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Maxwell's equations

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   Electromagnetism
   Electricity · Magnetism
           Electrostatics
   Electric charge
   Coulomb's law
   Electric field
   Gauss's law
   Electric potential
   Electric dipole moment
           Magnetostatics
   Ampère's law
   Magnetic field
   Magnetic dipole moment
          Electrodynamics
   Electric current
   Lorentz force law
   Electromotive force
   (EM) Electromagnetic induction
   Faraday-Lenz law
   Displacement current
   Maxwell's equations
   (EMF) Electromagnetic field
   (EM) Electromagnetic radiation
         Electrical Network
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   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.

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 Lorentz force that appears as equation (D) in the
   original eight is the solution to Faraday's law of electromagnetic
   induction that appears in the 'Heaviside Four', and 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

   Name Differential form Integral form
   Gauss's law: \nabla \cdot \mathbf{D} = \rho \oint_S \mathbf{D} \cdot
   \mathrm{d}\mathbf{A} = q = \int_V \rho\, \mathrm{d}V
   Gauss' law for magnetism
   (absence of magnetic monopoles): \nabla \cdot \mathbf{B} = 0 \oint_S
   \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0
   Faraday's law of induction: \nabla \times \mathbf{E} = -\frac{\partial
   \mathbf{B}} {\partial t} \oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l}
   = - \int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d}
   \mathbf{A}
   Ampère's Circuital Law
   (with Maxwell's extension): \nabla \times \mathbf{H} = \mathbf{J} +
   \frac{\partial \mathbf{D}} {\partial t} \oint_C \mathbf{H} \cdot
   \mathrm{d}\mathbf{l} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
   \int_S \frac{\partial\mathbf{D}}{\partial t} \cdot \mathrm{d}
   \mathbf{A}

   The following table provides the meaning of each symbol and the SI unit
   of measure:
   Symbol Meaning SI Unit of Measure
   \mathbf{E} electric field volt per meter or, equivalently,
   newton per coulomb
   μ magnetic permeability of the medium henries per meter, or newtons per
   ampere squared
   i[N] net electrical current enclosed by an Amperian line amperes
   q net electric charge enclosed by the Gaussian surface coulombs
   \mathbf{H} magnetic field
   also called the auxiliary field ampere per meter
   \mathbf{D} electric displacement field
   also called the electric flux density coulomb per square meter
   \mathbf{B} magnetic flux density
   also called the magnetic induction
   also called the magnetic field tesla, or equivalently,
   weber per square meter
   \ \rho \ free electric charge density,
   not including dipole charges bound in a material coulomb per cubic
   meter
   \mathbf{J} free current density,
   not including polarization or magnetization currents bound in a
   material ampere per square meter
   \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}V \ differential element of volume V enclosed by surface S
   cubic meters
   \mathrm{d} \mathbf{l} differential vector element of path length
   tangential to contour C enclosing surface S meters
   \nabla \cdot the divergence operator per meter
   \nabla \times the curl operator per meter

   Although SI units are given here for the various symbols, Maxwell's
   equations are unchanged in many systems of units (and require only
   minor modifications in all others). The most commonly used systems of
   units are SI, used for engineering, electronics and most practical
   physics experiments, and Planck units (also known as "natural units"),
   used in theoretical physics, quantum physics and cosmology. An older
   system of units, the cgs system, is also used.

   In order to complete the theory of electromagnetism we need to add
   another equation to Heaviside's group of 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} \times \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{P} = \chi_e \varepsilon_0 \mathbf{E}

          \mathbf{M} = \chi_m \mathbf{H}

   and the \mathbf{D} and \mathbf{B} fields are related to \mathbf{E} and
   \mathbf{H} by:

          \mathbf{D} \ \ = \ \ \varepsilon_0 \mathbf{E} + \mathbf{P} \ \ =
          \ \ (1 + \chi_e) \varepsilon_0 \mathbf{E} \ \ = \ \ \varepsilon
          \mathbf{E}

          \mathbf{B} \ \ = \ \ \mu_0 ( \mathbf{H} + \mathbf{M} ) \ \ = \ \
          (1 + \chi_m) \mu_0 \mathbf{H} \ \ = \ \ \mu \mathbf{H}

   where:

   χ[e] is the electrical susceptibility of the material,

   χ[m] is the magnetic susceptibility of the material,

   \varepsilon is the electrical permittivity of the material, and

   μ 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

          \nabla \cdot \varepsilon \mathbf{E} = \rho

          \nabla \cdot \mu \mathbf{H} = 0

          \nabla \times \mathbf{E} = - \mu \frac{\partial \mathbf{H}}
          {\partial t}

          \nabla \times \mathbf{H} = \mathbf{J} + \varepsilon
          \frac{\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 ε[0] and
   μ[0] (neglecting very slight nonlinearities due to quantum effects).

          \mathbf{D} = \varepsilon_0 \mathbf{E}

          \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:

          \nabla \cdot \mathbf{E} = 0

          \nabla \cdot \mathbf{H} = 0

          \nabla \times \mathbf{E} = - \mu_0 \frac{\partial\mathbf{H}}
          {\partial t}

          \nabla \times \mathbf{H} = \ \ \varepsilon_0 \frac{\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

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

   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
   currently accepted values for the speed of light, the permittivity, and
   the permeability are summarized in the following table:
   Symbol Name Numerical Value SI Unit of Measure Type
   c \ Speed of light 2.99792458 \times 10^8 meters per second defined
   \ \varepsilon_0 Permittivity 8.85419 \times 10^{-12} farads per meter
   derived
   \ \mu_0 \ Permeability 4 \pi \times 10^{-7} henries per meter defined

The Difference between the Magnetic Induction Vector B and the Magnetic Field
Vector H

   The difference between the \mathbf{B} vector and the \mathbf{H} vector
   can be traced back to Maxwell's 1855 paper entitled 'On Faraday's Lines
   of Force'. 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

   \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 ρ is electric charge density and v is movement. \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.

   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{D} is to \mathbf{E} . Ie. \mathbf{B}
   parallels with \mathbf{D} , whereas \mathbf{H} parallels with
   \mathbf{E} .

The Heaviside Versions in Detail

(1) Gauss's Law

   Gauss's law yields the sources (and sinks) of electric charge.

          \nabla \cdot \mathbf{D} = \rho

   where ρ is the free electric charge density (in units of C/m^3), not
   including dipole charges bound in a material, and \mathbf{D} is the
   electric displacement field (in units of C/m^2). 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[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, ε:

          \mathbf{D} = \varepsilon \mathbf{E} .

   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
   ε[0], and appears in:

          \nabla \cdot \mathbf{E} = \frac{\rho_t}{\varepsilon_0}

   where, again, \mathbf{E} is the electric field (in units of V/m), ρ[t]
   is the total charge density (including bound charges), and ε[0]
   (approximately 8.854 pF/m) is the permittivity of free space. ε can
   also be written as \varepsilon_0 \varepsilon_r , where ε[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.

          \nabla \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

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

   The equivalent integral form is:

          \oint_{C} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = -
          \frac{\mathrm{d}}{\mathrm{d} t}\int_{S} \mathbf{B} \cdot
          \mathrm{d}\mathbf{A}

   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 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. 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,

          \nabla \times \mathbf{H} = \mathbf{J} + \frac {\partial
          \mathbf{D}} {\partial t}

   where \mathbf{H} is the magnetic field strength (in units of A/m),
   related to the magnetic flux \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} = \rho_q\mathbf{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 that 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,
   μ[0], which is defined to be exactly 4π×10^-7 Wb/A•m. Also, the
   permittivity becomes the permittivity of free space ε[0]. Thus, in free
   space, the equation becomes:

          \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0
          \frac{\partial \mathbf{E}}{\partial t}

   Equivalent integral form:

          \oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0
          I_\mathrm{encircled} + \mu_0\varepsilon_0 \int_S \frac{\partial
          \mathbf{E}}{\partial t} \cdot \mathrm{d} \mathbf{A}

   \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: \begin{matrix}I_{\mathrm{through}\ A} = \int_S
   \mathbf{J}\cdot \mathrm{d}\mathbf{A}\end{matrix} ). 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

          \varepsilon_0 \int_S \frac{\partial \mathbf{E}}{\partial t}
          \cdot \mathrm{d} \mathbf{A}

   is sometimes called displacement current

   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:

          \nabla \cdot \mathbf{E} = 4\pi\rho

          \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} + \frac{4\pi}{c} \mathbf{J}

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

          \nabla \cdot \mathbf{E} = 0

          \nabla \cdot \mathbf{B} = 0

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

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

   In this system of units the relation between magnetic induction,
   magnetic field and total magnetization take the form:

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

   With the linear approximation:

          \mathbf{B} = (\ 1 + 4\pi\chi_m\ )\mathbf{H}

   χ[m] for vacuum is zero and therefore:

          \mathbf{B} = \mathbf{H}

   and in the ferro or ferri magnetic materials where χ[m] is much bigger
   than 1:

          \mathbf{B} = 4\pi\chi_m\mathbf{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} \ .

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):

          {4 \pi \over c }J^{\beta} = {\partial F^{\alpha\beta} \over
          {\partial x^{\alpha}} } \ \stackrel{\mathrm{def}}{=}\
          \partial_{\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} \
          \stackrel{\mathrm{def}}{=}\ {F_{\alpha\beta}}_{,\gamma} +
          {F_{\gamma\alpha}}_{,\beta} +{F_{\beta\gamma}}_{,\alpha} \
          \stackrel{\mathrm{def}}{=}\ \epsilon_{\delta\alpha\beta\gamma}
          {F^{\beta\gamma}}_{,\alpha}

   where \, J^{\alpha} is the 4-current, \, F^{\alpha\beta} is the field
   strength tensor, \, \epsilon_{\alpha\beta\gamma\delta} is the
   Levi-Civita symbol, and

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

   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

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

   where d denotes the exterior derivative - a differential operator
   acting on forms - and the source equation

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

   where the (dual) Hodge star operator * is a linear transformation from
   the space of 2 forms to the space of 4-2 forms defined by the metric in
   Minkowski space (or in four dimensions by its conformal class), and the
   fields are in natural units where 1 / 4πε[0] = 1. Here, the 3-form J is
   called the "electric current" or " current (3-)form" satisfying the
   continuity equation

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

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

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

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

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

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

          \bold{F} = F_{pq}\bold{\theta}^p\wedge\bold{\theta}^q .

   the constitutive relation takes the form

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

   where the field coefficient functions are antisymmetric in the indices
   and the constitutive coefficients are antisymmetric in the
   corresponding pairs. 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" (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

          \mathbf{J}_{tot} = \mathbf{J} +
          \frac{\partial\mathbf{D}}{\partial t}

   (B) Definition of the Magnetic Vector Potential

          \mu \mathbf{H} = \nabla \times \mathbf{A}

   (C) Ampère's Circuital Law

          \nabla \times \mathbf{H} = \mathbf{J}_{tot}

   (D) The Lorentz Force. Electric fields created by convection,
   induction, and by charges.

          \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 of Charge

          \nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\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).
          ρ 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).
          φ is the electric potential (which Maxwell also called "electric
          potential").
          σ 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} \times \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} \times \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 \nabla on the line bundle has a curvature \bold{F} =
   \nabla^2 which is a two form that automatically satisfies
   \mathrm{d}\bold{F} = 0 and can be interpreted as a field strength. If
   the line bundle is trivial with flat reference connection d we can
   write \nabla = \mathrm{d}+\bold{A} and F = 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 \nabla 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,

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

   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. When the Michelson-Morley experiment, conducted by Edward
   Morley and Albert Abraham Michelson, produced a null result for the
   change of the velocity of light due to the Earth's motion through the
   hypothesized ether, however, alternative explanations were sought 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) analysed 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: the magnetic field equations can be derived from
   consideration of the transformation of the electric field equations
   under relativistic transformations at low velocities. 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.

   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

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):

          { 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[γ] 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 xα and therefore have a basis of 1-forms d xα in every
   point of the open set where the coordinates are defined. Using this
   basis we have:
     * The field form

          \bold{F} = F_{\alpha\beta} \,\mathrm{d}\,x^{\alpha} \wedge
          \mathrm{d}\,x^{\beta}

     * The current form

          \bold{J} = {4 \pi \over c } J^{\alpha} \sqrt{-g} \,
          \epsilon_{\alpha\beta\gamma\delta} \mathrm{d}\,x^{\beta} \wedge
          \mathrm{d}\,x^{\gamma} \wedge \mathrm{d}\,x^{\delta}

     * the Bianchi identity

          \mathrm{d}\bold{F} = 2(\partial_{\gamma} F_{\alpha\beta} +
          \partial_{\beta} F_{\gamma\alpha} + \partial_{\alpha}
          F_{\beta\gamma})\mathrm{d}\,x^{\alpha}\wedge
          \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} = 0

     * the source equation

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

     * the continuity equation

          \mathrm{d}\bold{J} = { 4 \pi \over c } {J^{\alpha}}_{;\alpha}
          \sqrt{-g} \,
          \epsilon_{\alpha\beta\gamma\delta}\mathrm{d}\,x^{\alpha}\wedge
          \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} \wedge
          \mathrm{d}\,x^{\delta} = 0

   Here g is as usual the determinant of the metric tensor gαβ.

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