Maxwell's equations

For thermodynamic relations, see Maxwell relations.

Maxwell's equations are a set of four partial differential equations that relate the electric and magnetic fields to their sources, charge density and current density. These equations can be combined 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. The set of equations is named after James Clerk Maxwell.

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 description

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 article.

• Gauss' law describes how an electric field is generated by electric charges: The electric field tends to point away from positive charges and towards negative charges. More technically, it relates the electric flux through any hypothetical closed "Gaussian surface" to the electric charge within the surface.

• Gauss' law for magnetism states that there are no "magnetic charges" (also called magnetic monopoles), analogous to electric charges. Instead the magnetic field is generated by a configuration called a dipole, which has no magnetic charge but resembles a positive and negative charge inseparably bound together. Equivalent technical statements are that the total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.

An Wang's magnetic core memory (1954) is an application of Ampere's law. Each core stores one bit of data.

• Faraday's law describes how a changing magnetic field can create ("induce") an electric field. This aspect of electromagnetic induction is the operating principle behind many electric generators: A bar magnet is rotated to create a changing magnetic field, which in turn generates an electric field in a nearby wire. (Note: The "Faraday's law" that occurs in Maxwell's equations is a bit different than the version originally written by Michael Faraday. Both versions are equally true laws of physics, but they have different scope, for example whether "motional EMF" is included. See Faraday's law of induction for details.)

• Ampère's law with Maxwell's correction states that magnetic fields can be generated in two ways: by electrical current (this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's correction").

Maxwell's correction to Ampère's law is particularly important: It means that a changing magnetic field creates an electric field, and a changing electric field creates a magnetic field. Therefore, these equations allow self-sustaining "electromagnetic waves" to travel through empty space (see electromagnetic wave equation).

The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, exactly 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 1864, thereby unifying the previously-separate fields of electromagnetism and optics.

General formulation

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), 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 4.

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

Formulation in terms of free charge and current

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)	$\nabla \times \mathbf {H} =\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}$ ${\displaystyle }$	$\oint _{\partial S}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =I_{f,S}+{\frac {\partial \Phi _{D,S}}{\partial t}}$

Formulation in terms of total charge and current

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:

Definitions and units

Symbol	Meaning (first term is the most common)	SI Unit of Measure
$\mathbf {E} \$	electric field also called the electric field intensity	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, volt-second per square meter
$\mathbf {D} \$	electric displacement field also called the electric induction 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, also called the electric constant, a universal constant	farads per meter
$\mu _{0}\$	permeability of free space, also called 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 free electric charge within the three-dimensional volume V (not including bound charge)	coulombs
$\,Q(V)$	net electric charge within the three-dimensional volume V (including both free and bound charge)	coulombs
$\oint _{\partial S}\mathbf {E} \cdot \mathrm {d} \mathbf {l}$	line integral of the electric field along the boundary ∂S of a surface S (∂S is always a closed curve).	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	tesla-meters
$\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \mathbf {E} \;\cdot \mathrm {d} \mathbf {A}$	the electric flux (surface integral of the electric field) through the (closed) surface $\partial V$ (the boundary of the volume V)	joule-meter per coulomb
$\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset \mathbf {B} \;\cdot \mathrm {d} \mathbf {A}$	the magnetic flux (surface integral of the magnetic B-field) through the (closed) surface $\partial V$ (the boundary of the volume 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, volt-seconds
$\iint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} =\Phi _{E,S}$	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)	amperes
$\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {A} =I_{S}$	net electrical current passing through the surface S (including both free and bound current)	amperes

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

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

All four of what are now described as Maxwell's equations can be found in recognizable 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.

The term Maxwell's equations

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, concurrently with similar work by Willard Gibbs and Heinrich Hertz. These equations were also known variously as the Hertz-Heaviside equations and the Maxwell-Hertz equations, and are sometimes still known as the Maxwell–Heaviside equations.

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. 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; this effort was somewhat controversial, 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. 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, 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.

Maxwell's On Physical Lines of Force (1861)

The four modern day Maxwell's equations appeared throughout Maxwell's 1861 paper On Physical Lines of Force:

1. Equation (56) in Maxwell's 1861 paper is $\nabla \cdot \mathbf {B} =0$.
2. Equation (112) is Ampère's circuital law with Maxwell's displacement current added. It is the addition of displacement current that 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).
3. Equation (115) is Gauss's law.
4. Equation (54) is an equation that Oliver Heaviside referred to as 'Faraday's law'. This equation caters for the time varying aspect of electromagnetic induction, but not for the motionally induced aspect, whereas Faraday's original flux law caters for both aspects. Maxwell deals with the motionally dependent aspect of electromagnetic induction, v × B, at equation (77). Equation (77) which is the same as equation (D) in the original eight Maxwell's equations listed below, corresponds to all intents and purposes to the modern day force law F = q ( E + v × B ) which sits adjacent to Maxwell's equations and bears the name Lorentz force, even though Maxwell derived it 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.

Figure of Maxwell's molecular vortex model. For a uniform magnetic field, the field lines point outward from the display screen, as can be observed from the black dots in the middle of the hexagons. The vortex of each hexagonal molecule rotates counter-clockwise. The small green circles are clockwise rotating particles sandwiching between the molecular vortices.

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 $\mathbf {D}$. i.e. $\mathbf {B}$ parallels with $\mathbf {E}$, whereas $\mathbf {H}$ parallels with $\mathbf {D}$.

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

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," 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 and 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

$\nabla \cdot \mathbf {J} =-{\frac {\partial \rho }{\partial t}}$

or

$\nabla \cdot \mathbf {J} _{tot}=0$

Notation

$\mathbf {H}$ 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).

$\mathbf {D}$ 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)

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 matter

Bound charge and current

Main articles: Bound charge § Bound charge, and Bound current § Magnetization current

Left: A schematic view of how an assembly of microscopic dipoles appears like a macroscopically separated pair of charged sheets, as shown at top and bottom (these sheets are not intended to be viewed as originating the electric field that causes the dipole alignment, but as a representation equivalent to the dipole array); Right: How an assembly of microscopic current loops appears as a macroscopically circulating current loop. Inside the boundaries, the individual contributions tend to cancel, but at the boundaries no cancellation occurs.

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.)

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.)

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

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 ρ_b and J_b 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 2 are precisely equivalent.

Constitutive relations

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. 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

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

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

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:

• The strength of the fields (the case of nonlinearity, which occurs when ε and μ are functions of E and B; see, for example, Kerr and Pockels effects),
• The direction of the fields (the case of anisotropy, birefringence, or dichroism; which occurs when ε and μ are second-rank tensors),
• The frequency with which the fields vary (the case of dispersion, which occurs when ε and μ are functions of frequency; see, for example, Kramers–Kronig relations).

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:

$\mathbf {D} (\mathbf {r} ,t)=\epsilon _{0}\mathbf {E} (\mathbf {r} ,t)+\mathbf {P} (\mathbf {r} ,t)$

$\mathbf {H} (\mathbf {r} ,t)={\frac {1}{\mu _{0}}}\mathbf {B} (\mathbf {r} ,t)-\mathbf {M} (\mathbf {r} ,t)$

$\mathbf {P} (\mathbf {r} ,t)=\epsilon _{0}\int d^{3}\mathbf {r} 'dt'\;{\hat {\chi }}_{\mathrm {elec} }(\mathbf {r} ,\mathbf {r} ',t,t';\mathbf {E} )\,\mathbf {E} (\mathbf {r} ',t')$

$\mathbf {M} (\mathbf {r} ,t)={\frac {1}{\mu _{0}}}\int d^{3}\mathbf {r} 'dt'\;{\hat {\chi }}_{\mathrm {magn} }(\mathbf {r} ,\mathbf {r} ',t,t';\mathbf {B} )\,\mathbf {B} (\mathbf {r} ',t')$,

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 materials

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 electric 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 relations

See also: Computational electromagnetics

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. 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 continuous 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 (valid for excitations with wavelengths much larger than the scale of the inhomogeneity).

Theoretical results have their place, but often require fitting to experiment. Continuum-approximation properties of many real materials rely upon measurement, 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.

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

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 we are in the vacuum ($\rho =0\$ and $\mathbf {J} =0\$) around the charges and currents responsible for the described fields, then:

$\nabla \cdot \mathbf {E} =0$

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

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

$\nabla \times \mathbf {B} =\ \ \mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}.$

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:

$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 knew from an 1856 leyden jar experiment by Wilhelm Eduard Weber and Rudolf Kohlrausch, that c was very close to the measured speed of light in vacuum (known from early experiments), and concluded (correctly) that light is a form of electromagnetic radiation.

With magnetic monopoles

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. There is no known magnetic analog of an electron, however recently scientists have described behavior in a crystalline state of matter known as spin-ice which have macroscopic behavior like magnetic monopoles. (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. With the inclusion of a variable for these magnetic charges, say $\rho _{m}\,$, there will also be a "magnetic current" variable in the equations, $\mathbf {J} _{m}\,$. 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:	$\nabla \cdot \mathbf {E} =4\pi \rho _{e}$	$\nabla \cdot \mathbf {E} =4\pi \rho _{e}$
Gauss's law for magnetism:	$\nabla \cdot \mathbf {B} =0$	$\nabla \cdot \mathbf {B} =4\pi \rho _{m}$
Maxwell–Faraday equation (Faraday's law of induction):	$-\nabla \times \mathbf {E} ={\frac {\partial \mathbf {B} }{\partial t}}$	$-\nabla \times \mathbf {E} ={\frac {\partial \mathbf {B} }{\partial t}}+4\pi \mathbf {j} _{m}$
Ampère's law (with Maxwell's extension):	$\nabla \times \mathbf {B} ={\frac {\partial \mathbf {E} }{\partial t}}+4\pi \mathbf {j} _{e}$  $\$	$\nabla \times \mathbf {B} ={\frac {\partial \mathbf {E} }{\partial t}}+4\pi \mathbf {j} _{e}$
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 $\mathbf {\nabla } \cdot \mathbf {B} =0$.

Boundary conditions: using Maxwell's equations

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 and started in time using initial conditions.

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.

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. In other cases, the universe at large sometimes is approximated by an artificial absorbing boundary, or, for example for radiating antennas or communication satellites, these boundary conditions can take the form of asymptotic limits imposed upon the solution. 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. 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 and the book by Friedman.

CGS units

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:

$\nabla \cdot \mathbf {D} =4\pi \rho _{f}$

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

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

$\nabla \times \mathbf {H} ={\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}+{\frac {4\pi }{c}}\mathbf {J} _{f}$

where c is the speed of light in a vacuum. For the electromagnetic field in a vacuum, assuming that there is no current or electric charge present in the 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 electric 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} \$.

Some equations in the article are given in Gaussian units but not SI or vice-versa. Fortunately, there are general rules to convert from one to the other; see the article Gaussian units for details.

Special relativity

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

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 consisted 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 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). 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.

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 equations

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:

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

and

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

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

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. (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 $\varphi$, called the electric potential:

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

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

$\mathbf {E} =-\mathbf {\nabla } \varphi -{\frac {\partial \mathbf {A} }{\partial t}}$

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

$\nabla ^{2}\varphi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon _{0}}}$

$\left(\nabla ^{2}\mathbf {A} -{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {A} }{\partial t^{2}}}\right)-\mathbf {\nabla } \left(\mathbf {\nabla } \cdot \mathbf {A} +{\frac {1}{c^{2}}}{\frac {\partial \varphi }{\partial t}}\right)=-\mu _{0}\mathbf {J} .$

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

Four-potential

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

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

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

$A^{\mu }=\left(\varphi ,\mathbf {A} c\right)$

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

$\Box A^{\mu }=\mu _{0}j^{\mu }$,

where $\Box =\partial ^{2}=\partial _{\alpha }\partial ^{\alpha }$ is the d'Alembertian operator, or four-Laplacian, $\left({\partial ^{2} \over \partial t^{2}}-\nabla ^{2}\right)$, sometimes written $\Box ^{2}$, or $\Box \cdot \Box$, where $\Box$ is the four-gradient.

Differential forms

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-Gaussian units, not SI units are used. (To convert to SI, see here.) 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

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

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

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

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

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

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

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

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:

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

$\mathrm {d} {\mathbf {G}}={\mathbf {J}}$

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 ${\mathbf {\theta }}^{p}$,

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

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 formulation

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 A→A' = A-dα: see also gauge fixing and Fadeev-Popov ghosts.

Classical electrodynamics as the curvature of a line bundle

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 ${\mathbf {F}}=\nabla ^{2}$ which is a two-form that automatically satisfies $\mathrm {d} {\mathbf {F}}=0$ and can be interpreted as a field-strength. If the line bundle is trivial with flat reference connection d we can write $\nabla =\mathrm {d} +{\mathbf {A}}$ 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 iron 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 spacetime

Main article: Maxwell's equations in curved spacetime

Traditional formulation

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-Gaussian units):

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

and

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

Here,

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

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

Formulation in terms of differential forms

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-Gaussian units we define

• The antisymmetric infinitesimal field tensor $F_{\alpha \beta }$, corresponding to the field 2-form F

${\mathbf {F}}:={\frac {1}{2}}F_{\alpha \beta }\,\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }.$

• The current-vector infinitesimal 3-form J

${\mathbf {J}}:={4\pi \over c}j^{\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }.$

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

$\mathrm {d} {\mathbf {F}}=2(\partial _{\gamma }F_{\alpha \beta }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\alpha }F_{\beta \gamma })\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }=0,$

• the source equation

$\mathrm {d} *{\mathbf {F}}={F^{\alpha \beta }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\beta \gamma \delta \eta }\mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }\wedge \mathrm {d} \,x^{\eta }={\mathbf {J}},$

• the continuity equation

$\mathrm {d} {\mathbf {J}}={4\pi \over c}{j^{\alpha }}_{;\alpha }{\sqrt {-g}}\,\epsilon _{\alpha \beta \gamma \delta }\mathrm {d} \,x^{\alpha }\wedge \mathrm {d} \,x^{\beta }\wedge \mathrm {d} \,x^{\gamma }\wedge \mathrm {d} \,x^{\delta }=0.$

See also

• Abraham-Lorentz force
• Ampere's law
• Antenna (radio)
• Bremsstrahlung
• Computational electromagnetics
• Electrical generator
• Electromagnetic wave equation
• Finite-difference time-domain method
• Fresnel equations
• Green–Kubo relations
• Green's function (many-body theory)
• Interface conditions for electromagnetic fields
• Jefimenko's equations
• Kramers–Kronig relation
• Laser
• Linear response function
• Lorentz force
• Mathematical descriptions of the electromagnetic field
• Moving magnet and conductor problem
• Nonhomogeneous electromagnetic wave equation
• Photon dynamics in the double-slit experiment
• Photon polarization
• Photonic crystal
• Scattering-matrix method
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Theoretical and experimental justification for the Schrödinger equation
• Transformer
• Waveguide
• Wheeler-Feynman time-symmetric theory for electrodynamics

Further reading

Journal articles

• 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.)

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.

see

• Macrossan, M. N. (1986) "A note on relativity before Einstein", Brit. J. Phil. Sci., 37, 232-234

University level textbooks

Undergraduate

• Feynman, Richard P. (2005). The Feynman Lectures on Physics. Vol. 2 (2nd ed.). Addison-Wesley. ISBN 978-0805390650.
• Fleisch, Daniel (2008). A Student's Guide to Maxwell's Equations. Cambridge University Press. ISBN 978-0521877619.
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Hoffman, Banesh (1983). Relativity and Its Roots. W. H. Freeman.
• Krey, U.; Owen, A. (2007). Basic Theoretical Physics: A Concise Overview. Springer. ISBN 978-3-540-36804-5. See especially part II.
• Purcell, Edward Mills (1985). Electricity and Magnetism. McGraw-Hill. ISBN 0-07-004908-4.
• Reitz, John R.; Milford, Frederick J.; Christy, Robert W. (2008). Foundations of Electromagnetic Theory (4th ed.). Addison Wesley. ISBN 978-0321581747.
• Sadiku, Matthew N. O. (2006). Elements of Electromagnetics (4th ed.). Oxford University Press. ISBN 0-19-5300483.
• Schwarz, Melvin (1987). Principles of Electrodynamics. Dover. ISBN 0-486-65493-1.
• Stevens, Charles F. (1995). The Six Core Theories of Modern Physics. MIT Press. ISBN 0-262-69188-4.
• Tipler, Paul; Mosca, Gene (2007). Physics for Scientists and Engineers. Vol. 2 (6th ed.). W. H. Freeman. ISBN 978-1429201339.
• Ulaby, Fawwaz T. (2007). Fundamentals of Applied Electromagnetics (5th ed.). Pearson Education. ISBN 0-13-241326-4.

Graduate

• Jackson, J. D. (1999). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.
• Panofsky, Wolfgang K. H.; Phillips, Melba (2005). Classical Electricity and Magnetism (2nd ed.). Dover. ISBN 978-0486439242.

Older classics

• Lifshitz, Evgeny (1980). The Classical Theory of Fields (4th ed.). Butterworth-Heinemann. ISBN 0750627689. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Lifshitz, Evgeny (1984). Electrodynamics of Continuous Media (2nd ed.). Butterworth-Heinemann. ISBN 0750626348. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Maxwell, James Clerk (1873). A Treatise on Electricity and Magnetism. ISBN 0-486-60637-6. {{cite book}}: Unknown parameter |puclisher= ignored (help)
• Misner, Charles W. (1973). Gravitation. W. H. Freeman. ISBN 0-7167-0344-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help) Sets out the equations using differential forms.

Computational techniques

• Chew, W. C. (2001). Fast and Efficient Algorithms in Computational Electromagnetics. Artech House. ISBN 1-58053-152-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Harrington, R. F. (1993). Field Computation by Moment Methods. Wiley-IEEE Press. ISBN 0-78031-014-4.
• Jin, J. (2002). The Finite Element Method in Electromagnetics (2nd ed.). Wiley-IEEE Press. ISBN 0-47143-818-9.
• Lounesto, Pertti (1997). Clifford Algebras and Spinors. Cambridge University Press. Chapter 8 sets out several variants of the equations using exterior algebra and differential forms.
• Taflove, Allen (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method (3rd ed.). Artech House. ISBN 1-58053-832-0. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

External links

• Mathematical aspects of Maxwell's equation are discussed on the Dispersive PDE Wiki.

Modern treatments

• Electromagnetism, B. Crowell, Fullerton College
• Lecture series: Relativity and electromagnetism, R. Fitzpatrick, University of Texas at Austin
• Electromagnetic waves from Maxwell's equations on Project PHYSNET.
• MIT Video Lecture Series (36 x 50 minute lectures) (in .mp4 format) - Electricity and Magnetism Taught by Professor Walter Lewin.

Historical

• James Clerk Maxwell, A Treatise on Electricity And Magnetism Vols 1 and 2 1904—most readable edition with all corrections—Antique Books Collection suitable for free reading online.
• Maxwell, J.C., A Treatise on Electricity And Magnetism - Volume 1 - 1873 - Posner Memorial Collection - Carnegie Mellon University
• Maxwell, J.C., A Treatise on Electricity And Magnetism - Volume 2 - 1873 - Posner Memorial Collection - Carnegie Mellon University
• On Faraday's Lines of Force - 1855/56 Maxwell's first paper (Part 1 & 2) - Compiled by Blaze Labs Research (PDF)
• On Physical Lines of Force - 1861 Maxwell's 1861 paper describing magnetic lines of Force - Predecessor to 1873 Treatise
• Maxwell, James Clerk, "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.)
• Catt, Walton and Davidson. "The History of Displacement Current". Wireless World, March 1979.
• Reprint from Dover Publications (ISBN 0-486-60636-8)
• Full text of 1904 Edition including full text search.
• A Dynamical Theory Of The Electromagnetic Field - 1865 Maxwell's 1865 paper describing his 20 Equations in 20 Unknowns - Predecessor to the 1873 Treatise

Other

• Feynman's derivation of Maxwell equations and extra dimensions
• Simple explanation of the equations and their physical implications - David Morgan-Mar - Irregular Webcomic!
• According to an article in Physicsweb, the Maxwell equations rate as "The greatest equations ever".