satisfying Faraday's law and Ampere's law ''automatically'' also satisfies the two Gauss's laws, as long as the system's initial condition does, and assuming conservation of charge and the nonexistence of magnetic monopoles. This explanation was first introduced by Julius Adams Stratton in 1941. Although it is possible to simply ignore the two Gauss's laws in a numerical algorithm (apart from the initial conditions), the imperfect precision of the calculations can lead to ever-increasing violations of those laws. By introducing dummy variables characterizing these violations, the four equations become not overdetermined after all. The resulting formulation can lead to more accurate algorithms that take all four laws into account. Both identities $\nabla\cdot \nabla\times \mathbf \equiv 0, \nabla\cdot \nabla\times \mathbf \equiv 0$, which reduce eight equations to six independent ones, are the true reason of overdetermination. Or First-order partial differential equation#Definitions of linear dependence for differential systems, definitions of linear dependence for PDE can be referred. Equivalently, the overdetermination can be viewed as implying conservation of electric and magnetic charge, as they are required in the derivation described above but implied by the two Gauss's laws. For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But in differential equations, and especially PDEs, one needs appropriate boundary conditions, which depend in not so obvious ways on the equations. Even more, if one rewrites them in terms of vector and scalar potential, then the equations are underdetermined because of Gauge fixing.

# Maxwell's equations as the classical limit of QED

Maxwell's equations and the Lorentz force law (along with the rest of classical electromagnetism) are extraordinarily successful at explaining and predicting a variety of phenomena. However they do not account for quantum effects and so their domain of applicability is limited. Maxwell's equations are thought of as the classical limit of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the relativity theory, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum m ...
(QED). Some observed electromagnetic phenomena are incompatible with Maxwell's equations. These include photon–photon scattering and many other phenomena related to photons or virtual particle, virtual photons, "nonclassical light" and quantum entanglement of electromagnetic fields (see quantum optics). E.g. quantum cryptography cannot be described by Maxwell theory, not even approximately. The approximate nature of Maxwell's equations becomes more and more apparent when going into the extremely strong field regime (see Euler–Heisenberg Lagrangian) or to extremely small distances. Finally, Maxwell's equations cannot explain any phenomenon involving individual photons interacting with quantum matter, such as the photoelectric effect, Planck's law, the Duane–Hunt law, and Single-photon avalanche diode, single-photon light detectors. However, many such phenomena may be approximated using a halfway theory of quantum matter coupled to a classical electromagnetic field, either as external field or with the expected value of the charge current and density on the right hand side of Maxwell's equations.

# Variations

Popular variations on the Maxwell equations as a classical theory of electromagnetic fields are relatively scarce because the standard equations have stood the test of time remarkably well.

## Magnetic monopoles

Maxwell's equations posit that there is
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like c ...
, but no magnetic charge (also called
magnetic monopole In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is th ...
s), in the universe. Indeed, magnetic charge has never been observed, despite extensive searches,See
magnetic monopole In particle physics Particle physics (also known as high energy physics) is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is th ...
for a discussion of monopole searches. Recently, scientists have discovered that some types of condensed matter, including spin ice and topological insulators, which display ''emergent'' behavior resembling magnetic monopoles. (Se
sciencemag.org
an

) Although these were described in the popular press as the long-awaited discovery of magnetic monopoles, they are only superficially related. A "true" magnetic monopole is something where , whereas in these condensed-matter systems, while only .
and may not exist. If they did exist, both Gauss's law for magnetism and Faraday's law would need to be modified, and the resulting four equations would be fully symmetric under the interchange of electric and magnetic fields.

# Historical publications

On Faraday's Lines of Force – 1855/56
Maxwell's first paper (Part 1 & 2) – Compiled by Blaze Labs Research (PDF) * [//upload.wikimedia.org/wikipedia/commons/b/b8/On_Physical_Lines_of_Force.pdf On Physical Lines of Force – 1861] Maxwell's 1861 paper describing magnetic lines of Force – Predecessor to 1873 Treatise *
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branches of science, area of interest. In classica ... , "A Dynamical Theory of the Electromagnetic Field", ''Philosophical Transactions of the Royal Society of London'' 155, 459–512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.) *
A Dynamical Theory Of The Electromagnetic Field – 1865
Maxwell's 1865 paper describing his 20 Equations, link from Google Books. * J. Clerk Maxwell (1873) A Treatise on Electricity and Magnetism
Maxwell, J.C., A Treatise on Electricity And Magnetism – Volume 1 – 1873
– Posner Memorial Collection – Carnegie Mellon University
Maxwell, J.C., A Treatise on Electricity And Magnetism – Volume 2 – 1873
nbsp;– Posner Memorial Collection – Carnegie Mellon University The developments before relativity: * Joseph Larmor (1897) "On a dynamical theory of the electric and luminiferous medium", ''Phil. Trans. Roy. Soc.'' 190, 205–300 (third and last in a series of papers with the same name). * Hendrik Lorentz (1899) "Simplified theory of electrical and optical phenomena in moving systems", ''Proc. Acad. Science Amsterdam'', I, 427–43. * Hendrik Lorentz (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", ''Proc. Acad. Science Amsterdam'', IV, 669–78. * Henri Poincaré (1900) "La théorie de Lorentz et le Principe de Réaction", ''Archives Néerlandaises'', V, 253–78. * Henri Poincaré (1902) ''La Science et l'Hypothèse'' * Henri Poincaré (1905
"Sur la dynamique de l'électron"
''Comptes Rendus de l'Académie des Sciences'', 140, 1504–8.

*

* *
maxwells-equations.com
— An intuitive tutorial of Maxwell's equations.

## Modern treatments

B. Crowell, Fullerton College

R. Fitzpatrick, University of Texas at Austin
''Electromagnetic waves from Maxwell's equations''
o
Project PHYSNET

Taught by Professor Walter Lewin.

## Other

*

{{DEFAULTSORT:Maxwell's Equations Maxwell's equations, Electromagnetism Equations of physics Partial differential equations James Clerk Maxwell Functions of space and time Scientific laws