Maxwell's equations

Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together
with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic
circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power
generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields
are generated by charges, currents, and changes of the fields.[note 1] The equations are named after the physicist and
mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included
the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The
modern form of the equations in their most common formulation is credited to Oliver Heaviside.[1]

Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at
a constant speed in vacuum, c (299 792 458 m/s).[2] Known as electromagnetic radiation, these waves occur at various
wavelengths to produce a spectrum of radiation from radio waves to gamma rays.

In differential form and SI units, Maxwell's microscopic equations can be written as

With the electric field, the magnetic field, the

electric charge density and the current density.
is the vacuum permittivity and the vacuum
permeability.
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. 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 and magnetic
scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics,
or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately)
makes the compatibility of Maxwell's equations with special relativity manifest. Maxwell's equations in curved
spacetime, commonly used in high-energy and gravitational physics, are compatible with general relativity.[note 2] In fact,
Albert Einstein 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 limit
of the more precise theory of quantum electrodynamics.

History of the equations

Conceptual descriptions

Gauss's law

Electric field from

positive to
negative charges

Gauss's law describes the relationship between an electric field and electric charges: an electric field points away from
positive charges and towards negative charges, and the net outflow of the electric field through a closed surface 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.
Gauss's law for magnetism

Gauss's law for magnetism: magnetic

field lines never begin nor end but
form loops or extend to infinity as
shown here with the magnetic field
due to a ring of current.

Gauss's law for magnetism states that electric charges have no magnetic analogues, called magnetic monopoles; no
north or south magnetic poles exist in isolation.[3] Instead, the magnetic field of a material is attributed to a dipole, 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 through a
Gaussian surface is zero, and the magnetic field is a solenoidal vector field.[note 3]

Faraday's law

In a geomagnetic storm, a surge in the flux of charged

particles temporarily alters Earth's magnetic field,
which induces electric fields in Earth's atmosphere,
thus causing surges in electrical power grids. (Not to
scale.)

The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds
to curl of an electric field.[3] 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 is the operating principle behind many electric generators: for example, a rotating bar
magnet creates a changing magnetic field and generates an electric field in a nearby wire.

Ampère's law with Maxwell's addition

Magnetic-core memory (1954) is an

application of Ampère's law. Each
core stores one bit of data.

The original law of Ampère states that magnetic fields relate to electric current. Maxwell's addition states that
magnetic fields also relate to changing electric fields, which Maxwell called displacement current. The integral form
states that electric and displacement currents are associated with a proportional magnetic field along any enclosing
curve.

Maxwell's addition to Ampère's law is important because the laws of Ampère and Gauss must otherwise be adjusted
for static fields.[4] As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field.[3][5]
A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space.

The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and
currents,[note 4] matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio
waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby
unifying the theories of electromagnetism and optics.

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 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 formalism below, the work of Oliver
Heaviside,[6][7] 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; both are 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.[8]

Key to the notation

Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise
indicated. The equations introduce the electric field, E, a vector field, and the magnetic field, B, a pseudovector field,
each generally having a time and location dependence. The sources are

the total electric charge density (total charge per

unit volume), ρ, and
the total electric current density (total current per
unit area), J.
The universal constants appearing in the equations (the first two ones explicitly only in the SI units formulation) are:

the permittivity of free space, ε0, and

the permeability of free space, μ0, and

the speed of light,

Differential equations
In the differential equations,

the nabla symbol, ∇, denotes the three-

dimensional gradient operator, del,
the ∇⋅ symbol (pronounced "del dot") denotes the
divergence operator,
the ∇× symbol (pronounced "del cross") denotes
the curl operator.

Integral equations
In the integral equations,

Ω is any volume with closed boundary surface ∂Ω,

and
Σ is any surface with closed boundary curve ∂Σ,
The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces
and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-
independent, we can bring the differentiation under the integral sign in Faraday's law:
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 over the boundary

surface ∂Ω, with the loop indicating the surface is

closed

is a volume integral over the volume Ω,

is a line integral around the boundary curve

∂Σ, with the loop indicating the curve is closed.

is a surface integral over the surface Σ,

The total electric charge Q enclosed in Ω is the

volume integral over Ω of the charge density ρ
(see the "macroscopic formulation" section
below):
 where dV is the volume element.
The net electric current I is the surface integral of
the electric current density J passing through a
fixed surface, Σ:

where dS denotes the differential vector element

of surface area S, normal to surface Σ. (Vector
area is sometimes denoted by A rather than S, but
this conflicts with the notation for magnetic vector
potential).

Formulation in SI units convention

Name Integral equations Differential equations

Gauss's law

Gauss's law for magnetism

Maxwell–Faraday equation
(Faraday's law of induction)

Ampère's circuital law (with

Maxwell's addition)
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 ε0 and μ0 into the units of calculation, by convention. With a corresponding change
in convention for the Lorentz force law this yields the same physics, i.e. trajectories of charged particles, or work 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: the
Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and
dimension.[9]: vii Such modified definitions are conventionally used with the Gaussian (CGS) units. Using these
definitions and conventions, colloquially "in Gaussian units",[10] the Maxwell equations become:[11]

Name Integral equations Differential equations

Gauss's law

Gauss's law for magnetism

Maxwell–Faraday equation
(Faraday's law of induction)

Ampère's circuital law (with

Maxwell's addition)

The equations simplify slightly when a system of quantities is chosen in the speed of light, c, is used for
nondimensionalization, so that, for example, seconds and lightseconds are interchangeable, and c = 1.

Further changes are possible by absorbing factors of 4π. This process, called rationalization, affects whether
Coulomb's law or Gauss's law includes such a factor (see Heaviside–Lorentz units, used mainly in particle physics).

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.
Flux and divergence

Volume Ω and its closed boundary

∂Ω, containing (respectively
enclosing) a source (+) and sink (−) of
a vector field F. Here, F could be the E
field with source electric charges, but
not the B field, which has no magnetic
charges as shown. The outward unit
normal is n.

According to the (purely mathematical) Gauss divergence theorem, the electric flux through the boundary surface ∂Ω
can be rewritten as

The integral version of Gauss's equation can thus be rewritten as

Since Ω is arbitrary (e.g. an arbitrary small ball with

arbitrary center), this is satisfied if and only if 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 Gauss's law for magnetism in integral form gives

which is satisfied for all Ω if and only if everywhere.

Circulation and curl

Surface Σ with closed boundary ∂Σ. F

could be the E or B fields. Again, n is
the unit normal. (The curl of a vector
field does not literally look like the
"circulations", this is a heuristic
depiction.)

By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an
integral of the "circulation of the fields" (i.e. their curls) over a surface it bounds, i.e.

Hence the modified Ampere law in integral form can

be rewritten as
Since Σ 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
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: the circulation of a fluid is the line
integral of the fluid's flow velocity field around a closed loop, and the vorticity 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:

i.e.,

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

This 3D diagram shows a plane

linearly polarized wave propagating
from left to right, defined by
E = E0 sin(−ωt + k ⋅ r) and
B = B0 sin(−ωt + k ⋅ r) The oscillating
fields are detected at the flashing
point. The horizontal wavelength is λ.
E0 ⋅ B0 = 0 = E0 ⋅ k = B0 ⋅ k

In a region with no charges (ρ = 0) and no currents (J = 0), such as in a vacuum, Maxwell's equations reduce to:

Taking the curl ( ∇×) of the curl equations, and using the curl of the curl identity we obtain
The quantity has the dimension of (time/length)2. Defining , the equations above have the form
of the standard wave equations

Already during Maxwell's lifetime, it was found that the known values for and give , then
already known to be the speed of light in free space. This led him to propose that light and radio waves were
propagating electromagnetic waves, since amply confirmed. In the old SI system of units, the values of
and are defined constants, (which means that by definition
) 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, εr, and relative permeability, μr, the phase velocity of light becomes

which is usually[note 5] less than c.

In addition, E and B are perpendicular to each other and to the direction of wave propagation, and are in 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. In turn, that electric field creates a changing magnetic field through Maxwell's addition to Ampère's law.
This perpetual cycle allows these waves, now known as electromagnetic radiation, to move through space at velocity c.
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.[12]: 5

"Maxwell's macroscopic equations", also known as Maxwell's equations in matter, are more similar to those that
Maxwell introduced himself.

Differential
Integral equations Differential equations
Name equations
(SI convention) (Gaussian convention)
(SI convention)

Gauss's law

Ampère's circuital law (with

Maxwell's addition)

Gauss's law for magnetism

Maxwell–Faraday equation
(Faraday's law of induction)

In the macroscopic equations, the influence of bound charge Qb and bound current Ib is incorporated into the
displacement field D and the magnetizing field H, while the equations depend only on the free charges Qf and free
currents If. This reflects a splitting of the total electric charge Q and current I (and their densities ρ and J) into free and
bound parts:
The cost of this splitting is that the additional fields D and H need to be determined through phenomenological
constituent equations relating these fields to the electric field E and the magnetic field B, 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;[note 6] and the macroscopic equations, dealing with
free charge and current, practical to use within materials.

Bound charge and current

Left: A schematic view of how an assembly of

microscopic dipoles produces opposite surface
charges as shown at top and bottom. Right: How an
assembly of microscopic current loops add
together to produce a macroscopically circulating
current loop. Inside the boundaries, the individual
contributions tend to cancel, but at the boundaries
no cancelation occurs.

When an electric field is applied to a dielectric material its molecules respond by forming microscopic electric dipoles
– their 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 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 P of the material, its dipole moment per unit volume. If P is uniform, a macroscopic separation of
charge is produced only at the surfaces where P enters and leaves the material. For non-uniform P, a charge is also
produced in the bulk.[13]

Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the
angular momentum of the components of the atoms, most notably their electrons. The 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 currents can be described using the
magnetization M.[14]
The very complicated and granular bound charges and bound currents, therefore, can be represented on the
macroscopic scale in terms of P and M, 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 definitions of the auxiliary fields are:

where P is the polarization field and M is the magnetization field, which are defined in terms of microscopic bound
charges and bound currents respectively. The macroscopic bound charge density ρb and bound current density Jb in
terms of polarization P and magnetization M are then defined as

If we define the total, bound, and free charge and current density by

and use the defining relations above to eliminate D,

and H, 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 displacement field
D and the electric field E, as well as the magnetizing field H and the magnetic field B. Equivalently, we have to specify
the dependence of the polarization P (hence the bound charge) and the magnetization M (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.[15]: 44–45

For materials without polarization and magnetization, the constitutive relations are (by definition)[9]: 2

where ε0 is the permittivity of free space and μ0 the

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[15]: 44–45

where ε is the permittivity and μ the permeability of

the material. For the displacement field D 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 , 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, ε and μ are constant
throughout the material, while for inhomogeneous
materials they depend on location within the
material (and perhaps time).[16]: 463
For isotropic materials, ε and μ are scalars, while
for anisotropic materials (e.g. due to crystal
structure) they are tensors.[15]: 421 [16]: 463
Materials are generally dispersive, so ε and μ
depend on the frequency of any incident EM
 waves.[15]: 625 [16]: 397
Even more generally, in the case of non-linear materials (see for example nonlinear optics), D and P are not necessarily
proportional to E, similarly H or M is not necessarily proportional to B. In general D and H depend on both E and B, 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 E and B 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

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 φ and the vector potential A. 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 main article for details of each formulation. SI units are used throughout.
Vector calculus
Formulation Homogeneous equations Inhomogeneous equations

Fields

3D Euclidean space + time

Potentials (any gauge)

3D Euclidean space + time

Potentials (Lorenz gauge)

3D Euclidean space + time

Tensor calculus
Homogeneous
Formulation Inhomogeneous equations
equations

Fields

space + time

spatial metric independent of

time

Potentials

space (with § topological

restrictions) + time

spatial metric independent of

time

Potentials (Lorenz gauge)

space (with topological

restrictions) + time

spatial metric independent of

time
Differential forms
Homogeneous
Formulation Inhomogeneous equations
equations

Fields

any space + time

Potentials (any gauge)

any space (with § topological restrictions) +

time

Potential (Lorenz Gauge)

any space (with topological restrictions) + time

spatial metric independent of time

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, the electric and magnetic fields are treated on equal footing and are recognized 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 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 below describes one formalism.

Tensor calculus
Formulation Homogeneous equations Inhomogeneous equations

Fields
Minkowski space

Potentials (any gauge)

Minkowski space

Potentials (Lorenz gauge)

Minkowski space

Fields
any spacetime

Potentials (any gauge)

any spacetime
(with §topological restrictions)

Potentials (Lorenz gauge)

any spacetime
(with topological restrictions)

Differential forms
Formulation Homogeneous equations Inhomogeneous equations

Fields
any spacetime

Potentials (any gauge)

any spacetime
(with topological restrictions)

Potentials (Lorenz gauge)

any spacetime
(with topological restrictions)

In the tensor calculus formulation, the

electromagnetic tensor Fαβ is an antisymmetric
covariant order 2 tensor; the four-potential, Aα, is a
covariant vector; the current, Jα, is a vector; the
square brackets, [ ], denote antisymmetrization of
indices; ∂α is the partial derivative with respect to
the coordinate, xα. In Minkowski space
coordinates are chosen with respect to an inertial
frame; (xα) = (ct, x, y, z), so that the metric tensor
used to raise and lower indices is
ηαβ = diag(1, −1, −1, −1). The d'Alembert operator
on Minkowski space is ◻ = ∂α∂α as in the vector
formulation. In general spacetimes, the coordinate
system xα is arbitrary, the covariant derivative ∇, α
the Ricci tensor, Rαβ and raising and lowering of
indices are defined by the Lorentzian metric, gαβ
and the d'Alembert operator is defined as
◻ = ∇α ∇ . The topological restriction is that the
α

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
1
space times, F = 2 Fαβdxα ∧ dx β is the
electromagnetic tensor considered as a 2-form,
A = Aαdxα is the potential 1-form,
is the current 3-form, d 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 conformally invariant, but the Lorenz gauge
condition breaks conformal invariance. The
operator is the
d'Alembert–Laplace–Beltrami operator on 1-forms
on an arbitrary 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 formulation and a matrix representation of Maxwell's equations.
Historically, a quaternionic formulation[17][18] 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 equation and the 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. Some general remarks follow.

As for any differential equation, boundary conditions[19][20][21] and initial conditions[22] are necessary for a 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.[23] 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 artificial absorbing boundary representing the rest of the
universe,[24][25] or periodic boundary conditions, or walls that isolate a small region from the outside world (as with a
waveguide or cavity resonator).[26]

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 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.[19][21][27][28][29] For more details, see Computational electromagnetics.
Overdetermination of Maxwell's
equations
Maxwell's equations seem overdetermined, in that they involve six unknowns (the three components of E and B) 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.[30][31] This
explanation was first introduced by Julius Adams Stratton in 1941.[32]

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.[33]

Both identities , which reduce eight equations to six independent ones, are the true
[34][35]
reason of overdetermination.

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 partial differential equations (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 and quantum

mechanics
Maxwell's equations are valid in both the classical and the Quantum realm. In the Heisenberg representation of
Quantum Mechanics, the equations of the E and B operators are precisely Maxwell's equations. Of course since the
fields are quantum operators, there are many aspects which differ from the classical fields. For example, the E field
acts like the momentum conjugate to the spatial components of the vector potential A. This of course leads to many
aspects of the quantum electromagnetic field with differ from them as classical fields but they still obey the same
evolution equations as the classical field does.
Of course once one examines the effects of the electromagnetic fields on charged matter, and those effects then
change the electromagnetic field are examined, the field equations become non-linear, and the quantum behaviour of
non-linear field can be very different from the classical behaviour of the non-linear fields. That however does not alter
the fact that if one remains in the linear regime, the fields obey 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, but no magnetic charge (also called magnetic monopoles), in
the universe. Indeed, magnetic charge has never been observed, despite extensive searches,[note 7] 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.[9]: 273–275

See also

Algebra of physical space Electronics

portal
Fresnel equations
Physics
Gravitoelectromagnetism portal

Interface conditions for

electromagnetic fields
 Moving magnet and conductor problem
Riemann–Silberstein vector
Spacetime algebra
Wheeler–Feynman absorber theory

Explanatory notes

1. Electric and magnetic fields, according to the theory of

relativity, are the components of a single
electromagnetic field.

2. In general relativity, however, they must enter, through

its stress–energy tensor, into Einstein field equations
that include the spacetime curvature.

3. 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" (https://
zenodo.org/record/4518772#.YCJU_WhKjIU) , IEEE
Magnetics Letters, vol. 8, art. 1306005, 2017.

4. The quantity we would now call 1/√ ε0μ0 , with units of

velocity, was directly measured before Maxwell's
equations, in an 1855 experiment by Wilhelm Eduard
Weber and Rudolf Kohlrausch. They charged a leyden
jar (a kind of capacitor), and measured the
electrostatic force associated with the potential; then,
they discharged it while measuring the magnetic force
from the current in the discharge wire. Their result was
 3.107 × 108 m/s, 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 (https://books.google.com/books?id=uwgNAtq
SHuQC&pg=PA115) .

5. There are cases (anomalous dispersion) where the

phase velocity can exceed c, but the "signal velocity"
will still be < c

6. 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.

7. See magnetic monopole 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. (See
sciencemag.org (http://www.sciencemag.org/cgi/cont
ent/abstract/1178868) and nature.com (http://www.n
ature.com/nature/journal/v461/n7266/full/nature0850
 0.html) .) 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 ∇ ⋅ B ≠ 0,
whereas in these condensed-matter systems, ∇ ⋅ B = 0
while only ∇ ⋅ H ≠ 0.

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ISBN 9783527617241

Further reading

Imaeda, K. (1995), "Biquaternionic Formulation of

Maxwell's Equations and their Solutions", in
Ablamowicz, Rafał; Lounesto, Pertti (eds.), Clifford
Algebras and Spinor Structures, Springer, pp. 265–
280, doi:10.1007/978-94-015-8422-7_16 (https://d
oi.org/10.1007%2F978-94-015-8422-7_16) ,
ISBN 978-90-481-4525-6
Historical publications

On Faraday's Lines of Force (https://web.archive.o

rg/web/20081217035457/http://blazelabs.com/O
n%20Faraday%27s%20Lines%20of%20Force.pd
f) – 1855/56. Maxwell's first paper (Part 1 & 2) –
Compiled by Blaze Labs Research (PDF).
On Physical Lines of Force (https://upload.wikime
dia.org/wikipedia/commons/b/b8/On_Physical_Li
nes_of_Force.pdf) – 1861. Maxwell's 1861 paper
describing magnetic lines of force – Predecessor
to 1873 Treatise.
James Clerk Maxwell, "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 (https://books.google.com/books?id=5
HE_cmxXt2MC&q=Proceedings+of+the+Royal
+Society+Of+London+Vol+XIII) – 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 (http://posner.library.
cmu.edu/Posner/books/book.cgi?call=537_M
46T_1873_VOL._1) – 1873 – Posner
Memorial Collection – Carnegie Mellon
University.
Maxwell, J. C., "A Treatise on Electricity And
Magnetism" – Volume 2 (http://posner.library.
cmu.edu/Posner/books/book.cgi?call=537_M
46T_1873_VOL._2) – 1873 – Posner
 Memorial Collection – Carnegie Mellon
University.
The developments before relativity:

Larmor Joseph (1897). "On a dynamical theory of

the electric and luminiferous medium. Part 3,
Relations with material media" (https://en.wikisou
rce.org/wiki/Dynamical_Theory_of_the_Electric_an
d_Luminiferous_Medium_III) . Phil. Trans. R. Soc.
190: 205–300.

Lorentz Hendrik (1899). "Simplified theory of

electrical and optical phenomena in moving
systems" (https://en.wikisource.org/wiki/Simplifie
d_Theory_of_Electrical_and_Optical_Phenomena_i
n_Moving_Systems) . Proc. Acad. Science
Amsterdam. I: 427–443.

Lorentz Hendrik (1904). "Electromagnetic

phenomena in a system moving with any velocity
less than that of light" (https://en.wikisource.org/
wiki/Electromagnetic_phenomena) . Proc. Acad.
Science Amsterdam. IV: 669–678.

Henri Poincaré (1900) "La théorie de Lorentz et le

Principe de Réaction" (in French), Archives
Néerlandaises, V, 253–278.

Henri Poincaré (1902) "La Science et l'Hypothèse"

(in French).
Henri Poincaré (1905) "Sur la dynamique de
l'électron" (http://www.soso.ch/wissen/hist/SRT/P
-1905-1.pdf) (in French), Comptes Rendus de
l'Académie des Sciences, 140, 1504–1508.

Catt, Walton and Davidson. "The History of

Displacement Current" (http://www.electromagneti
sm.demon.co.uk/z014.htm) Archived (https://we
b.archive.org/web/20080506120012/http://www.e
lectromagnetism.demon.co.uk/z014.htm) 2008-
05-06 at the Wayback Machine. Wireless World,
March 1979.
External links

"Maxwell equations" (https://ww Wikimedia

Commons
w.encyclopediaofmath.org/index.
has media
php?title=Maxwell_equations) , related to
Maxwell's
Encyclopedia of Mathematics,
equations.
EMS Press, 2001 [1994]
Wikiquote
maxwells-equations.com (http:// has
quotations
www.maxwells-equations.com)
related to
— An intuitive tutorial of Maxwell's
equations.
Maxwell's equations.
Wikiversity
The Feynman Lectures on discusses
Physics Vol. II Ch. 18: The basic
Maxwell
Maxwell Equations (https://feyn integrals
manlectures.caltech.edu/II_18.ht for
students.
ml)
Wikiversity Page on Maxwell's Equations
Modern treatments

Electromagnetism (ch. 11) (http://lightandmatter.c

om/area1sn.html) , B. Crowell, Fullerton College
Lecture series: Relativity and electromagnetism (h
ttps://web.archive.org/web/20030803151533/htt
p://farside.ph.utexas.edu/~rfitzp/teaching/jk1/lec
tures/node6.html) , R. Fitzpatrick, University of
Texas at Austin
Electromagnetic waves from Maxwell's equations (h
ttp://www.physnet.org/modules/pdf_modules/m2
10.pdf) on Project PHYSNET (http://www.physne
t.org) .
MIT Video Lecture Series (36 × 50 minute lectures)
(in .mp4 format) – Electricity and Magnetism (http
s://web.archive.org/web/20090324084439/http://
ocw.mit.edu/OcwWeb/Physics/8-02Electricity-and
 -MagnetismSpring2002/VideoAndCaptions/index.
htm) Taught by Professor Walter Lewin.

Other

Silagadze, Z. K. (2002). "Feynman's derivation of

Maxwell equations and extra dimensions". Annales
de la Fondation Louis de Broglie. 27: 241–256.
arXiv:hep-ph/0106235 (https://arxiv.org/abs/hep-p
h/0106235) . Bibcode:2001hep.ph....6235S (http
s://ui.adsabs.harvard.edu/abs/2001hep.ph....6235
S) .
Nature Milestones: Photons – Milestone 2 (1861)
Maxwell's equations (http://www.nature.com/miles
tones/milephotons/full/milephotons02.html)

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