MAXWELL'S EQUATIONS

In electromagnetism, 'Maxwell's equations' are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. They describe the interrelationship between electric fields, magnetic fields, electric charge, and electric current.
Although Maxwell himself was not the originator of the individual equations, he derived them again independently in conjunction with his molecular vortex model of Faraday's lines of force, and he was the person who first grouped these equations all together into a coherent set. Most importantly, he introduced an extra term to Ampère's circuital law. This extra term is the time derivative of electric field and is known as Maxwell's displacement current. Maxwell's modified version of Ampère's circuital law enables the set of equations to be combined together to derive the electromagnetic wave equation.
Although Maxwell's equations were known before special relativity, they can be derived from Coulomb's law and special relativity if one assumes invariance of electric charge.^[1][2]
This in turn leads to a very interesting parallel with gravity in that the same reasoning can be applied to Newton's law of gravitation leading to a gravitational equivalent of Maxwell's equations. See ''gravitomagnetism'' for more information.

Contents

History of Maxwell's Equations

Summary of the Modern Heaviside Versions

General case

In linear materials

In vacuum, without charges or currents

The Heaviside Versions in Detail

'(1)' Gauss's Law

'(2)' The Divergence of the Magnetic Field

'(3)' Faraday's Law of Electromagnetic Induction

'(4)' Ampère's Circuital Law

Maxwell's equations in CGS units

Formulation of Maxwell's equations in special relativity

Maxwell's equations in terms of differential forms

Conceptual insight from this formulation

The original eight Maxwell's Equations

Classical electrodynamics as the curvature of a line bundle

Links to relativity

Maxwell's equations in curved spacetime

Traditional formulation

Formulation in terms of differential forms

See also

References

Journal articles

University level textbooks

Undergraduate

Graduate

Computational techniques

Citations

External links

Modern treatments

Historical

Feynman’s derivation of Maxwell equations

Other

History of Maxwell's Equations

Maxwell's equations are a set of four equations that can all be found at various places in Maxwell's 1861 paper On Physical Lines of Force. They express (i) how electric charges produce electric fields (Gauss's law), (ii) the experimental absence of magnetic monopoles, (iii) how electric currents and changing electric fields produce magnetic fields (Ampère's circuital law), and (iv) how changing magnetic fields produce electric fields (Faraday's law of induction).
Apart from Maxwell's amendment to Ampère's circuital law, none of these equations are original. However, Maxwell uniquely re-derived them hydrodynamically and mechanically using his vortex model of Faraday's lines of force.
In the year 1884 Oliver Heaviside selected these four equations, and in conjunction with Willard Gibbs, he put them into modern vector notation. This gives rise to the claim by some scientists that Maxwell's equations are in actual fact Heaviside's equations.
This matter is further confused by the fact that the term 'Maxwell's Equations' is also used to describe a set of eight equations labelled '(A)' to '(H)' in Maxwell's 1865 paper ''A Dynamical Theory of the Electromagnetic Field''. It therefore helps when referring to 'Maxwell's Equations' to specify whether we are talking about the original eight equations or the modified 'Heaviside Four'.
The two sets of equations are physically equivalent to all intents and purposes although Gauss's Law is the only actual equation that appears in both sets. The Maxwell/Ampère equation in the 'Heaviside Four' is an amalgamation of two equations in the original eight.

Summary of the Modern Heaviside Versions

Symbols in 'bold' represent vector quantities, whereas symbols in ''italics'' represent scalar quantities.

General case

The Equations are given in SI units. See below for CGS units.
{| class="wikitable" border="1" cellpadding="8" cellspacing="0"
! Name
! Differential form
! Integral form
|-
| Gauss's law:
| $$
abla cdot mathbf{E} = rac {
ho} {epsilon_0}
|
|-
| Gauss' law for magnetism
(absence of magnetic monopoles):
| $$
abla cdot mathbf{B} = 0
| $oint\_S\; mathbf\{B\}\; cdot\; mathrm\{d\}mathbf\{A\}\; =\; 0$
|-
| Faraday's law of induction:
| $$
abla imes mathbf{E} = -rac{partial mathbf{B}} {partial t}
|
|-
| Ampère's Circuital Law
(with Maxwell's correction):
| $$
abla imes mathbf{B} = mu_0 mathbf{J} + mu_0 epsilon_0 rac{partial mathbf{E}} {partial t}
|

|}
The following table provides the meaning of each symbol and the SI unit of measure:
{| class="wikitable" border="1" cellpadding="8" cellspacing="0"
! Symbol
! Meaning (first term is the most common)
! SI Unit of Measure
|-
|$$
abla cdot
| the divergence operator
|rowspan=2 | per meter (factor contributed by applying either operator)
|-
| $$
abla imes
| the curl operator
|-
|
| partial derivative with respect to time
| per second (factor contributed by applying the operator)
|-
| $mathbf\{E\}$
| electric field
also called the electric flux density
| 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
|-
| $$
ho
| electric charge density
| coulomb per cubic meter
|-
| $epsilon\_0$
| Permittivity of free space, a universal constant
| farads per meter
|-
| $oint\_S\; mathbf\{E\}\; cdot\; mathrm\{d\}mathbf\{A\}$
| The flux of the electric field over any closed Gaussian surface S
| joule-meter per coulomb
|-
| $mathbf\{Q\}\_S$
| net unbalanced electric charge enclosed by the Gaussian surface S, including so-called ''Bound charges''
| coulombs
|-
| $oint\_S\; mathbf\{B\}\; cdot\; mathrm\{d\}mathbf\{A\}$
| The flux of the magnetic field field over any closed surface S
| Tesla meter-squared or webber
|-
| $oint\_\{partial\; S\}\; mathbf\{E\}\; cdot\; mathrm\{d\}mathbf\{l\}$
| line integral of the electric field along the boundary (therefore necessarily a closed curve) of the surface S
| Joule per coulomb
|-
| $mathbf\{Phi\}\_\{B,S\}\; =\; int\_S\; mathbf\{B\}\; cdot\; mathrm\{d\}\; mathbf\{A\}$
| magnetic flux over any surface S (not necessarily closed)
| webber
|-
| $mu\_0$
| magnetic permeability of free space, a universal constant
| henries per meter, or newtons per ampere squared
|-
|$mathbf\{J\}$
| current density
| ampere per square meter
|-
| $oint\_\{partial\; S\}\; mathbf\{B\}\; cdot\; mathrm\{d\}mathbf\{l\}$
| line integral of the magnetic field over the closed boundary of the surface S
| tesla-meter
|-
| $mathbf\{I\}\_S\; =\; int\_S\; mathbf\{J\}\; cdot\; mathrm\{d\}\; mathbf\{A\}$
| net electrical current passing through the surface S
| amperes
|-
| $mathbf\{Phi\}\_\{E,S\}\; =\; int\_S\; mathbf\{E\}\; cdot\; mathrm\{d\}\; mathbf\{A\}$
| Electric flux over any surface S, not necessarily closed
|-
| $mathrm\{d\}mathbf\{A\}$
| 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 contour
| meters
|}
The equations are given here 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. For example, the electric field and the magnetic field have the same unit (gauss) in the Gaussian system. 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^[3]), Lorentz-Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics).
In order to connect the theory of classical electrodynamics to mechanics we need to add another equation to the four Maxwell's Equations. The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:
: $mathbf\{F\}\; =\; q\; (mathbf\{E\}\; +\; mathbf\{v\}\; imes\; mathbf\{B\}),$
where $q$ is the charge on the particle and $mathbf\{v\}$ is the particle velocity. This is slightly different when expressed in the cgs system of units below.
This extra equation appeared in cartesian format as equation '(D)' of the original eight 'Maxwell's Equations'.
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. Below the microscopic, Maxwell's equations, ignoring quantum effects, are simply those of a vacuum — but one must include all atomic charges and so on, which is generally an intractable problem.

In linear materials

In linear materials, the polarization density $mathbf\{P\}$ (in coulombs per square meter) and magnetization density $mathbf\{M\}$ (in amperes per meter) are given by:
:
:$mathbf\{M\}\; =\; chi\_m\; mathbf\{H\}$
and the $mathbf\{D\}$ and $mathbf\{B\}$ fields are related to $mathbf\{E\}$ and $mathbf\{H\}$ by:
:
= arepsilon mathbf{E}
:$mathbf\{B\}\; =\; mu\_0\; (mathbf\{H\}\; +\; mathbf\{M\})\; =\; (1\; +\; chi\_m)\; mu\_0\; mathbf\{H\}$
= mu mathbf{H}
where:
$chi\_e$ is the electrical susceptibility of the material,
$chi\_m$ is the magnetic susceptibility of the material,
is the electrical permittivity of the material, and
$mu$ is the magnetic permeability of the material
(This can actually be extended to handle nonlinear materials as well, by making 'ε' and 'μ' depend upon the field strength; see e.g. the Kerr and Pockels effects.)
In non-dispersive, isotropic media, ε and μ are time-independent scalars, and Maxwell's equations reduce to
:$$
abla cdot arepsilon mathbf{E} =
ho
:$$
abla cdot mu mathbf{H} = 0
:$$
abla imes mathbf{E} = - mu rac{partial mathbf{H}} {partial t}
:$$
abla imes mathbf{H} = mathbf{J} + arepsilon rac{partial mathbf{E}} {partial t}
In a uniform (homogeneous) medium, ε and μ are constants independent of position, and can thus be furthermore interchanged with the spatial derivatives.
More generally, ε and μ can be rank-2 tensors (3×3 matrices) describing birefringent (anisotropic) materials. Also, although for many purposes the time/frequency-dependence of these constants can be neglected, every real material exhibits some material dispersion by which ε and/or μ depend upon frequency (and causality constrains this dependence to obey the Kramers-Kronig relations).

In vacuum, without charges or currents

The vacuum is a linear, homogeneous, isotropic, dispersionless medium, and the proportionality constants in the vacuum are denoted by ε₀ and μ₀.
:
:$mathbf\{B\}\; =\; mu\_0\; mathbf\{H\}$
Since there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:
:$$
abla cdot mathbf{E} = 0
:$$
abla cdot mathbf{B} = 0
:$$
abla imes mathbf{E} = - rac{partialmathbf{B}} {partial t}
:$$
abla imes mathbf{B} = mu_0arepsilon_0 rac{partial mathbf{E}} {partial t}
These equations have a solution in terms of travelling 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, travelling at the speed
:
Maxwell discovered that this quantity ''c'' is simply the speed of light in vacuum, and thus that light is a form of electromagnetic radiation. The current SI values for the speed of light, the electric and the magnetic constant are summarized in the following table:
{| class="wikitable" border="1" cellspacing="0" cellpadding="8"
! Symbol
! Name
! Numerical Value
! SI Unit of Measure
! Type
|-
|-
| $c$
| Speed of light in vacuum
|$2.99792458\; imes\; 10^8$
| meters per second
| defined
|-
|
| electric constant
| $8.85419\; imes\; 10^\{-12\}$
| farads per meter
| derived
|-
|$mu\_0$
| magnetic constant
|$4\; pi\; imes\; 10^\{-7\}$
| henries per meter
| defined
|}

The Heaviside Versions in Detail

'(1)' Gauss's Law

Gauss's law yields the sources (and sinks) of electric charge.
: $$
abla cdot mathbf{D} =
ho
where $\{$
ho} is the ''free'' electric charge density (in units of C/m³), not including dipole charges bound in a material, and $mathbf\{D\}$ is the electric displacement field (in units of C/m²). The solution to 'Gauss's Law' is Coulomb's law for stationary charges in vacuum.
The equivalent integral form (by the divergence theorem), also known as Gauss' law, is:
: $oint\_S\; mathbf\{D\}\; cdot\; mathrm\{d\}mathbf\{A\}\; =\; Q\_mathrm\{enclosed\}$
where $mathrm\{d\}mathbf\{A\}$ is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and $Q\_mathrm\{enclosed\}$ is the free charge enclosed by the surface.
In a ''linear material'', $mathbf\{D\}$ is directly related to the electric field $mathbf\{E\}$ via a material-dependent constant called the permittivity, $epsilon$:
:.
Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as $epsilon\_0$, and appears in:
:$$
abla cdot mathbf{E} = rac{
ho_t}{arepsilon_0}
where, again, $mathbf\{E\}$ is the electric field (in units of V/m), $$
ho_t is the total charge density (including bound charges), and $epsilon\_0$ (approximately 8.854 pF/m) is the permittivity of free space. $epsilon$ can also be written as , where $epsilon\_r$ is the material's relative permittivity or its ''dielectric constant''.
Compare Poisson's equation.

'(2)' The Divergence of the Magnetic Field

The divergence of a magnetic field is always zero and hence magnetic field lines are solenoidal.
:$$
abla cdot mathbf{B} = 0
$mathbf\{B\}$ is the magnetic flux density (in units of teslas, T), also called the magnetic induction.
Equivalent integral form:
: $oint\_S\; mathbf\{B\}\; cdot\; mathrm\{d\}mathbf\{A\}\; =\; 0$
$mathrm\{d\}mathbf\{A\}$ is the area of a differential square on the surface $A$ with an outward facing surface normal defining its direction.
Like the electric field's integral form, this equation only works if the integral is done over a closed surface.
This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, this is a mathematical formulation of the statement that there are no magnetic monopoles.

'(3)' Faraday's Law of Electromagnetic Induction

: $$
abla imes mathbf{E} = -rac {partial mathbf{B}}{partial t}
The equivalent integral form is (according to Stoke's Theorem):
:
where
$scriptstyle\; mathbf\{E\}$ is the electric field,
$scriptstyle\; C=partial\; S$ is the boundary of the surface $S$.
If a conducting wire, following the contour $C$, is introduced into the field, the so-called electromotive force in this wire is equal to the value of these integrals (over the fields in absence of the wire!).
The negative sign was established experimentally by Faraday in 1831, a common modern textbook interpretation is that it is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's law.
This equation relates the electric and magnetic fields, but it also has a number of practical applications. In circuit theory it takes the form of the relationship of induced voltage due to a changing current in an inductance, sometimes called a reverse or back emf. This equation describes how electric motors and electric generators work. Specifically, it demonstrates that a voltage can be generated by varying the magnetic flux passing through a given area over time, such as by uniformly rotating a loop of wire through a fixed magnetic field. In a motor or generator, the fixed excitation is provided by the field circuit and the varying voltage is measured across the armature circuit. In some types of motors/generators, the field circuit is mounted on the rotor and the armature circuit is mounted on the stator, but other types of motors/generators reverse the configuration.
Maxwell's equations apply to a right-handed coordinate system. To apply them unmodified to a left handed system would reverse the polarity of magnetic fields (not inconsistent, but confusingly against convention).

'(4)' Ampère's Circuital Law

Ampère's Circuital Law describes the source of the magnetic field,
:$$
abla imes mathbf{H} = mathbf{J} + rac {partial mathbf{D}} {partial t}
where $mathbf\{H\}$ is the magnetic field strength (in units of A/m), related to the magnetic flux density $mathbf\{B\}$ by a constant called the permeability, μ ($mathbf\{B\}=mu\; mathbf\{H\}$), and $mathbf\{J\}$ is the 'current density', defined by: $mathbf\{J\}\; =$
ho_qmathbf{v} where $mathbf\{v\}$ is a vector field called the drift velocity that describes the velocities of the charge carriers which have a density described by the scalar function ρ_q. The second term on the right hand side of Ampère's Circuital Law is known as the displacement current.
It was Maxwell who added the displacement current term to Ampère's Circuital Law at equation (112) in his 1861 paper On Physical Lines of Force. This addition means that either Maxwell's original eight equations, or the modified Heaviside four equations can be combined together to obtain the electromagnetic wave equation.
Maxwell used the displacement current in conjunction with the original eight equations in his 1864 paper ''A Dynamical Theory of the Electromagnetic Field'' to derive the electromagnetic wave equation in a much more cumbersome fashion than that which is employed when using the 'Heaviside Four'. Most modern textbooks derive the electromagnetic wave equation using the 'Heaviside Four'.
In free space, the permeability μ is the permeability of free space, μ₀, which is defined to be ''exactly'' 4π×10^-7 Wb/A•m. Also, the permittivity becomes the permittivity of free space ε₀. Thus, in free space, the equation becomes:
:$$
abla imes mathbf{B} = mu_0 mathbf{J} + mu_0arepsilon_0 rac{partial mathbf{E}}{partial t}
Equivalent integral form:
:
''$mathbf\{l\}$'' is the edge of the open surface ''A'' (any surface with the curve ''$mathbf\{l\}$'' as its edge will do), and ''I''_encircled is the current encircled by the curve ''$mathbf\{l\}$'' (the current through any surface is defined by the equation: ). In some situations, this integral form of Ampere-Maxwell Law appears in:
:$oint\_C\; mathbf\{B\}\; cdot\; mathrm\{d\}mathbf\{l\}\; =\; mu\_0\; (I\_mathrm\{enc\}\; +\; I\_mathrm\{d,enc\})$
for
:
is sometimes called displacement current. The displacement current concept was Maxwell's greatest innovation in electromagnetic theory. It states that a magnetic field appears during the charge or discharge of a capacitor. With this concept, and the Faraday law equation, Maxwell was able to derive the wave equations, and by showing that the prediced wave velocity was the same as the measured velocity of light, Maxwell asserted that light waves are electromagnetic waves.
If the electric flux density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's law.

Maxwell's equations in 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), the equations take the following form:
:$$
abla cdot mathbf{D} = 4pi
ho
:$$
abla cdot mathbf{B} = 0
:$$
abla imes mathbf{E} = -rac{1}{c} rac{partial mathbf{B}} {partial t}
:$$
abla imes mathbf{H} = rac{1}{c} rac{partial mathbf{D}} {partial t} + rac{4pi}{c} mathbf{J}
Where ''c'' is the speed of light in a vacuum. For the electromagnetic field in a vacuum, the equations become:
:$$
abla cdot mathbf{E} = 0
:$$
abla cdot mathbf{B} = 0
:$$
abla imes mathbf{E} = -rac{1}{c} rac{partial mathbf{B}} {partial t}
:$$
abla imes mathbf{B} = rac{1}{c} rac{partial mathbf{E}}{partial t}
In this system of units the relation between magnetic induction, magnetic field and total magnetization take the form:
:$mathbf\{B\}\; =\; mathbf\{H\}\; +\; 4pimathbf\{M\}$
With the linear approximation:
: $mathbf\{B\}\; =\; (\; 1\; +\; 4pichi\_m\; )mathbf\{H\}$
$chi\_m$ for vacuum is zero and therefore:
: $mathbf\{B\}\; =\; mathbf\{H\}$
and in the ferro or ferri magnetic materials where $chi\_m$ is much bigger than 1:
: $mathbf\{B\}\; =\; 4pichi\_mmathbf\{H\}$
The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:
:
ight),
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 $mathbf\{E\}$.

Formulation of Maxwell's equations in special relativity

Main articles: Formulation of Maxwell's equations in special relativity

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, the vacuum Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form (cgs units):
:,
and
:
where is the 4-current, is the field strength tensor, is the Levi-Civita symbol, and
:
abla
ight)
is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations.
The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and Ampere's law with Maxwell's correction. The second equation is an expression of the homogenous equations, Faraday's law of induction and the absence of magnetic monopoles.

Maxwell's equations in terms of differential forms

In a vacuum, where ε and μ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. 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
:
where d denotes the exterior derivative - a differential operator acting on forms - and the source equation
:$mathrm\{d\}$
★ {old{F}}=old{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 (or in four dimensions by its conformal class), and the fields are in natural units where $1/4piepsilon\_0=1$. Here, the 3-form 'J' is called the "electric current" or "current (3-)form" satisfying the continuity equation
:
As the exterior derivative is defined on any manifold, this formulation of electromagnetism works for any 4-dimensional oriented manifold with a Lorentz metric, e.g. on the curved space-time of 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
:$C:Lambda^2$
iold{F}mapsto old{G}inLambda^{(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:
:
:
where the current 3-form 'J' still satisfies the continuity equation d'J'= 0.
When the fields are expressed as linear combinations (of exterior products) of basis forms ,
:.
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. The Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking
:$C\_\{pq\}^\{mn\}\; =\; g^\{ma\}g^\{nb\}\; epsilon\_\{abpq\}\; sqrt\{-g\}$
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 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. The price one pays for this simplification, however, is a need for knowledge of more technical mathematics.

Conceptual insight from this formulation

On the conceptual side, from a 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 that 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.

The original eight Maxwell's Equations

In Part III of A Dynamical Theory of the Electromagnetic Field which is entitled "GENERAL EQUATIONS OF THE ELECTROMAGNETIC FIELD" [1] (page 480 of the article and page 2 of the pdf link), Maxwell formulated eight equations labelled A to H. These eight equations were to become known as Maxwell's equations. Nowadays however, references to Maxwell's equations invariably refer to the Heaviside versions. Heaviside's versions of Maxwell's equations actually only contain one of the original eight, i.e. equation G Gauss's Law. Another of Heaviside's four equations is an amalgamation of Maxwell's Law of Total Currents (equation A) with Ampère's Circuital Law (equation C). This amalgamation, which Maxwell himself had actually originally made at equation (112) in his 1861 paper "On Physical Lines of Force", is the one that modifies Ampère's Circuital Law to include Maxwell's Displacement current.
The eight original Maxwell's equations will now be listed in modern vector notation,
'(A)' The Law of Total Currents
:
'(B)' Definition of the Magnetic Vector Potential
:$mu\; mathbf\{H\}\; =$
abla imes mathbf{A}
'(C)' Ampère's Circuital Law
:$$
abla imes mathbf{H} = mathbf{J}_{tot}
'(D)' The Lorentz Force. Electric fields created by convection, induction, and by charges.
:
abla phi
'(E)' The Electric Elasticity Equation
:
'(F)' Ohm's Law
:
'(G)' Gauss's Law
:$$
abla cdot mathbf{D} =
ho
'(H)' Equation of Continuity of Charge
:$$
abla cdot mathbf{J} = -rac{partial
ho}{partial t}
Notation
: $mathbf\{H\}$ is the magnetic field, which Maxwell called the "magnetic intensity".
: $mathbf\{J\}$ is the electric current density (with $mathbf\{J\}\_\{tot\}$ being the total current including displacement current).
: $mathbf\{D\}$ is the displacement field (called the "electric displacement" by Maxwell).
: $$
ho 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 the electric field (called the "electromotive force" by Maxwell, not to be confused with the scalar quantity that is now called electromotive force).
: $phi$ 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).
Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive permittivity ε and permeability μ, although he also discussed the possibility of anisotropic materials.
It is of particular interest to note that Maxwell includes a $mu\; mathbf\{v\}\; imes\; mathbf\{H\}$ term in his expression for the "electromotive force" at equation D , which corresponds to the magnetic force per unit charge on a moving conductor with velocity $mathbf\{v\}$. This means that equation D is effectively the Lorentz force. This equation first appeared at equation (77) in Maxwell's 1861 paper "On Physical Lines of Force" quite some time before Lorentz thought of it. Nowadays, the Lorentz force sits alongside Maxwell's Equations as an additional electromagnetic equation that is not included as part of the set.
When Maxwell derives the electromagnetic wave equation in his 1864 paper, he uses equation D as opposed to using Faraday's law of electromagnetic induction as in modern textbooks. Maxwell however drops the $mu\; mathbf\{v\}\; imes\; mathbf\{H\}$ term from equation D when he is deriving the electromagnetic wave equation, and he considers the situation only from the rest frame.

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 $$
abla on the line bundle has a curvature
abla^2 which is a two form that automatically satisfies and can be interpreted as a field strength. If the line bundle is trivial with flat reference connection ''d'' we can write $$
abla = mathrm{d}+old{A} and 'F' = ''d'' 'A' 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 super conducting tube. Because
of the Meissner effect the superconductor perfectly shields off the magnetic field so the
magnetic field strength is zero outside of the tube. Since there is no electric field either,
the Maxwell tensor 'F = 0' in the space time region outside the tube, during the experiment. This means by definition that the connection $$
abla is flat there.
However the connection depends on
the magnetic field through the tube since the holonomy along a non contractible curve
encircling the super conducting tube is the magnetic flux through the tube in the proper units.
This can be detected quantum mechanically with a double split 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.)''

Links to relativity

In the late 19th century, because of the appearance of a velocity,
:
in the equations, Maxwell's equations were only thought to express electromagnetism in the rest frame of the luminiferous aether (the postulated medium for light, whose interpretation was considerably debated). The symbols represent the permittivity and permeability of free space. The prevailing theory of the aether was that it was a medium that supported electromagnetic waves and that it was at rest relative to the sun, in accordance with the Copernican hypothesis. Maxwell's work suggested to the American scientist A.A. Michelson that the velocity of the earth through the stationary aether could be detected by 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. Two alternative explanations for this result were investigated. Michelson conducted experiments which sought to prove that the aether was dragged by the earth according to the Stokes aether theory. Another solution was suggested by George FitzGerald, Joseph Larmor and Hendrik Lorentz. 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 the group property of the Lorentz transformation (Poincaré 1905). This culminated in Einstein's theory of special relativity, which postulated the absence of any absolute rest frame, dismissed the aether as unnecessary, and established the invariance of Maxwell's equations in all inertial frames of reference.
The electromagnetic field equations have an intimate link with special relativity, because the equations of special relativity are derived from Maxwell's equations by the Lorentz invariance requirement. The magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities, and the same may be done with the electric field equations. Einstein motivated the special 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 frame is the magnet frame or the conductor frame.[2]
In relativity, the equations are written in an even more compact, "manifestly covariant" form, in terms of the rank-2 antisymmetric field-strength 4-tensor that unifies the electric and magnetic fields into a single object.
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 is an active area of research in particle physics.

Maxwell's equations in curved spacetime

Main articles: Maxwell's equations in curved spacetime

Traditional formulation

Matter and energy generate curvature in 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):
:,
and
:.
Here,
:
is a Christoffel symbol that characterizes the curvature of spacetime and $$
D_{gamma}
is the covariant derivative.

Formulation in terms of differential forms

The above formulation is related to the differential form formulation of the Maxwell equations as follows. We have implicitly chosen local coordinates and therefore have a basis of 1-forms in every point of the open set where the coordinates are defined. Using this basis we have:

★ The field form
:

★ The current form
:

★ the Bianchi identity
:

★ the source equation
:$mathrm\{d\}$
★ old{F} = {F^{lphaeta}}_{;lpha}sqrt{-g} , epsilon_{etagammadeltaeta}mathrm{d},x^{gamma} wedge mathrm{d},x^{delta} wedge mathrm{d},x^{eta} = old{J}

★ the continuity equation
:
Here ''g'' is as usual the determinant of the metric tensor .

See also

★ Electromagnetic wave equation

★ Sinusoidal plane-wave solutions of the electromagnetic wave equation

★ Nonhomogeneous electromagnetic wave equation

★ Interface conditions for electromagnetic fields

★ Photon dynamics in the double-slit experiment

★ Photon polarization

★ Computational electromagnetics

★ Finite-difference time-domain method

★ Jefimenko's equations

★ Moving magnet and conductor problem

★ Abraham-Lorentz force

★ Theoretical and experimental justification for the Schrödinger equation

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

References

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 Neerlandaies'', 'V', 253-78.

★ Henri Poincaré (1901) ''Science and Hypothesis''

★ Henri Poincaré (1905) "Sur la dynamique de l'electron", ''Comptes Rendues'', '140', 1504-8.
see

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

University level textbooks

Undergraduate

★ Elements of Electromagnetics (4th ed.), Sadiku, Matthew N. O., , , Oxford University Press, 2006, ISBN 0-19-5300483

★ Introduction to Electrodynamics (3rd ed.), Griffiths, David J., , , Prentice Hall, 1998, ISBN 0-13-805326-X

★ Hoffman, Banesh, 1983. ''Relativity and Its Roots''. W. H. Freeman.

★ Lounesto, Pertti, 1997. ''Clifford Algebras and Spinors''. Cambridge Univ. Press. Chpt. 8 sets out several variants of the equations, using exterior algebra and differential forms.

★ Electricity and Magnetism, Edward Mills Purcell, , , McGraw-Hill, 1985, ISBN 0-07-004908-4

★ Stevens, Charles F., 1995. ''The Six Core Theories of Modern Physics''. MIT Press. ISBN 0-262-69188-4.

★ Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.), Tipler, Paul, , , W. H. Freeman, 2004, ISBN 0-7167-0810-8

Graduate

★ J. D. Jackson, 1999. ''Classical Electrodynamics (3rd ed.)''. Wiley. ISBN 0-471-30932-X.

★ Lev Landau, 1987. ''The Classical Theory of Fields'' (Course of Theoretical Physics: Volume 2). Oxford: Butterworth-Heinemann.

★ James Clerk Maxwell, 1873. ''A Treatise on Electricity and Magnetism''. Dover. ISBN 0-486-60637-6.

★ Charles W. Misner, Kip Thorne, John Archibald Wheeler, 1973. ''Gravitation''. W.H. Freeman. ISBN 0-7167-0344-0. Sets out the equations using differential forms.

Computational techniques

★ Field Computation by Moment Methods, R. F. Harrington, , , Wiley-IEEE Press, 1993, ISBN 0-78031-014-4

★ Fast and Efficient Algorithms in Computational Electromagnetics, W. C. Chew, J.-M. Jin, E. Michielssen, and J. Song, , , Artech House Publishers, 2001, ISBN 1-58053-152-0

★ The Finite Element Method in Electromagnetics, 2nd. ed., J. Jin, , , Wiley-IEEE Press, 2002, ISBN 0-47143-818-9

★ Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed., Allen Taflove and Susan C. Hagness, , , Artech House Publishers, 2005, ISBN 1-58053-832-0

Citations

1. L. D. Landau, E. M. Lifshitz, ''The Classical Theory of Fields''
2. http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm
J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics". ''Phys. Scr.'' '74' 702-717
3. Introduction to Electrodynamics by Griffiths

External links

Modern treatments

★ Electromagnetism, B. Crowell, Fullerton College

★ Lecture series: Relativity and electromagnetism, R. Fitzpatrick, University of Texas at Austin

★ Maxwell's Equations: Electricity and Magnetism (PDF file), R. Young, Project PHYSNET

Historical

★ A Treatise on Electricity And Magnetism - Volume 1 - 1873 - Posner Memorial Collection - Carnegie Mellon University

★ A Treatise on Electricity And Magnetism - Volume 2 - 1873 - Posner Memorial Collection - Carnegie Mellon University

★ Original Maxwell Equations - Maxwell's 20 Equations in 20 Unknowns - PDF

★ On Physical Lines of Force - 1861 Maxwell's 1861 paper describing magnetic lines of Force - Predecessor to 1873 Treatise

★ A Dynamical Theory Of The Electromagnetic Field - 1865 Maxwell's 1865 paper describing his 20 Equations in 20 Unknowns - Predecessor to the 1873 Treatise

★ http://www.zpenergy.com/downloads/Maxwell_1864_1.pdf

★ http://www.zpenergy.com/downloads/Maxwell_1864_2.pdf

★ http://www.zpenergy.com/downloads/Maxwell_1864_3.pdf

★ http://www.zpenergy.com/downloads/Maxwell_1864_4.pdf

★ http://www.zpenergy.com/downloads/Maxwell_1864_5.pdf

★ http://www.zpenergy.com/downloads/Maxwell_1864_6.pdf

★ Catt, Walton and Davidson. "The History of Displacement Current". ''Wireless World'', March 1979.

Feynman’s derivation of Maxwell equations

★ Feynman's derivation of Maxwell equations and extra dimensions

Other

★ 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".