Maxwell's equations

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In electromagnetism, Maxwell's equations are a set of four equations that were first presented as a distinct group in 1884 by Oliver Heaviside in conjunction with Willard Gibbs. These equations had appeared throughout James Clerk Maxwell's 1861 paper entitled On Physical Lines of Force.

Those equations describe the interrelationship between electric field, magnetic field, electric charge, and electric current. Although Maxwell himself was the originator of only one of these equations (by virtue of modifying an already existing equation), he derived them all again independently in conjunction with his molecular vortex model of Faraday's "lines of force".


Contents

[edit] History

Maxwell's equations are a set of four equations originally appearing separately in Maxwell's 1861 paper On Physical Lines of Force as equation (54) Faraday's law, equation (56) div B = 0, equation (112) Ampère's law with Maxwell's correction, and equation (113) Gauss's law. They express respectively how changing magnetic fields produce electric fields, the experimental absence of magnetic monopoles, how electric currents and changing electric fields produce magnetic fields (Ampère's circuital law with Maxwell's correction), and how electric charges produce electric fields.

The most significant aspect of Maxwell's work in electromagnetism is the term he introduced into Ampère's circuital law; the time derivative of the electric field, known as Maxwell's displacement current.

In Maxwell's 1865 paper, A Dynamical Theory of the Electromagnetic Field Maxwell's modified version of Ampère's circuital law enabled him to derive the electromagnetic wave equation, hence demonstrating that light is an electromagnetic wave.

Apart from Maxwell's amendment to Ampère's circuital law, none of these equations was original. Maxwell however uniquely re-derived them hydrodynamically and mechanically using his vortex model of Faraday's lines of force.

In 1884 Oliver Heaviside, in conjunction with Willard Gibbs, grouped these equations together and restated them in modern vector notation. It is important however to note that in doing so, Heaviside used partial time derivative notation as opposed to the total time derivative notation used by Maxwell at equation (54). The consequence of this is that we lose the vXB term that appeared in Maxwell's follow up equation (77). Nowadays, the vXB term sits beside the group known as Maxwell's equations and bears the name Lorentz Force.

This whole matter is confused because the term Maxwell's equations is also used for a set of eight equations in Maxwell's 1865 paper, A Dynamical Theory of the Electromagnetic Field, and this confusion is compounded by the fact that six of those eight equations are each written as three separate equations (one for each of the Cartesian axes), hence allowing even Maxwell to refer to them as twenty equations in twenty unknowns.

The two sets of Maxwell's equations are nearly physically equivalent, although the vXB term at equation (D) of the original eight is absent from the modern Heaviside four. The Maxwell-Ampère equation in Heaviside's restatement is an amalgamation of two equations in the set of eight that Maxwell published in his 1865 paper.

[edit] Summary of the modern Heaviside versions

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

[edit] Case without dielectric or magnetic materials

The Equations are given in SI units. See below for CGS units.

Name Differential form Integral form
Gauss's law: \nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon_0} \oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac {Q_S}{\epsilon_0}
Gauss's 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_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {d \Phi_{B,S}}{dt}
Ampère's Circuital Law
(with Maxwell's correction):		 \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}} {\partial t} \oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 I_S + \mu_0 \epsilon_0 \frac {d \Phi_{E,S}}{dt}

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

Symbol Meaning (first term is the most common) SI Unit of Measure
\nabla \cdot the divergence operator per meter (factor contributed by applying either operator)
\nabla \times the curl operator
\frac {\partial}{\partial t} 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
\ \rho \ electric charge density coulomb per cubic meter
ε0 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
QS 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 over any closed surface S tesla meter-squared or weber
\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
\Phi_{B,S} = \int_S \mathbf{B} \cdot \mathrm{d} \mathbf{A} magnetic flux over any surface S (not necessarily closed) weber
μ0 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
I_S = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} net electrical current passing through the surface S amperes
\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[1]), 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} \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.

[edit] 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:

χe is the electrical susceptibility of the material,

χm 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 \mathbf{D} = \rho_{\text{free}}
\nabla \cdot \mathbf{B} = 0
\nabla \times \mathbf{E} = - \frac{\partial \mathbf{B}} {\partial t}
\nabla \times \mathbf{H} = \mathbf{J}_{\text{free}} + \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).

[edit] 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{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{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}

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[2]

c_0 = \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \ .

The travelling wave solution is found by substitution of one of the curl equations into the other, producing:

\nabla \times \left( \nabla \times \mathbf{E} \right ) =\nabla \times \left( - \frac{\partial\mathbf{B}} {\partial t} \right ) = \ \ - \mu_0\varepsilon_0 \frac{\partial ^{\ 2} \mathbf{E}} {\partial t ^2} \ ,

which reduces to the electromagnetic wave equation due to an identity in vector calculus. The equation is satisfied in one dimension, for example, by a solution of the form E = E( x − c0t ), that is, by a solution that is unchanged when t advances to t + Δt at a position x that advances to x + c0 Δt.

Maxwell discovered that this quantity c0 is 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:[3]

Symbol Name Numerical Value SI Unit of Measure Type
c_0 \ Speed of light in vacuum 2.99792458 \times 10^8 meters per second defined
\ \varepsilon_0 electric constant 8.854187817\ \times 10^{-12} farads per meter derived   \ \stackrel{\mathrm{def}}{=}\ \frac {1} {\mu_0 {c_0}^2 }
\ \mu_0 \ magnetic constant 4 \pi \times 10^{-7} henries per meter defined

Because these constants have defined values, they are not subject to alteration due to experimental observation. For example, if length is measured in units λ and time in units τ, the distance x in units of λ becomes x = λ ζ and the time t becomes t = τ η, where ζ is the number of length units in x and η is the number of time units in t. The above curl equation for the travelling wave becomes (see nondimensionalization):

\nabla_{\xi} \times \left( \nabla_{\xi} \times \mathbf{E} \right ) = \ \ - \left(\frac {\lambda}{c_0 \tau} \right)^2 \frac{\partial ^{\ 2} \mathbf{E}} {\partial \eta ^2} \ ,

and because the SI units are related by λ = c0τ this equation does not depend any longer on the speed of light. Experiment could in principle, however, alter the standard meter, for example, as a result of greater measurement accuracy.

[edit] 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. 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.

Fully symmetric equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges.[1] With the inclusion of a variable for these magnetic charges, say \rho_m \,, there will also be "magnetic current" variable in the equations, \vec{J}_m \,. The extended Maxwell's equations, simplified by nondimensionalization, are as follows:

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

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

[edit] The Heaviside versions in detail

[edit] 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/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 Qenclosed is the free charge enclosed by the surface. (Here it is assumed that the surface itself is uncharged; otherwise there is an extra contribution weighted by a factor 1/2.)

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 ε0, 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), ρt is the total charge density (including bound charges), and ε0 (approximately 8.854 pF/m) is the permittivity of free space. ε can also be written as \varepsilon_0 \varepsilon_r, where εr is the material's relative permittivity or its dielectric constant.

Compare Poisson's equation.

[edit] 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.

[edit] Faraday's law of electromagnetic induction

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

The equivalent integral form is (according to Stokes' Theorem):

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

where

\scriptstyle \mathbf{E} is the electric field,

\scriptstyle C=\partial S is the boundary of the surface S.

(Precisely, Stokes' theorem would only lead to - \int_S \frac{\partial\mathbf B}{\partial t} \cdot \mathrm{d}\mathbf{A}, but then relativistic invariance comes into play and allows to pull the differential in front of the integral, in agreement with the observation that Faraday's induction voltage also appears if the magnetic field does not change at all, but only the integration surface S. In the differential version the relativistic invariance can only be seen from the whole set of equations. More details can be found, for example, in U. Krey, A. Owen: Basic Theoretical Physics - A Concise Overview, Berlin and elsewhere, Springer 2007.)


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).

[edit] 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 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} = \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 ρq. 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 magnetic constant, μ0, which is defined to be exactly 4π×10-7 Wb/A•m. Also, the permittivity becomes the electric constant ε0, also a defined quantity. 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 Iencircled 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. 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 predicted wave velocity was the same as the measured velocity of light, Maxwell asserted that light waves are electromagnetic waves.

Here again, due to relativistic invariance, the expression \int\frac{\partial}{\partial t} ... can be replaced by \frac{\mathrm d}{\mathrm dt} \left ( \int ... \right ).

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.

[edit] Maxwell's original eight equations

In Part III of A Dynamical Theory of the Electromagnetic Field, entitled "General Equations of the Electromagnetic Field" [2] (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. Today, however, references to Maxwell's equations usually refer to the Heaviside restatements. Heaviside's versions of Maxwell's equations actually contain only one of the original eight, Gauss's law (Maxwell's equation G). 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 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 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) 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 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).
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.

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 in Maxwell's set.

When Maxwell derives the electromagnetic wave equation in his 1864 paper, he uses equation D rather than Faraday's law of electromagnetic induction, which is used in modern textbooks. However, Maxwell 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.

[edit] 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{D} = 4\pi\rho
\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}

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}

χm for vacuum is zero and therefore:

\mathbf{B} = \mathbf{H}

and in the ferro or ferri magnetic materials where χm 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} \ .

[edit] Formulation of Maxwell's equations in special relativity

Main article: 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. Upper and lower components of a vector, vα and vα respectively, are interchanged with the fundamental matrix g, e.g., g=diag(+1,-1,-1,-1).

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 homogeneous equations, Faraday's law of induction and the absence of magnetic monopoles.

It has been suggested that the vXB component of the Lorentz force can be derived from Coulomb's law and special relativity if one assumes invariance of electric charge.[4][5]

[edit] 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 natural coordinate and metric independent differential operator acting on forms — and the source equation

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

where the (dual) Hodge star operator * is a linear transformation from the space of 2 forms to the space of 4-2 forms defined by the metric in Minkowski space (in four dimensions even by any metric conformal to this metric), and the fields are in natural units where 1 / 4πε0 = 1. Here, the 3-form J is called the "electric current" or "current (3-)form" satisfying the continuity equation

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

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

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

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

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

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

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

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

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

the constitutive relation takes the form

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

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

C_{pq}^{mn} = 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.

[edit] 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.

[edit] 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 superconducting 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 superconducting 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.)

[edit] Links to relativity

In the late 19th century, because of the appearance of a velocity,

c_0=\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. The prevailing theory of the aether was that it was a medium that supported electromagnetic waves. 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 force is calculated in the rest frame of the magnet or that of the conductor.[3]

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:

F = \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)

Here mksA units are used; in cgs units, one would have to replace c by 1.

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.

[edit] Maxwell's equations in curved spacetime

Main article: Maxwell's equations in curved spacetime

[edit] 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 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 Dγ is the covariant derivative.

[edit] 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 xα which gives a basis of 1-forms dxα in every point of the open set where the coordinates are defined. Using this basis and cgs units we define

  • The anti symmetric field tensor Fαβ
\bold{F} := \frac{1}{2}F_{\alpha\beta} \,\mathrm{d}\,x^{\alpha} \wedge \mathrm{d}\,x^{\beta}
  • The current vector Jα
\bold{J} := {4 \pi \over c } J^{\alpha} \sqrt{-g} \, \epsilon_{\alpha\beta\gamma\delta} \mathrm{d}\,x^{\beta} \wedge \mathrm{d}\,x^{\gamma} \wedge \mathrm{d}\,x^{\delta}

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

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

[edit] See also

[edit] References

[edit] Journal articles

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

[edit] University level textbooks

[edit] Undergraduate

  • Sadiku, Matthew N. O. (2006). Elements of Electromagnetics (4th ed.). Oxford University Press. ISBN 0-19-5300483. 
  • 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.
  • 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.
  • Edward Mills Purcell (1985). Electricity and Magnetism. McGraw-Hill. ISBN 0-07-004908-4. 
  • Stevens, Charles F., 1995. The Six Core Theories of Modern Physics. MIT Press. ISBN 0-262-69188-4.
  • Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8. 
  • Schwarz, Melvin (1987). Principles of Electrodynamics. Dover Publications. ISBN 0-486-65493-1. 
  • Ulaby, Fawwaz T. (2007). Fundamentals of Applied Electromagnetics (5th ed.). Pearson Education, Inc.. ISBN 0-13-241326-4. 
  • Krey, U., Owen, A. (2007), Basic Theoretical Physics - A Concise Overview, esp. part II, Springer, ISBN 978-3-540-36804-5

[edit] Graduate

[edit] Computational techniques

[edit] Footnotes

  1. ^ Introduction to Electrodynamics by Griffiths
  2. ^ Current practice is to use c0 to denote the speed of light in vacuum ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose.
  3. ^ NIST:Latest value of the constants
  4. ^ Cite error: Invalid <ref> tag; no text was provided for refs named Landau
  5. ^ Cite error: Invalid <ref> tag; no text was provided for refs named Field

[edit] External links

[edit] Modern treatments

[edit] Historical

[edit] Feynman’s derivation of Maxwell equations

[edit] Other

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