For thermodynamic relations, see Maxwell relations.

Template:Electromagnetism In electromagnetism, Maxwell's equations are a set of four partial differential equations that describe the properties of the electric and magnetic fields and relate them to their sources, charge density and current density. These equations are used to show that light is an electromagnetic wave. Individually, the equations are known as Gauss's law, Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction.

These four equations, together with the Lorentz force law are the complete set of laws of classical electromagnetism. The Lorentz force law itself was actually derived by Maxwell under the name of "Equation for Electromotive Force" and was one of an earlier set of eight equations by Maxwell.

Conceptual descriptionEdit

This section will conceptually describe each of the four Maxwell's equations, and also how they link together to explain the origin of electromagnetic radiation such as light. The exact equations are set out in later sections of this entry.

  • Gauss's law relates electric charge contained within a closed surface (Gaussian surface) to the surrounding electric field. It describes with mathematical clarity how the divergence of an electrical field is affected by charges (electric field lines diverge from positive charges and are drawn towards negative charges). It also states that the total electric flux through a Gaussian surface is unrelated to the shape and size of that surface.
  • Gauss's law for magnetism states that the total magnetic flux through a Gaussian surface is zero. This is due to real world magnetic charges coming in pairs (referred to as dipoles), with the two charges giving rise to opposite magnetic field divergences which cancel each other out. The theoretical single magnetic charge is referred to as a magnetic monopole. Gauss's Law for Magnetism is also a mathematical form of the assertion that magnetic monopoles do not exist.
File:Magnetic core.jpg
  • Faraday's law of induction describes how a changing magnetic field can create an electric field. This is, for example, the operating principle behind many electric generators: Mechanical force, such as water falling on turbines in a hydroelectric dam, spins a huge magnet, and the changing magnetic field creates an electric field which drives electricity through the power grid.

Maxwell's correction to Ampère's law was particularly important. In 1864, Maxwell derived the electromagnetic wave equation by linking the displacement current to the time-varying electric field that is associated with electromagnetic induction. This is described in his A Dynamical Theory of the Electromagnetic Field, where he wrote:

"The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws."[1]

The extension to displacement current applies in the pure vacuum. This has been interpreted by some to mean that a changing electric field can produce a magnetic field, and vice-versa. Under this interpretation it follows that even with no electric charges or currents present, it is possible to have stable, self-perpetuating waves of oscillating electric and magnetic fields, with each field driving the other. The physical parameters of transverse elasticity and density, which Maxwell used to calculate the speed of these electromagnetic waves, have been replaced by two easily-measurable physical constants, the electric constant and the magnetic constant.

The speed calculated for electromagnetic radiation exactly matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell thus unified the hitherto separate fields of electromagnetism and optics.

General formulationEdit

The equations in this section are given in SI units. Unlike the equations of mechanics (for example), Maxwell's equations are not unchanged in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places. Other than SI (used in engineering), the units commonly used are Gaussian units (based on the cgs system and considered to have some theoretical advantages over SI[2]), Lorentz-Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). See below for CGS-Gaussian units.

Two equivalent, general formulations of Maxwell's equations follow. The first separates bound charge and bound current (which arise in the context of dielectric and/or magnetized materials) from free charge and free current (the more conventional type of charge and current). This separation is useful for calculations involving dielectric or magnetized materials. The second formulation treats all charge equally, combining free and bound charge into total charge (and likewise with current). This is the more fundamental or microscopic point of view, and is particularly useful when no dielectric or magnetic material is present. More details, and a proof that these two formulations are mathematically equivalent, are given in section 3.

Symbols in bold represent vector quantities, whereas symbols in italics represent scalar quantities. The definitions of terms used in the two tables of equations are given in another table immediately following.

Table 1: Formulation in terms of free charge and currentEdit
Name Differential form Integral form
Gauss's law: $ \nabla \cdot \mathbf{D} = \rho_f $ $ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A = Q_{f}(V) $
Gauss's law for magnetism: $ \nabla \cdot \mathbf{B} = 0 $ $ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0 $
Maxwell–Faraday equation
(Faraday's law of induction):
$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} $ $ \oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t} $
Ampère's circuital law
(with Maxwell's correction):
$ \nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t} $ $ \oint_{\partial S} \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,S} + \frac {\partial \Phi_{D,S}}{\partial t} $

Here V is a three-dimensional volume, S a two-dimensional oriented surface; Q(V) the electrical charge (free rsp. total) contained in V, and $ I_{S} $ the electrical current flowing through S. $ \,\Phi_{B,S} $ is the flux of the magnetic field flowing through S.

Table 2: Formulation in terms of total charge and currentEdit

For the total quantities one has an analogous table; H and D are no longer present:

Name Differential form Integral form
Gauss's law: $ \nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0} $ $ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf E\;\cdot\mathrm{d}\mathbf A = \frac{Q(V)}{\varepsilon_0} $
Gauss's law for magnetism: $ \nabla \cdot \mathbf{B} = 0 $ $ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0 $
Maxwell–Faraday equation
(Faraday's law of induction):
$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} $ $ \oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t} $
Ampère's circuital law
(with Maxwell's correction):
$ \nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}\ $ $ \oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_S + \mu_0 \varepsilon_0 \frac {\partial \Phi_{E,S}}{\partial t} $

The following table provides the meaning of each symbol and the SI unit of measure:

Table 3: Definitions and unitsEdit
Symbol Meaning (first term is the most common) SI Unit of Measure
$ \mathbf{E} \ $ electric field volt per meter or, equivalently,
newton per coulomb
$ \mathbf{B} \ $ magnetic field
also called the magnetic induction
also called the magnetic field density
also called the magnetic flux density
tesla, or equivalently,
weber per square meter
voltsecond per square meter
$ \mathbf{D} \ $ electric displacement field
also called the electric flux density
coulombs per square meter or, equivalently,
newton per volt-meter
$ \mathbf{H} \ $ magnetizing field
also called auxiliary magnetic field
also called magnetic field intensity
also called magnetic field
ampere per meter
$ \mathbf{\nabla \cdot} $ the divergence operator per meter (factor contributed by applying either operator)
$ \mathbf{\nabla \times} $ the curl operator
$ \frac {\partial}{\partial t} $ partial derivative with respect to time per second (factor contributed by applying the operator)
$ \mathrm{d}\mathbf{A} $ differential vector element of surface area A, with infinitesimally

small magnitude and direction normal to surface S

square meters
$ \mathrm{d} \mathbf{l} $ differential vector element of path length tangential to the path/curve meters
$ \varepsilon_0 \ $ permittivity of free space, officially the electric constant,
a universal constant
farads per meter
$ \mu_0 \ $ permeability of free space, officially the magnetic constant,
a universal constant
henries per meter, or Newtons per ampere squared
$ \ \rho_f \ $ free charge density (not including bound charge) coulombs per cubic meter
$ \ \rho \ $ total charge density (including both free and bound charge) coulombs per cubic meter
$ \mathbf{J}_f $ free current density (not including bound current) amperes per square meter
$ \mathbf{J} $ total current density (including both free and bound current) amperes per square meter
$ \,Q_f (V) $ net unbalanced free electric charge in the interior of an
arbitrary closed surface $ S=\partial V $ (not including bound charge)
$ \,Q(V) $ net unbalanced electric charge in the interior of an
arbitrary closed surface $ S=\partial V $ (including both free and bound charge)
$ \oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} $ line integral of the electric field along the boundary ∂S
(therefore necessarily a closed curve) of the surface S
joules per coulomb
$ \oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} $ line integral of the magnetic field over
the closed boundary ∂S of the surface S
$ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf E\;\cdot\mathrm{d}\mathbf A $ the flux of the electric field through any closed surface $ S=\partial V $ joule-meter per coulomb
$ \iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A $ the flux of the magnetic field through any closed surface $ S=\partial V $ tesla meters-squared or webers
$ \iint_S \mathbf{B} \cdot \mathrm{d} \mathbf{A} = \Phi_{B,S} $ magnetic flux through any surface S (not necessarily closed) webers or equivalently,
$ \iint_S \mathbf{E} \cdot \mathrm{d} \mathbf{A} = \Phi_{E,S} $ electric flux through any surface S, not necessarily closed joule-meters per coulomb
$ \iint_S \mathbf{D} \cdot \mathrm{d} \mathbf{A} = \Phi_{D,S} $ flux of electric displacement field through any surface S, not necessarily closed coulombs
$ \iint_S \mathbf{J}_f \cdot \mathrm{d} \mathbf{A} = I_{f,s} $ net free electrical current passing through
the surface S (not including bound current)
$ \iint_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} = I_{S} $ net electrical current passing through the
surface S (including both free and bound current)

Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo quantum mechanical effects as well). It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material. At microscopic level, Maxwell's equations, ignoring quantum effects, describe fields, charges and currents in free space — but at this level of detail one must include all charges, even those at an atomic level, generally an intractable problem.

History Edit

Although James Clerk Maxwell is said by some not to be the originator of these equations in modern notation, he nevertheless derived three of them again independently in conjunction with his molecular vortex model of Faraday's "lines of force", along with the full version of Faraday's law of induction. In doing so he made an important addition to Ampère's circuital law.

In addition, all four of what are now described as Maxwell's equations can be found in recognisable form (albeit without any trace of a vector notation, let alone ) in his 1861 paper On Physical Lines of Force, in his 1865 paper A Dynamical Theory of the Electromagnetic Field, and also in vol. 2 of Maxwell's "A Treatise on Electricity & Magnetism", published in 1873, in Chapter IX, entitled "General Equations of the Electromagnetic Field". This book by Maxwell pre-dates publications by Heaviside, Hertz and others.

Maxwell also developed Faraday's law of induction into another equation, which used to be listed as a 'Maxwell's equation' but is nowadays known as the Lorentz force law.[3]

The term Maxwell's equations Edit

The term Maxwell's equations originally applied to a set of eight equations published by Maxwell in 1865, but nowadays applies to modified versions of four of these equations that were grouped together in 1884 by Oliver Heaviside,[4] concurrently with similar work by Willard Gibbs and Heinrich Hertz.[5] These equations were also known variously as the Hertz–Heaviside equations and the Maxwell–Hertz equations,[4] and are sometimes still known as the Maxwell–Heaviside equations.[6]

Maxwell's contribution to Science in producing these equations lies in the correction he made to Ampère's circuital law in his 1861 paper On Physical Lines of Force. He added the displacement current term to Ampère's circuital law and this enabled him to derive the electromagnetic wave equation in his later 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrate the fact that light is an electromagnetic wave. This fact was then later confirmed experimentally by Heinrich Hertz in 1887.

The concept of fields was introduced by, among others, Faraday. Albert Einstein wrote, "The precise formulation of the time-space laws was the work of Maxwell. Imagine his feelings when the differential equations he had formulated proved to him that electromagnetic fields spread in the form of polarised waves, and at the speed of light! To few men in the world has such an experience been vouchsafed . . it took physicists some decades to grasp the full significance of Maxwell's discovery, so bold was the leap that his genius forced upon the conceptions of his fellow-workers" ('Science', May 24, 1940).

The equations were called by some the Hertz–Heaviside equations, but later Einstein referred to them as the Maxwell–Hertz equations.[4] However, in 1940 Einstein referred to the equations as "Maxwell's equations" in "The Fundamentals of Theoretical Physics" published in the Washington periodical Science, May 24, 1940.

Heaviside worked to eliminate the potentials (electrostatic potential and vector potential) that Maxwell had used as the central concepts in his equations;[4] this effort was somewhat controversial,[7] though it was understood by 1884 that the potentials must propagate at the speed of light like the fields, unlike the concept of instantaneous action-at-a-distance like the then conception of gravitational potential.[5] Modern analysis of, for example, radio antennas, makes full use of Maxwell's vector and scalar potentials to separate the variables, a common technique used in formulating the solutions of differential equations. However the potentials can be introduced by algebraic manipulation of the four fundamental equations.

The net result of Heaviside's work was the symmetrical "duplex" set of four equations,[4] all of which originated in Maxwell's previous publications, in particular Maxwell's 1861 paper On Physical Lines of Force, the 1865 paper A Dynamical Theory of the Electromagnetic Field and the Treatise. The fourth was a partial time derivative version of Faraday's law of induction that doesn't include motionally induced EMF; this version is often termed the Maxwell–Faraday equation or Faraday's law in differential form to keep clear the distinction from Faraday's law of induction, though it expresses the same law.[8][9]

Maxwell's On Physical Lines of Force (1861) Edit

Three of Maxwell's four equations appeared throughout Maxwell's 1861 paper On Physical Lines of Force, but all appear in Maxwell's 'Treatise':

(i) The equation (56) of Maxwell's 1861 paper is $ \nabla \cdot \mathbf{B} = 0 $.

(ii) The equation (112) is Ampère's circuital law with Maxwell's correction. It is this correction called displacement current which is the most significant aspect of Maxwell's work in electromagnetism as it enabled him to later derive the electromagnetic wave equation in his 1865 paper A Dynamical Theory of the Electromagnetic Field, and hence show that light is an electromagnetic wave. It is therefore this aspect of Maxwell's work which gives the equations their full significance. (Interestingly, Kirchhoff derived the telegrapher's equations in 1857 without using displacement current. But he did use Poisson's equation and the equation of continuity which are the mathematical ingredients of the displacement current. Nevertheless, Kirchhoff believed his equations to be applicable only inside an electric wire and so he is not credited with having discovered that light is an electromagnetic wave).

(iii) The equation (115) is Gauss's law.

(iv) Maxwell's fourth equation introduced a restricted partial time derivative version of Faraday's law of induction. (A full version of Faraday's law of induction had appeared at equation (54) of Maxwell's 1861 paper). It is important however to note that Heaviside's partial time derivative notation, as opposed to the total time derivative notation used by Maxwell at equations (54), resulted in the loss of the v × B term that appeared in Maxwell's equation (77). Nowadays, the v × B term appears in the force law F = q ( E + v × B ) which sits adjacent to Maxwell's equations and bears the name Lorentz force. The Lorentz Force corresponds in effect to Maxwell's equation (77), but it appeared in this paper when Lorentz was still a young boy.

The difference between the $ \mathbf{B} $ and the $ \mathbf{H} $ vectors can be traced back to Maxwell's 1855 paper entitled On Faraday's Lines of Force which was read to the Cambridge Philosophical Society. The paper presented a simplified model of Faraday's work, and how the two phenomena were related. He reduced all of the current knowledge into a linked set of differential equations.

It is later clarified in his concept of a sea of molecular vortices that appears in his 1861 paper On Physical Lines of Force - 1861. Within that context, $ \mathbf{H} $ represented pure vorticity (spin), whereas $ \mathbf{B} $ was a weighted vorticity that was weighted for the density of the vortex sea. Maxwell considered magnetic permeability µ to be a measure of the density of the vortex sea. Hence the relationship,

(1) Magnetic induction current causes a magnetic current density

$ \mathbf{B} = \mu \mathbf{H} $

was essentially a rotational analogy to the linear electric current relationship,

(2) Electric convection current

$ \mathbf{J} = \rho \mathbf{v} $

where $ \rho $ is electric charge density. $ \mathbf{B} $ was seen as a kind of magnetic current of vortices aligned in their axial planes, with $ \mathbf{H} $ being the circumferential velocity of the vortices. With µ representing vortex density, it follows that the product of µ with vorticity $ \mathbf{H} $ leads to the magnetic field denoted as $ \mathbf{B} $.

The electric current equation can be viewed as a convective current of electric charge that involves linear motion. By analogy, the magnetic equation is an inductive current involving spin. There is no linear motion in the inductive current along the direction of the $ \mathbf{B} $ vector. The magnetic inductive current represents lines of force. In particular, it represents lines of inverse square law force.

The extension of the above considerations confirms that where $ \mathbf{B} $ is to $ \mathbf{H} $, and where $ \mathbf{J} $ is to ρ, then it necessarily follows from Gauss's law and from the equation of continuity of charge that $ \mathbf{E} $ is to $ \mathbf{D} $. ie. $ \mathbf{B} $ parallels with $ \mathbf{E} $, whereas $ \mathbf{H} $ parallels with $ \mathbf{D} $.

Maxwell's A Dynamical Theory of the Electromagnetic Field (1864) Edit

Main article: A Dynamical Theory of the Electromagnetic Field

In 1864 Maxwell published A Dynamical Theory of the Electromagnetic Field in which he showed that light was an electromagnetic phenomenon. Confusion over the term "Maxwell's equations" is exacerbated because it is also sometimes used for a set of eight equations that appeared in Part III of Maxwell's 1864 paper A Dynamical Theory of the Electromagnetic Field, entitled "General Equations of the Electromagnetic Field",[10] a confusion compounded by the writing of six of those eight equations as three separate equations (one for each of the Cartesian axes), resulting in twenty equations in twenty unknowns. (As noted above, this terminology is not common: Modern references to the term "Maxwell's equations" refer to the Heaviside restatements.)

The eight original Maxwell's equations can be written in modern vector notation as follows:

(A) The law of total currents
$ \mathbf{J}_{tot} = \mathbf{J} + \frac{\partial\mathbf{D}}{\partial t} $
(B) The equation of magnetic force
$ \mu \mathbf{H} = \nabla \times \mathbf{A} $
(C) Ampère's circuital law
$ \nabla \times \mathbf{H} = \mathbf{J}_{tot} $
(D) Electromotive force created by convection, induction, and by static electricity. (This is in effect the Lorentz force)
$ \mathbf{E} = \mu \mathbf{v} \times \mathbf{H} - \frac{\partial\mathbf{A}}{\partial t}-\nabla \phi $
(E) The electric elasticity equation
$ \mathbf{E} = \frac{1}{\epsilon} \mathbf{D} $
(F) Ohm's law
$ \mathbf{E} = \frac{1}{\sigma} \mathbf{J} $
(G) Gauss's law
$ \nabla \cdot \mathbf{D} = \rho $
(H) Equation of continuity
$ \nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\partial t} $
$ \mathbf{H} $ is the magnetizing field, which Maxwell called the "magnetic intensity".
$ \mathbf{J} $ is the electric current density (with $ \mathbf{J}_{tot} $ being the total current including displacement current).[11]
$ \mathbf{D} $ is the displacement field (called the "electric displacement" by Maxwell).
$ \rho $ is the free charge density (called the "quantity of free electricity" by Maxwell).
$ \mathbf{A} $ is the magnetic vector potential (called the "angular impulse" by Maxwell).
$ \mathbf{E} $ is called the "electromotive force" by Maxwell. The term electromotive force is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern term electric field.
$ \Phi $ is the electric potential (which Maxwell also called "electric potential").
$ \sigma $ is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

It is interesting to note the $ \mu \mathbf{v} \times \mathbf{H} $ term that appears in equation D. Equation D is therefore effectively the Lorentz force, similarly to equation (77) of his 1861 paper (see above).

When Maxwell derives the electromagnetic wave equation in his 1865 paper, he uses equation D to cater for electromagnetic induction rather than Faraday's law of induction which is used in modern textbooks. (Faraday's law itself does not appear among his equations.) However, Maxwell drops the $ \mu \mathbf{v} \times \mathbf{H} $ term from equation D when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame.

A Treatise on Electricity and Magnetism (1873)Edit


In A Treatise on Electricity and Magnetism, an 1873 textbook on electromagnetism written by James Clerk Maxwell, the equations are compiled into two sets.

The first set is

$ \mathbf{E} = - \nabla \phi - \frac{\partial \mathbf{A}}{\partial t} $
$ \mathbf{B} = \nabla \times \mathbf{A}. $

The second set is

$ \nabla \cdot \mathbf{D} = \rho $
$ \nabla \times \mathbf{H} - \frac{\partial \mathbf{D}}{\partial t} = \mathbf{J}. $

Maxwell's equations and matterEdit

Bound charge and current Edit

Main article: Bound charge#Bound charge
File:Polarization and magnetization.svg

If an electric field is applied to a dielectric material, each of the molecules responds by forming a microscopic electric dipole -- its atomic nucleus will move a tiny distance in the direction of the field, while its electrons will move a tiny distance in the opposite direction. This is called polarization of the material. In an idealized situation like that shown in the figure, the distribution of charge that results from these tiny movements turns out to be identical (outside the material) to having a layer of positive charge on one side of the material, and a layer of negative charge on the other side – a macroscopic separation of charge – even though all of the charges involved are "bound" to individual molecules. The volume polarization P is a result of bound charge. (Mathematically, once physical approximation has established the electric dipole density P based upon the underlying behavior of atoms, the surface charge that is equivalent to the material with its internal polarization is provided by the divergence theorem applied to a region straddling the interface between the material and the surrounding vacuum.)[12][13]

Somewhat similarly, in all materials the constituent atoms exhibit magnetic moments that are intrinsically linked to the angular momentum of the atoms' components, 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 magnetic moment is traveling a large distance. The "bound currents" can be described using M. (Mathematically, once physical approximation has established the magnetic dipole density based upon the underlying behavior of atoms, the surface current that is equivalent to the material with its internal magnetization is provided by Stokes' theorem applied to a path straddling the interface between the material and the surrounding vacuum.)[14][15]

These ideas suggest that for some situations the microscopic details of the atomic and electronic behavior can be treated in a simplified fashion that ignores many details on a fine scale that may be unimportant to understanding matters on a grosser scale. That notion underlies the bound/free partition of behavior.

Proof that the two general formulations are equivalent Edit

In this section, a simple proof is outlined which shows that the two alternate general formulations of Maxwell's equations given in Section 1 are mathematically equivalent.

The relation between polarization, magnetization, bound charge, and bound current is as follows:

$ \rho_b = -\nabla\cdot\mathbf{P} $
$ \mathbf{J}_b = \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t} $
$ \mathbf{D} = \epsilon_0\mathbf{E} + \mathbf{P} $
$ \mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M}) $
$ \rho = \rho_b + \rho_f \ $
$ \mathbf{J} = \mathbf{J}_b + \mathbf{J}_f $

where P and M are polarization and magnetization, and ρb and Jb are bound charge and current, respectively. Plugging in these relations, it can be easily demonstrated that the two formulations of Maxwell's equations given in Section 1 are precisely equivalent.

Constitutive relations Edit

In order to apply Maxwell's equations (the formulation in terms of free/bound charge and current using D and H), it is necessary to specify the relations between D and E, and H and B.

Finding relations between these fields is another way to say that to solve Maxwell's equations by employing the free/bound partition of charges and currents, one needs the properties of the materials relating the response of bound currents and bound charges to the fields applied to these materials.[16] These relations may be empirical (based directly upon measurements), or theoretical (based upon statistical mechanics, transport theory or other tools of condensed matter physics). The detail employed may be macroscopic or microscopic, depending upon the level necessary to the problem under scrutiny. These material properties specifying the response of bound charge and current to the field are called constitutive relations, and correspond physically to how much polarization and magnetization a material acquires in the presence of electromagnetic fields.

Once the responses of bound currents and charges are related to the fields, Maxwell's equations can be fully formulated in terms of the E- and B-fields alone, with only the free charges and currents appearing explicitly in the equations.

Case without magnetic or dielectric materials Edit

In the absence of magnetic or dielectric materials, the relations are simple:

$ \mathbf{D} = \epsilon_0\mathbf{E}, \;\;\; \mathbf{H} = \mathbf{B}/\mu_0 $

where ε0 and μ0 are two universal constants, called the permittivity of free space and permeability of free space, respectively.

Case of linear materials Edit

In a "linear", isotropic, nondispersive, uniform material, the relations are also straightforward:

$ \mathbf{D} = \epsilon\mathbf{E}, \;\;\; \mathbf{H} = \mathbf{B}/\mu $

where ε and μ are constants (which depend on the material), called the permittivity and permeability, respectively, of the material.

General case Edit

For real-world materials, the constitutive relations are not simple proportionalities, except approximately. The relations can usually still be written:

$ \mathbf{D} = \epsilon\mathbf{E}, \;\;\; \mathbf{H} = \mathbf{B}/\mu $

but ε and μ are not, in general, simple constants, but rather functions. For example, ε and μ can depend upon:

If further there are dependencies on:

  • The position inside the material (the case of a nonuniform material, which occurs when the response of the material varies from point to point within the material, an effect called spatial inhomogeneity; for example in a domained structure, heterostructure or a liquid crystal, or most commonly in the situation where there are simply multiple materials occupying different regions of space),
  • The history of the fields — in a linear time-invariant material, this is equivalent to the material dispersion mentioned above (a frequency dependence of the ε and μ), which after Fourier transforming turns into a convolution with the fields at past times, expressing a non-instantaneous response of the material to an applied field; in a nonlinear or time-varying medium, the time-dependent response can be more complicated, such as the example of a hysteresis response,

then the constitutive relations take a more complicated form:[17][18]

$ \mathbf{D}(\mathbf{r}, t) = \epsilon_0 \mathbf{E}(\mathbf{r}, t) + \mathbf{P}(\mathbf{r}, t) $
$ \mathbf{H}(\mathbf{r}, t) = \frac{1}{\mu_0} \mathbf{B}(\mathbf{r}, t) - \mathbf{M}(\mathbf{r}, t) $
$ \mathbf{P}(\mathbf{r}, t) = \epsilon_0 \int d^3 \mathbf{r}' d t'\; \hat{\chi}_{\mathrm{elec}} (\mathbf{r}, \mathbf{r}', t, t'; \mathbf{E})\, \mathbf{E}(\mathbf{r}', t') $
$ \mathbf{M}(\mathbf{r}, t) = \frac{1}{\mu_0} \int d^3 \mathbf{r}' d t' \; \hat{\chi}_{\mathrm{magn}} (\mathbf{r}, \mathbf{r}', t, t'; \mathbf{B})\, \mathbf{B}(\mathbf{r}', t') $,

in which the permittivity and permeability functions are replaced by integrals over the more general electric and magnetic susceptibilities.

It may be noted that man-made materials can be designed to have customized permittivity and permeability, such as metamaterials and photonic crystals.

Maxwell's equations in terms of E and B for linear materialsEdit

Substituting in the constitutive relations above, Maxwell's equations in linear, dispersionless, time-invariant materials (differential form only) are:

$ \nabla \cdot (\epsilon \mathbf{E}) = \rho_f $
$ \nabla \cdot \mathbf{B} = 0 $
$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} $
$ \nabla \times (\mathbf{B} / \mu) = \mathbf{J}_f + \epsilon \frac{\partial \mathbf{E}} {\partial t}. $

These are formally identical to the general formulation in terms of E and B (given above), except that the permittivity of free space was replaced with the permittivity of the material (see also displacement field, electric susceptibility and polarization density), the permeability of free space was replaced with the permeability of the material (see also magnetization, magnetic susceptibility and magnetic field), and only free charges and currents are included (instead of all charges and currents). Unless that material is homogeneous in space, ε and μ cannot be factored out of the derivative expressions on the left-hand sides.

Calculation of constitutive relationsEdit

The fields in Maxwell's equations are generated by charges and currents. Conversely, the charges and currents are affected by the fields through the Lorentz force equation:

$ \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}), $

where q is the charge on the particle and v is the particle velocity. (It also should be remembered that the Lorentz force is not the only force exerted upon charged bodies, which also may be subject to gravitational, nuclear, etc. forces.) Therefore, in both classical and quantum physics, the precise dynamics of a system form a set of coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of statistical mechanics.[19] This remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents, which enter Maxwell's equations through the constitutive equations, as described next.

Commonly, real materials are approximated as "continuum" media with bulk properties such as the refractive index, permittivity, permeability, conductivity, and/or various susceptibilities. These lead to the macroscopic Maxwell's equations, which are written (as given above) in terms of free charge/current densities and D, H, E, and B ( rather than E and B alone ) along with the constitutive equations relating these fields. For example, although a real material consists of atoms whose electronic charge densities can be individually polarized by an applied field, for most purposes behavior at the atomic scale is not relevant and the material is approximated by an overall polarization density related to the applied field by an electric susceptibility.

Continuum approximations of atomic-scale inhomogeneities cannot be determined from Maxwell's equations alone, but require some type of quantum mechanical analysis such as quantum field theory as applied to condensed matter physics. See, for example, density functional theory, Green–Kubo relations and Green's function (many-body theory). Various approximate transport equations have evolved, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier-Stokes equations. Some examples where these equations are applied are magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, plasma modeling. An entire physical apparatus for dealing with these matters has developed. A different set of homogenization methods (evolving from a tradition in treating materials such as conglomerates and laminates) are based upon approximation of an inhomogeneous material by a homogeneous effective medium[20][21] (valid for excitations with wavelengths much larger than the scale of the inhomogeneity).[22][23][24][25]

Theoretical results have their place, but often require fitting to experiment. Continuum-approximation properties of many real materials rely upon measurement,[26] for example, ellipsometry measurements.

In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects. For example: optical nonlinearities can be neglected for low field strengths; material dispersion is unimportant where frequency is limited to a narrow bandwidth; material absorption can be neglected for wavelengths where a material is transparent; and metals with finite conductivity often are approximated at microwave or longer wavelengths as perfect metals with infinite conductivity (forming hard barriers with zero skin depth of field penetration).

And, of course, some situations demand that Maxwell's equations and the Lorentz force be combined with other forces that are not electromagnetic. An obvious example is gravity. A more subtle example, which applies where electrical forces are weakened due to charge balance in a solid or a molecule, is the Casimir force from quantum electrodynamics.[27]

The connection of Maxwell's equations to the rest of the physical world is via the fundamental charges and currents. These charges and currents are a response of their sources to electric and magnetic fields and to other forces. The determination of these responses involves the properties of physical materials.

In vacuum Edit

Further information: Electromagnetic wave equation and Sinusoidal plane-wave solutions of the electromagnetic wave equation

Starting with the equations appropriate in the case without dielectric or magnetic materials, and assuming that there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:[28]

$ \nabla \cdot \mathbf{E} = 0 $
$ \nabla \cdot \mathbf{B} = 0 $
$ \nabla \times \mathbf{E} = - \frac{\partial\mathbf{B}} {\partial t} $
$ \nabla \times \mathbf{B} = \ \ \mu_0\varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}. $

One set of solutions to these equations is in the form of traveling sinusoidal plane waves, with the electric and magnetic field directions orthogonal to one another and the direction of travel, and with the two fields in phase, traveling at the speed[29]

$ c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}. \ $

In fact, Maxwell's equations explains specifically how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. That electric field, in turn, creates a changing magnetic field through Maxwell's correction to Ampère's law. This perpetual cycle allows these waves, known as electromagnetic radiation, to move through space, always at velocity c.

Maxwell discovered that this quantity c equals the speed of light in vacuum (known from early experiments), and concluded (correctly) that light is a form of electromagnetic radiation.

With magnetic monopoles Edit

Maxwell's equations of electromagnetism relate the electric and magnetic fields to the motions of electric charges. The standard form of the equations provide for an electric charge, but posit no magnetic charge (in accordance with the fact that magnetic charge has never been seen and may not exist). Except for this, the equations are symmetric under interchange of electric and magnetic field. In fact, symmetric equations can be written when all charges are zero, and this is how the wave equation is derived (see immediately above).

Fully symmetric equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.[30] With the inclusion of a variable for these magnetic charges, say $ \rho_m \, $, there will also be "magnetic current" variable in the equations, $ \vec{J}_m \, $. The extended Maxwell's equations, simplified by nondimensionalization via Planck units, are as follows:

Name Without magnetic monopoles With magnetic monopoles (hypothetical)
Gauss's law: $ \vec{\nabla} \cdot \vec{E} = 4 \pi \rho_e $ $ \vec{\nabla} \cdot \vec{E} = 4 \pi \rho_e $
Gauss's law for magnetism: $ \vec{\nabla} \cdot \vec{B} = 0 $ $ \vec{\nabla} \cdot \vec{B} = 4 \pi \rho_m $
Maxwell–Faraday equation
(Faraday's law of induction):
$ -\vec{\nabla} \times \vec{E} = \frac{\partial \vec{B}} {\partial t} $ $ -\vec{\nabla} \times \vec{E} = \frac{\partial \vec{B}}{\partial t} + 4 \pi\vec{j}_m $
Ampère's law
(with Maxwell's extension):
   $ \vec{\nabla} \times \vec{B} = \frac{\partial \vec{E}} {\partial t} + 4 \pi \vec{j}_e $  $ \ $    $ \vec{\nabla} \times \vec{B} = \frac{\partial \vec{E}} {\partial t} + 4 \pi \vec{j}_e $
Note: the Bivector notation embodies the sign swap, and these four equations can be written as only one equation.

If magnetic charges do not exist, or if they exist but where they are not present in a region, then the new variables are zero, and the symmetric equations reduce to the conventional equations of electromagnetism such as $ \vec{\nabla}\cdot\vec{B} = 0 $. Classically, the question is "Why does the magnetic charge always seem to be zero?"

Boundary conditions: using Maxwell's equationsEdit

Although Maxwell's equations apply throughout space and time, practical problems are finite and solutions to Maxwell's equations inside the solution region are joined to the remainder of the universe through boundary conditions[31][32][33] and started in time using initial conditions.[34]

In particular, in a region without any free currents or free charges, the electromagnetic fields in the region originate elsewhere, and are introduced via boundary and/or initial conditions. An example of this type is a an electromagnetic scattering problem, where an electromagnetic wave originating outside the scattering region is scattered by a target, and the scattered electromagnetic wave is analyzed for the information it contains about the target by virtue of the interaction with the target during scattering.[35]

In some cases, like waveguides or cavity resonators, the solution region is largely isolated from the universe, for example, by metallic walls, and boundary conditions at the walls define the fields with influence of the outside world confined to the input/output ends of the structure.[36] In other cases, the universe at large sometimes is approximated by an artificial absorbing boundary,[37][38][39] or, for example for radiating antennas or communication satellites, these boundary conditions can take the form of asymptotic limits imposed upon the solution.[40] In addition, for example in an optical fiber or thin-film optics, the solution region often is broken up into subregions with their own simplified properties, and the solutions in each subregion must be joined to each other across the subregion interfaces using boundary conditions.[41][42][43] A particular example of this use of boundary conditions is the replacement of a material with a volume polarization by a charged surface layer, or of a material with a volume magnetization by a surface current, as described in the section Bound charge and current.

Following are some links of a general nature concerning boundary value problems: Examples of boundary value problems, Sturm-Liouville theory, Dirichlet boundary condition, Neumann boundary condition, mixed boundary condition, Cauchy boundary condition, Sommerfeld radiation condition. Needless to say, one must choose the boundary conditions appropriate to the problem being solved. See also Kempel[44] and the extraordinary book by Friedman.[45]

CGS units Edit

The above equations are given in the International System of Units, or SI for short. In a related unit system, called CGS (short for centimeter-gram-second), there are several ways of defining the unit of electric current in terms of the CGS base units (cm, g, s). In one of those variants, called Gaussian units, the equations take the following form:[46]

$ \nabla \cdot \mathbf{D} = 4\pi\rho_f $
$ \nabla \cdot \mathbf{B} = 0 $
$ \nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t} $
$ \nabla \times \mathbf{H} = \frac{1}{c} \frac{\partial \mathbf{D}} {\partial t} + \frac{4\pi}{c} \mathbf{J}_f $

where c is the speed of light in a vacuum. For the electromagnetic field in a vacuum, the equations become:

$ \nabla \cdot \mathbf{E} = 0 $
$ \nabla \cdot \mathbf{B} = 0 $
$ \nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t} $
$ \nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t}. $

In this system of units the relation between displacement field, electric field and polarization density is:

$ \mathbf{D} = \mathbf{E} + 4\pi\mathbf{P}. $

And likewise the relation between magnetic induction, magnetic field and total magnetization is:

$ \mathbf{B} = \mathbf{H} + 4\pi\mathbf{M}. $

In the linear approximation, the electric susceptibility and magnetic susceptibility can be defined so that:

$ \mathbf{P} = \chi_e \mathbf{E} $,     $ \mathbf{M} = \chi_m \mathbf{H}. $

(Note that although the susceptibilities are dimensionless numbers in both cgs and SI, they have different values in the two unit systems, by a factor of 4π.) The permittivity and permeability are:

$ \ \epsilon = 1+4\pi\chi_e $,     $ \ \mu = 1+4\pi\chi_m $

so that

$ \mathbf{D} = \epsilon \mathbf{E} $,     $ \mathbf{B} = \mu \mathbf{H}. $

In vacuum, one has the simple relations $ \ \epsilon=\mu=1 $, D=E, and B=H.

The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:

$ \mathbf{F} = q \left(\mathbf{E} + \frac{\mathbf{v}}{c} \times \mathbf{B}\right), $

where $ q \ $ is the charge on the particle and $ \mathbf{v} \ $ is the particle velocity. This is slightly different from the SI-unit expression above. For example, here the magnetic field $ \mathbf{B} \ $ has the same units as the electric field $ \mathbf{E} \ $.

Special relativity Edit

Main article: Classical electromagnetism and special relativity

Maxwell's equations have a close relation to special relativity: Not only were Maxwell's equations a crucial part of the historical development of special relativity, but also, special relativity has motivated a compact mathematical formulation of Maxwell's equations, in terms of covariant tensors.

Historical developments Edit

Main article: History of special relativity

Maxwell's electromagnetic wave equation only applied in what he believed to be the rest frame of the luminiferous medium because he didn't use the v×B term of his equation (D) when he derived it. Maxwell's idea of the luminiferous medium was that it comprised of aethereal vortices aligned solenoidally along their rotation axes.

The American scientist A.A. Michelson set out to determine the velocity of the earth through the luminiferous medium aether using a light wave interferometer that he had invented. When the Michelson-Morley experiment was conducted by Edward Morley and Albert Abraham Michelson in 1887, it produced a null result for the change of the velocity of light due to the Earth's motion through the hypothesized aether. This null result was in line with the theory that was proposed in 1845 by George Stokes which suggested that the aether was entrained with the Earth's orbital motion.

Hendrik Lorentz objected to Stokes' aether drag model and along with George FitzGerald and Joseph Larmor, he suggested another approach. Both Larmor (1897) and Lorentz (1899, 1904) derived the Lorentz transformation (so named by Henri Poincaré) as one under which Maxwell's equations were invariant. Poincaré (1900) analyzed the coordination of moving clocks by exchanging light signals. He also established mathematically the group property of the Lorentz transformation (Poincaré 1905).

This culminated in Albert Einstein's revolutionary theory of special relativity, which postulated the absence of any absolute rest frame, dismissed the aether as unnecessary (a bold idea, which did not come to Lorentz nor to Poincaré), and established the invariance of Maxwell's equations in all inertial frames of reference, in contrast to the famous Newtonian equations for classical mechanics. But the transformations between two different inertial frames had to correspond to Lorentz' equations and not - as formerly believed - to those of Galileo (called Galilean transformations).[47] Indeed, Maxwell's equations played a key role in Einstein's famous paper on special relativity; for example, in the opening paragraph of the paper, he motivated his theory by noting that a description of a conductor moving with respect to a magnet must generate a consistent set of fields irrespective of whether the force is calculated in the rest frame of the magnet or that of the conductor.[48]

General relativity has also had a close relationship with Maxwell's equations. For example, Kaluza and Klein showed in the 1920s that Maxwell's equations can be derived by extending general relativity into five dimensions. This strategy of using higher dimensions to unify different forces continues to be an active area of research in particle physics.

Covariant formulation of Maxwell's equationsEdit

Main article: Covariant formulation of classical electromagnetism

In special relativity, in order to more clearly express the fact that Maxwell's equations in vacuum take the same form in any inertial coordinate system, Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form. The purely spatial components of the following are in SI units.

One ingredient in this formulation is the electromagnetic tensor, a rank-2 covariant antisymmetric tensor combining the electric and magnetic fields:

$ F_{\alpha \beta} = \left( \begin{matrix} 0 & \frac{-E_x}{c} & \frac{-E_y}{c} & \frac{-E_z}{c} \\ \frac{E_x}{c} & 0 & B_z & -B_y \\ \frac{E_y}{c} & -B_z & 0 & B_x \\ \frac{E_z}{c} & B_y & -B_x & 0 \end{matrix} \right) $

and the result of raising its indices

$ F^{\mu \nu} \, \stackrel{\mathrm{def}}{=} \, \eta^{\mu \alpha} \, F_{\alpha \beta} \, \eta^{\beta \nu} = \left( \begin{matrix} 0 & \frac{E_x}{c} & \frac{E_y}{c} & \frac{E_z}{c} \\ \frac{-E_x}{c} & 0 & B_z & -B_y \\ \frac{-E_y}{c} & -B_z & 0 & B_x \\ \frac{-E_z}{c} & B_y & -B_x & 0 \end{matrix} \right). $

The other ingredient is the four-current: $ J^{\alpha} = (c\rho,\vec{J}) $ where $ \rho $ is the charge density and J is the current density.

With these ingredients, Maxwell's equations can be written:

$ \mu_{0} \, J^{\beta} \, = \, {\partial F^{\beta\alpha} \over {\partial x^{\alpha}} } \, \stackrel{\mathrm{def}}{=} \, \partial_{\alpha} F^{\beta\alpha} \, \stackrel{\mathrm{def}}{=} \, {F^{\beta\alpha}}_{,\alpha} \, $


$ 0 = \partial_{\gamma} F_{\alpha\beta} + \partial_{\beta} F_{\gamma\alpha} + \partial_{\alpha} F_{\beta\gamma} \ \stackrel{\mathrm{def}}{=}\ {F_{\alpha\beta}}_{,\gamma} + {F_{\gamma\alpha}}_{,\beta} +{F_{\beta\gamma}}_{,\alpha}. $

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss's law and Ampere's law with Maxwell's correction. The second equation is an expression of the two homogeneous equations, Faraday's law of induction and Gauss's law for magnetism. The second equation is equivalent to

$ 0 = \epsilon^{\delta\alpha\beta\gamma} {F_{\beta\gamma}}_{,\alpha} $

where $ \, \epsilon^{\alpha\beta\gamma\delta} $ is the contravariant version of the Levi-Civita symbol, and

$ { \partial \over { \partial x^{\alpha} } } \ \stackrel{\mathrm{def}}{=}\ \partial_{\alpha} \ \stackrel{\mathrm{def}}{=}\ {}_{,\alpha} \ \stackrel{\mathrm{def}}{=}\ \left(\frac{1}{c}\frac{\partial}{\partial t}, \nabla\right) $

is the 4-gradient. In the tensor equations above, repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations. Upper and lower components of a vector, $ v^\alpha $ and $ v_\alpha $ respectively, are interchanged with the fundamental tensor g, e.g., g=η=diag(-1,+1,+1,+1).

Alternative covariant presentations of Maxwell's equations also exist, for example in terms of the four-potential; see Covariant formulation of classical electromagnetism for details.

Potentials Edit

Main article: Mathematical descriptions of the electromagnetic field

Maxwell's equations can be written in an alternative form, involving the electric potential (also called scalar potential) and magnetic potential (also called vector potential), as follows.[18] (The following equations are valid in the absence of dielectric and magnetic materials; or if such materials are present, they are valid as long as bound charge and bound current are included in the total charge and current densities.)

First, Gauss's law for magnetism states:

$ \nabla\cdot\mathbf{B} = 0. $

By Helmholtz's theorem, B can be written in terms of a vector field A, called the magnetic potential:

$ \mathbf{B} = \nabla \times \mathbf{A}. $

Second, plugging this into Faraday's law, we get:

$ \nabla\times \left( \mathbf{E} + \frac{\partial \mathbf{A}}{\partial t} \right) = 0. $

By Helmholtz's theorem, the quantity in parentheses can be written in terms of a scalar function $ \Phi $, called the electric potential:

$ \mathbf{E} + \frac{\partial \mathbf{A}}{\partial t} = -\nabla \Phi. $

Combining these with the remaining two Maxwell's equations yields the four relations:

$ \mathbf E = - \mathbf \nabla \Phi - \frac{\partial \mathbf A}{\partial t} $
$ \mathbf B = \mathbf \nabla \times \mathbf A $
$ \nabla^2 \Phi + \frac{\partial}{\partial t} \left ( \mathbf \nabla \cdot \mathbf A \right ) = - \frac{\rho}{\varepsilon_0} $
$ \left ( \nabla^2 \mathbf A - \frac{1}{c^2} \frac{\partial^2 \mathbf A}{\partial t^2} \right ) - \mathbf \nabla \left ( \mathbf \nabla \cdot \mathbf A + \frac{1}{c^2} \frac{\partial \Phi}{\partial t} \right ) = - \mu_0 \mathbf J. $

These equations, taken together, are as powerful and complete as Maxwell's equations. Moreover, if we work only with the potentials and ignore the fields, the problem has been reduced somewhat, as the electric and magnetic fields each have three components which need to be solved for (six components altogether), while the electric and magnetic potentials have only four components altogether.

Many different choices of A and $ \Phi $ are consistent with a given E and B, making these choices physically equivalent – a flexibility known as gauge freedom. Suitable choice of A and $ \Phi $ can simplify these equations, or can adapt them to suit a particular situation. For more information, see the article gauge freedom.


The two equations that represent the potentials can be reduced to one manifestly Lorentz invariant equation, using four-vectors: the four-current defined by

$ \mathbf{J} = \left( \rho c, \mathbf{j} \right) $

formed from the current density j and charge density ρ, and the electromagnetic four-potential defined by

$ \mathbf{\Psi} = \left( \Phi , \mathbf{A} c \right) $

formed from the vector potential A and the scalar potential $ \Phi \, $. The resulting single equation, due to Arnold Sommerfeld, a generalization of an equation due to Bernhard Riemann and known as the Riemann–Sommerfeld equation[49] or the covariant form of the Maxwell–Lorentz equations,[50] is:

$ \mathbf{\Box}^2 \mathbf{\Psi} = -\mu_0 \mathbf{J} $

using the four-gradient operator, which is defined as

$ \mathbf{\Box} = \left( -\frac{1}{c} \frac{\partial}{\partial t}, \mathbf \nabla \right). $

Differential forms Edit

In free space, where ε = ε0 and μ = μ0 are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. In what follows, cgs units, not SI units are used, however. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. Maxwell's equations then reduce to the Bianchi identity

$ \mathrm{d}\bold{F}=0 $

where d denotes the exterior derivative — a natural coordinate and metric independent differential operator acting on forms — and the source equation

$ \mathrm {d} * {\bold{F}}=\bold{J} $

where the (dual) Hodge star operator * is a linear transformation from the space of 2-forms to the space of (4-2)-forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric), and the fields are in natural units where $ 1/4\pi\epsilon_0=1 $. Here, the 3-form J is called the "electric current form" or "current 3-form" satisfying the continuity equation

$ \mathrm{d}{\bold{J}}=0. $

The current 3-form can be integrated over a 3-dimensional space-time region. The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any manifold, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. In particular the differential form version of the Maxwell equations are a convenient and intuitive formulation of the Maxwell equations in general relativity.

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call

$ C:\Lambda^2\ni\bold{F}\mapsto \bold{G}\in\Lambda^{(4-2)} $

the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become:

$ \mathrm{d}\bold{F} = 0 $
$ \mathrm{d}\bold{G} = \bold{J} $

where the current 3-form J still satisfies the continuity equation dJ= 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms $ \bold{\theta}^p $,

$ \bold{F} = \frac{1}{2}F_{pq}\bold{\theta}^p\wedge\bold{\theta}^q $.

the constitutive relation takes the form

$ G_{pq} = C_{pq}^{mn}F_{mn} $

where the field coefficient functions are antisymmetric in the indices and the constitutive coefficients are antisymmetric in the corresponding pairs. In particular, the Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking

$ C_{pq}^{mn} = \frac{1}{2}g^{ma}g^{nb} \epsilon_{abpq} \sqrt{-g} $

which up to scaling is the only invariant tensor of this type that can be defined with the metric.

In this formulation, electromagnetism generalises immediately to any 4-dimensional oriented manifold or with small adaptations any manifold, requiring not even a metric. Thus the expression of Maxwell's equations in terms of differential forms leads to a further notational and conceptual simplification. Whereas Maxwell's Equations could be written as two tensor equations instead of eight scalar equations, from which the propagation of electromagnetic disturbances and the continuity equation could be derived with a little effort, using differential forms leads to an even simpler derivation of these results.

Conceptual insight from this formulationEdit

On the conceptual side, from the point of view of physics, this shows that the second and third Maxwell equations should be grouped together, be called the homogeneous ones, and be seen as geometric identities expressing nothing else than: the field F derives from a more "fundamental" potential A. While the first and last one should be seen as the dynamical equations of motion, obtained via the Lagrangian principle of least action, from the "interaction term" A J (introduced through gauge covariant derivatives), coupling the field to matter.

Often, the time derivative in the third law motivates calling this equation "dynamical", which is somewhat misleading; in the sense of the preceding analysis, this is rather an artifact of breaking relativistic covariance by choosing a preferred time direction. To have physical degrees of freedom propagated by these field equations, one must include a kinetic term F *F for A; and take into account the non-physical degrees of freedom which can be removed by gauge transformation AA' = A-dα: see also gauge fixing and Fadeev-Popov ghosts.

Classical electrodynamics as the curvature of a line bundle Edit

An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or principal bundles with fibre U(1). The connection $ \nabla $ on the line bundle has a curvature $ \bold{F} = \nabla^2 $ which is a two-form that automatically satisfies $ \mathrm{d}\bold{F} = 0 $ and can be interpreted as a field-strength. If the line bundle is trivial with flat reference connection d we can write $ \nabla = \mathrm{d}+\bold{A} $ and F = dA with A the 1-form composed of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov-Bohm effect. In this experiment, a static magnetic field runs through a long magnetic wire (e.g. an Fe wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment. This means by definition that the connection $ \nabla $ is flat there.

However, as mentioned, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. This can be detected quantum-mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern. (See Michael Murray, Line Bundles, 2002 (PDF web link) for a simple mathematical review of this formulation. See also R. Bott, On some recent interactions between mathematics and physics, Canadian Mathematical Bulletin, 28 (1985) no. 2 pp 129–164.)

Curved spacetimeEdit

Main article: Maxwell's equations in curved spacetime

Traditional formulationEdit

Matter and energy generate curvature of spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum will also generate curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs units):

$ { 4 \pi \over c }j^{\beta} = \partial_{\alpha} F^{\alpha\beta} + {\Gamma^{\alpha}}_{\mu\alpha} F^{\mu\beta} + {\Gamma^{\beta}}_{\mu\alpha} F^{\alpha \mu} \ \stackrel{\mathrm{def}}{=}\ D_{\alpha} F^{\alpha\beta} \ \stackrel{\mathrm{def}}{=}\ {F^{\alpha\beta}}_{;\alpha} \,\! $,


$ 0 = \partial_{\gamma} F_{\alpha\beta} + \partial_{\beta} F_{\gamma\alpha} + \partial_{\alpha} F_{\beta\gamma} = D_{\gamma} F_{\alpha\beta} + D_{\beta} F_{\gamma\alpha} + D_{\alpha} F_{\beta\gamma}. $


$ {\Gamma^{\alpha}}_{\mu\beta} \! $

is a Christoffel symbol that characterizes the curvature of spacetime and $ D_{\gamma} $ is the covariant derivative.

Formulation in terms of differential formsEdit

The formulation of the Maxwell equations in terms of differential forms can be used without change in general relativity. The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates $ x^{\alpha} $ which gives a basis of 1-forms $ dx^{\alpha} $ in every point of the open set where the coordinates are defined. Using this basis and cgs units we define

  • The antisymmetric infinitesimal field tensor $ F_{\alpha\beta} $, corresponding to the field 2-form F
$ \bold{F} := \frac{1}{2}F_{\alpha\beta} \,\mathrm{d}\,x^{\alpha} \wedge \mathrm{d}\,x^{\beta}. $
  • The current-vector infinitesimal 3-form J
$ \bold{J} := {4 \pi \over c } j^{\alpha} \sqrt{-g} \, \epsilon_{\alpha\beta\gamma\delta} \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} \wedge \mathrm{d}\,x^{\delta}. $

Here g is as usual the determinant of the metric tensor $ g_{\alpha\beta} $. A small computation that uses the symmetry of the Christoffel symbols (i.e. the torsion-freeness of the Levi Civita connection) and the covariant constantness of the Hodge star operator then shows that in this coordinate neighborhood we have:

  • the Bianchi identity
$ \mathrm{d}\bold{F} = 2(\partial_{\gamma} F_{\alpha\beta} + \partial_{\beta} F_{\gamma\alpha} + \partial_{\alpha} F_{\beta\gamma})\mathrm{d}\,x^{\alpha}\wedge \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} = 0 $
  • the source equation
$ \mathrm{d} * \bold{F} = {F^{\alpha\beta}}_{;\alpha}\sqrt{-g} \, \epsilon_{\beta\gamma\delta\eta}\mathrm{d}\,x^{\gamma} \wedge \mathrm{d}\,x^{\delta} \wedge \mathrm{d}\,x^{\eta} = \bold{J} $
  • the continuity equation
$ \mathrm{d}\bold{J} = { 4 \pi \over c } {j^{\alpha}}_{;\alpha} \sqrt{-g} \, \epsilon_{\alpha\beta\gamma\delta}\mathrm{d}\,x^{\alpha}\wedge \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} \wedge \mathrm{d}\,x^{\delta} = 0. $

See also Edit


  1. See Maxwell 1864, p. 499.
  2. David J Griffiths (1999). Introduction to electrodynamics (Third Edition ed.), Prentice Hall. p. 559-562. ISBN 013805326X, 
  3. Ironically it is an equation which Maxwell himself was absolutely responsible for even though it doesn't count as a "Maxwell's equation". This extra equation appeared in an original list of eight Maxwell's equations in his 1865 paper entitled A Dynamical Theory of the Electromagnetic Field. Maxwell derived it from Faraday's law when Lorentz was still a young boy and he termed it the equation of electromotive force.
  4. 4.0 4.1 4.2 4.3 4.4 but are now universally known as "Maxwell's equations". Paul J. Nahin (2002). Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age, JHU Press. p. 108–112. ISBN 9780801869099,,M1. 
  5. 5.0 5.1 Jed Z. Buchwald (1994). The creation of scientific effects: Heinrich Hertz and electric waves, University of Chicago Press. p. 194. ISBN 9780226078885, 
  6. Myron Evans (2001). Modern nonlinear optics, John Wiley and Sons. p. 240. ISBN 9780471389316, 
  7. Oliver J. Lodge (November 1888). "Sketch of the Electrical Papers in Section A, at the Recent Bath Meeting of the British Association". Electrical Engineer 7: 535. 
  8. J. R. Lalanne, F. Carmona, and L. Servant (1999). Optical spectroscopies of electronic absorption, World Scientific. p. 8. ISBN 9789810238612, 
  9. Roger F. Harrington (2003). Introduction to Electromagnetic Engineering, Courier Dover Publications. p. 49–56. ISBN 9780486432410, 
  10. page 480.
  11. Here it is noted that a quite different quantity, the "magnetic polarization", $ \mu_0\mathbf M\,, $ by decision of an international IUPAP commission has been given the same name $ \mathbf J $. So for the electric current density, a name with small letters, $ \mathbf j\,, $ would be better. But even then the mathematitians would still use the large-letter-name $ \mathbf J $ for the corresponding current-twoform (see below).
  12. MS Longair (2003). Theoretical Concepts in Physics (2 ed.), Cambridge University Press. p. 127. ISBN 052152878X, 
  13. Kenneth Franklin Riley, Michael Paul Hobson, Stephen John Bence (2006). Mathematical methods for physics and engineering (3 ed.), Cambridge University Press. p. 404. ISBN 0521861535, 
  14. MS Longair (2003). Theoretical Concepts in Physics (2 ed.), Cambridge University Press. pp. 119 and 127. ISBN 052152878X, 
  15. Kenneth Franklin Riley, Michael Paul Hobson, Stephen John Bence (2006). Mathematical methods for physics and engineering (3 ed.), Cambridge University Press. p. 406. ISBN 0521861535, 
  16. The free charges and currents respond to the fields through the Lorentz force law and this response is calculated at a fundamental level using mechanics. The response of bound charges and currents is dealt with using grosser methods subsumed under the notions of magnetization and polarization. Depending upon the problem, one may choose to have no free charges whatsoever.
  17. Halevi, Peter (1992). Spatial dispersion in solids and plasmas. Amsterdam: North-Holland. ISBN 978-0444874054. 
  18. 18.0 18.1 Jackson, John David (1999). Classical Electrodynamics (3rd ed. ed.). New York: Wiley. ISBN 0-471-30932-X. 
  19. These complications show there is merit in separating the Lorentz force from the main four Maxwell equations. The four Maxwell's equations express the fields' dependence upon current and charge, setting apart the calculation of these currents and charges. As noted in this subsection, these calculations may well involve the Lorentz force only implicitly. Separating these complicated considerations from the Maxwell's equations provides a useful framework.
  20. Aspnes, D.E., "Local-field effects and effective-medium theory: A microscopic perspective," Am. J. Phys. 50, p. 704-709 (1982).
  21. Habib Ammari & Hyeonbae Kang (2006). Inverse problems, multi-scale analysis and effective medium theory : workshop in Seoul, Inverse problems, multi-scale analysis, and homogenization, June 22-24, 2005, Seoul National University, Seoul, Korea. Providence RI: American Mathematical Society. ISBN 0821839683,,M1. 
  22. O. C. Zienkiewicz, Robert Leroy Taylor, J. Z. Zhu, Perumal Nithiarasu (2006). The Finite Element Method (Sixth Edition ed.). Oxford UK: Butterworth-Heinemann. p. 550 ff. ISBN 0750663219,,M1. 
  23. See, e.g.: N. Bakhvalov and G. Panasenko, Homogenization: Averaging Processes in Periodic Media (Kluwer: Dordrecht, 1989); V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer: Berlin, 1994).
  24. Vitaliy Lomakin, Steinberg BZ, Heyman E, & Felsen LB (2003). "Multiresolution Homogenization of Field and Network Formulations for Multiscale Laminate Dielectric Slabs". IEEE Transactions on Antennas and Propagation 51 (10): 2761 ff. doi:10.1109/TAP.2003.816356, 
  25. AC Gilbert (Ronald R Coifman, Editor) (2000). Topics in Analysis and Its Applications: Selected Theses. Singapore: World Scientific Publishing Company. p. 155. ISBN 9810240945,,M1. 
  26. Edward D. Palik & Ghosh G (1998). Handbook of Optical Constants of Solids. London UK: Academic Press. ISBN 0125444222, 
  27. F Capasso, JN Munday, D. Iannuzzi & HB Chen Casimir forces and quantum electrodynamical torques: physics and nanomechanics
  28. The term "vacuum" often is used in this connection. However, it should be noted that free space is meant, that is, an idealized, physically unobtainable reference state that is not the same as any physically realizable vacuum, such as terrestrial ultra-high vacuum or outer space, nor any vacuum realizable in principle, such as the quantum vacuum, or the QCD vacuum.
  29. The ISO recommends using c0 as the international standard symbol for the speed of light; see NIST Special Publication 330, Appendix 2, p. 45 . However, this article will follow the practice common among physicists and engineers and use c instead.
  30. "IEEEGHN: Maxwell's Equations". Retrieved on 2008-10-19.
  31. Peter Monk (2003). Finite Element Methods for Maxwell's Equations. Oxford UK: Oxford University Press. p. 1 ff. ISBN 0198508883, 
  32. Thomas B. A. Senior & John Leonidas Volakis (1995). Approximate Boundary Conditions in Electromagnetics. London UK: Institution of Electrical Engineers. p. 261 ff. ISBN 0852968493,,M1. 
  33. T Hagstrom (Björn Engquist & Gregory A. Kriegsmann, Eds.) (1997). Computational Wave Propagation. Berlin: Springer. p. 1 ff. ISBN 0387948740, 
  34. Henning F. Harmuth & Malek G. M. Hussain (1994). Propagation of Electromagnetic Signals. Singapore: World Scientific. p. 17. ISBN 9810216890,,M1. 
  35. See for example: Fioralba Cakoni (2006). "The inverse scattering problem for an imperfect conductor". Qualitative methods in inverse scattering theory, Springer Science & Business. p. 61. ISBN 3540288449,,M1. , Khosrow Chadan et al. (1997). An introduction to inverse scattering and inverse spectral problems, Society for Industrial and Applied Mathematics. p. 45. ISBN 0898713870, .
  36. S. F. Mahmoud (1991). Electromagnetic Waveguides: Theory and Applications applications. London UK: Institution of Electrical Engineers. Chapter 2. ISBN 0863412327,,M1. 
  37. Jean-Michel Lourtioz (2005). Photonic Crystals: Towards Nanoscale Photonic Devices. Berlin: Springer. p. 84. ISBN 354024431X, 
  38. S. G. Johnson, Notes on Perfectly Matched Layers, online MIT course notes (Aug. 2007).
  39. Taflove A & Hagness S C (2005). Computational Electrodynamics: The Finite-difference Time-domain Method. Boston MA: Artech House. Chapters 6 & 7. ISBN 1580538320, 
  40. David M Cook (2003). The Theory of the Electromagnetic Field. Mineola NY: Courier Dover Publications. p. 335 ff. ISBN 0486425673, 
  41. Korada Umashankar (1989). Introduction to Engineering Electromagnetic Fields. Singapore: World Scientific. p. §10.7; pp. 359ff. ISBN 9971509210, 
  42. Joseph V. Stewart (2001). Intermediate Electromagnetic Theory. Singapore: World Scientific. Chapter III, pp. 111 ff Chapter V, Chapter VI. ISBN 9810244703,,M1. 
  43. Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett. p. 333ff and Chapter 3: pp. 89ff. ISBN 0-7637-3827-1,,M1. 
  44. John Leonidas Volakis, Arindam Chatterjee & Leo C. Kempel (1989). Finite element method for electromagnetics : antennas, microwave circuits, and scattering applications. New York: Wiley IEEE. p. 79 ff. ISBN 0780334256,,M1. 
  45. Bernard Friedman (1990). Principles and Techniques of Applied Mathematics. Mineola NY: Dover Publications. ISBN 0486664449, 
  46. Littlejohn, Robert (Fall 2007). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory" (pdf). Physics 221A, University of California, Berkeley lecture notes. Retrieved on 2008-05-06.
  47. Details can be found in U. Krey, A. Owen, Basic Theoretical Physics - A Concise Overview, Springer, Berlin and elsewhere, 2007, ISBN 978-3-540-36804-5
  48. "On the Electrodynamics of Moving Bodies". Retrieved on 2008-10-19.
  49. Carver A. Mead (2002). Collective Electrodynamics: Quantum Foundations of Electromagnetism, MIT Press. p. 37–38. ISBN 9780262632607, 
  50. Frederic V. Hartemann (2001). High-field electrodynamics, CRC Press. p. 102. ISBN 9780849323782,,M1. 

Further reading Edit

Journal articlesEdit

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 theorie de Lorentz et la Principe de Reaction", Archives Néerlandaises, V, 253-78.
  • Henri Poincaré (1901) Science and Hypothesis
  • Henri Poincaré (1905) "Sur la dynamique de l'électron", Comptes rendus de l'Académie des Sciences, 140, 1504-8.


University level textbooksEdit


  • Tipler, Paul; Mosca, Gene (2007), Physics for Scientists and Engineers: Volume 2 (6th ed.), W. H. Freeman, ISBN 978-1429201339 
  • Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.), Prentice Hall. ISBN 0-13-805326-X. 
  • Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (2008), Foundations of Electromagnetic Theory (4th ed.), Addison Wesley, ISBN 978-0321581747 
  • Edward Mills Purcell (1985). Electricity and Magnetism, McGraw-Hill. ISBN 0-07-004908-4. 
  • Schwarz, Melvin (1987). Principles of Electrodynamics, Dover Publications. ISBN 0-486-65493-1. 
  • Stevens, Charles F., 1995. The Six Core Theories of Modern Physics. MIT Press. ISBN 0-262-69188-4.
  • Krey, U., Owen, A. (2007), Basic Theoretical Physics - A Concise Overview, esp. part II, Springer, ISBN 978-3-540-36804-5
  • Ulaby, Fawwaz T. (2007). Fundamentals of Applied Electromagnetics (5th ed.), Pearson Education, Inc.. ISBN 0-13-241326-4. 
  • Sadiku, Matthew N. O. (2006). Elements of Electromagnetics (4th ed.), Oxford University Press. ISBN 0-19-5300483. 
  • Fleisch, Daniel (2008). A Student's Guide to Maxwell's Equations, Cambridge University Press. ISBN 978-0521877619. 
  • Hoffman, Banesh, 1983. Relativity and Its Roots. W. H. Freeman.


Older classicsEdit

Computational techniquesEdit

  • R. F. Harrington (1993). Field Computation by Moment Methods, Wiley-IEEE Press. ISBN 0-78031-014-4. 
  • W. C. Chew, J.-M. Jin, E. Michielssen, and J. Song (2001). Fast and Efficient Algorithms in Computational Electromagnetics, Artech House Publishers. ISBN 1-58053-152-0. 
  • J. Jin (2002). The Finite Element Method in Electromagnetics, 2nd. ed., Wiley-IEEE Press. ISBN 0-47143-818-9. 
  • Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed., Artech House Publishers. ISBN 1-58053-832-0. 

External linksEdit

Modern treatmentsEdit



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