Maxwell\'s equations

For thermodynamic relations, see Maxwell relations. Maxwell's relations are a set of equations in Thermodynamics which are derivable from the definitions of the Thermodynamic potentials.

Electromagnetism

Electricity · Magnetism

Electrodynamics
· Free space · Lorentz force law · EMF · Electromagnetic induction · Faraday’s law · Displacement current · Maxwell’s equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potentials · Maxwell tensor · Eddy current ·

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In classical electromagnetism, Maxwell's equations are a set of four partial differential equations that describe the properties of the electric and magnetic fields and relate them to their sources, charge density and current density. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation Electromotive force ( emf, \mathcal{E} is a term used to characterize electrical devices such as Voltaic cells thermoelectric devices electrical Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of Displacement current is a quantity that arises in a changing electric field In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric The electromagnetic field is a physical field produced by electrically charged objects. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. The Liénard-Wiechert potential describes the electromagnetic effect of a moving Electric charge. The Maxwell Stress Tensor (also known as Maxwell's Stress Tensor is used to calculate the stresses on objects in magnetic or electrical fields An eddy current (also known as Foucault current) is an electrical phenomenon discovered by French physicist Léon Foucault in Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume. Current density is a measure of the Density of flow of a conserved charge. Maxwell used the equations to show that light is an electromagnetic wave. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. Individually, the equations are known as Gauss' law, Gauss' law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed

These four equations, together with the Lorentz force law (derived by Maxwell),^[1] are the complete set of laws of classical electromagnetism. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation

1 General formulations of Maxwell's equations
2 History
3 Details and special cases of Maxwell's equations
4 Materials and dynamics
- 4.1 Boundary conditions
5 The Heaviside versions in detail
6 Maxwell's equations in CGS units
7 Maxwell's equations and special relativity
- 7.1 Historical developments
- 7.2 Covariant formulation of Maxwell's equations
8 Maxwell's equations in terms of differential forms
- 8.1 Conceptual insight from this formulation
9 Classical electrodynamics as the curvature of a line bundle
10 Maxwell's equations in curved spacetime
- 10.1 Traditional formulation
- 10.2 Formulation in terms of differential forms
11 Footnotes and references
12 See also
13 Further reading
- 13.1 Journal articles
- 13.2 University level textbooks
14 External links

General formulations of Maxwell's equations

The equations in this section are given in SI units. Unlike the equations of mechanics (for example), Maxwell's equations are not unchanged in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places. Other than SI (used in engineering), the units commonly used are Gaussian units (based on the cgs system and considered to have some theoretical advantages over SI^[2]), Lorentz-Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). The centimetre-gram-second system ( CGS) is a system of physical units. Lorentz–Heaviside units (or Heaviside–Lorentz units for Maxwell's equations are often used in relativistic calculations Planck units are Units of measurement named after the German physicist Max Planck, who first proposed them in 1899 See below for CGS-Gaussian units. The centimetre-gram-second system ( CGS) is a system of physical units.

Two equivalent, general formulations of Maxwell's equations follow. The first separates bound charge and bound current (which arise in the context of dielectric and/or magnetized materials) from free charge and free current (the more conventional type of charge and current). In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Magnetization is defined as the quantity of Magnetic moment per unit volume A dielectric is a nonconducting substance ie an insulator. The term was coined by William Whewell in response to a request from Michael Faraday. Magnetization is defined as the quantity of Magnetic moment per unit volume In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Magnetization is defined as the quantity of Magnetic moment per unit volume This separation is useful for calculations involving dielectric or magnetized materials. The second formulation treats all charge equally, combining free and bound charge into total charge (and likewise with current). This is the more fundamental or microscopic point of view, and is particularly useful when no dielectric or magnetic material is present. More detail, and a proof that these two formulations are mathematically equivalent, are given in section 3.

Symbols in bold represent vector quantities, whereas symbols in italics represent scalar quantities. In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or The definitions of terms used in the two tables of equations are given in another table immediately following.

Table 1: Formulation in terms of free charge and current

Name	Differential form	Integral form
Gauss's law:	$\nabla \cdot \mathbf{D} = \rho_f$	$\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = Q_{f,S}$
Gauss's law for magnetism:	$\nabla \cdot \mathbf{B} = 0$	$\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$
Maxwell-Faraday equation (Faraday's law of induction):	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$	$\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t}$
Ampère's Circuital Law (with Maxwell's correction):	$\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}$	$\oint_{\partial S} \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,S} + \frac {\partial \Phi_{D,S}}{\partial t}$

Table 2: Formulation in terms of total charge and current

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:	$\nabla \cdot \mathbf{B} = 0$	$\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$
Maxwell-Faraday equation (Faraday's law of induction):	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$	$\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {\partial \Phi_{B,S}}{\partial t}$
Ampère's Circuital Law (with Maxwell's correction):	$\nabla \times \mathbf{B} = \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t}\$ $\$	$\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = I_S + \frac {\partial \Phi_{E,S}}{\partial t}$

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

Table 3: Definitions and units

Symbol	Meaning (first term is the most common)	SI Unit of Measure
$\mathbf{\nabla \cdot}$	the divergence operator	per meter (factor contributed by applying either operator)
$\mathbf{\nabla \times}$	the curl operator	per meter (factor contributed by applying either operator)
$\frac {\partial}{\partial t}$	partial derivative with respect to time	per second (factor contributed by applying the operator)
$\mathbf{E} \$	electric field	volt per meter or, equivalently, newton per coulomb
$\mathbf{B} \$	magnetic field also called the magnetic induction also called the magnetic field density also called the magnetic flux density	tesla, or equivalently, weber per square meter volt•second per square meter
$\mathbf{D} \$	electric displacement field	coulombs per square meter or, equivalently, newton per volt-meter
$\mathbf{H} \$	magnetizing field also called auxiliary magnetic field also called magnetic field intensity also called magnetic field	ampere per meter
$\epsilon_0 \$	permittivity of free space, officially the electric constant, a universal constant	farads per meter
$\mu_0 \$	permeability of free space, officially the magnetic constant, a universal constant	henries per meter, or newtons per ampere squared
$\ \rho_f \$	free charge density (not including bound charge)	coulomb per cubic meter
$\ \rho \$	total charge density (including both free and bound charge)	coulomb per cubic meter
$\oint_S \mathbf{E \cdot \mathrm{d} A}$	the flux of the electric field over any closed gaussian surface S	joule-meter per coulomb
$Q_{f,S} \$	net unbalanced free electric charge enclosed by the Gaussian surface S (not including bound charge)	coulombs
$Q_{S} \$	net unbalanced electric charge enclosed by the Gaussian surface S (including both free and bound charge)	coulombs
$\oint_S \mathbf{B \cdot \mathrm{d} A}$	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 ∂S (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
$\mathbf{J}_f$	free current density (not including bound current)	ampere per square meter
$\mathbf{J}$	total current density (including both free and bound current)	ampere per square meter
$\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l}$	line integral of the magnetic field over the closed boundary ∂S of the surface S	tesla-meter
$I_{f,S} = \int_S \mathbf{J}_f \cdot \mathrm{d} \mathbf{A}$	net free electrical current passing through the surface S (not including bound current)	amperes
$I_{S} = \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A}$	net electrical current passing through the surface S (including both free and bound current)	amperes
$\Phi_{E,S} = \int_S \mathbf{E} \cdot \mathrm{d} \mathbf{A}$	electric flux through any surface S, not necessarily closed	joule-meter per coulomb
$\Phi_{D,S} = \int_S \mathbf{D} \cdot \mathrm{d} \mathbf{A}$	flux of electric displacement field through any surface S, not necessarily closed	coulombs
$\mathrm{d}\mathbf{A}$	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

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). In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Mathematics, an operator is a function which operates on (or modifies another function cURL is a Command line tool for transferring files with URL syntax. In Mathematics, an operator is a function which operates on (or modifies another function In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can The volt (symbol V) is the SI derived unit of electric Potential difference or Electromotive force. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical The coulomb (symbol C) is the SI unit of Electric charge. It is named after Charles-Augustin de Coulomb. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges The tesla (symbol T) is the SI derived unit of Magnetic field B (which is also known as "magnetic flux density" and "magnetic In Physics, the weber (symbol Wb ˈveɪbɚ ˈwiːbɚ is the SI unit of Magnetic flux. M^2 redirects here For other uses see M². CM2 redirects here The volt (symbol V) is the SI derived unit of electric Potential difference or Electromotive force. The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units M^2 redirects here For other uses see M². CM2 redirects here In Physics, the electric displacement field (also called electrical field/flux density is a Vector field \mathbf{D} that appears in Maxwell's equations The coulomb (symbol C) is the SI unit of Electric charge. It is named after Charles-Augustin de Coulomb. M^2 redirects here For other uses see M². CM2 redirects here The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical The volt (symbol V) is the SI derived unit of electric Potential difference or Electromotive force. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges The ampere, in practice often shortened to amp, (symbol A is a unit of Electric current, or amount of Electric charge per second The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical This is about the capacitance unit of measure For the charge unit see Faraday (unit. In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also The henry (symbol H is the SI unit of Inductance. It is named after Joseph Henry (1797-1878 the American scientist who discovered electromagnetic In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses The coulomb (symbol C) is the SI unit of Electric charge. It is named after Charles-Augustin de Coulomb. CM3 redirects here If you were looking for the 3rd game in the Cooking Mama series abbreviated as CM3 see here. The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses The coulomb (symbol C) is the SI unit of Electric charge. It is named after Charles-Augustin de Coulomb. CM3 redirects here If you were looking for the 3rd game in the Cooking Mama series abbreviated as CM3 see here. In Electromagnetism, electric flux is Flux of the Electric field. A Gaussian surface is a closed two-dimensional Surface through which a Flux or Electric field is to be calculated In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic In Mathematics a closed surface (2- Manifold) is a Closed manifold of dimension two with a single Connected component. In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated Magnetic flux, represented by the Greek letter Φ ( Phi) is a measure of quantity of Magnetism, taking into account the strength and the extent of a Magnetic In Physics, the weber (symbol Wb ˈveɪbɚ ˈwiːbɚ is the SI unit of Magnetic flux. Magnetization is defined as the quantity of Magnetic moment per unit volume Current density is a measure of the Density of flow of a conserved charge. Magnetization is defined as the quantity of Magnetic moment per unit volume Current density is a measure of the Density of flow of a conserved charge. Magnetization is defined as the quantity of Magnetic moment per unit volume Magnetization is defined as the quantity of Magnetic moment per unit volume In Mathematics, a line integral (sometimes called a path integral or curve integral) is an Integral where the function to be integrated Magnetization is defined as the quantity of Magnetic moment per unit volume Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. Magnetization is defined as the quantity of Magnetic moment per unit volume Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. Magnetization is defined as the quantity of Magnetic moment per unit volume Magnetization is defined as the quantity of Magnetic moment per unit volume In Electromagnetism, electric flux is Flux of the Electric field. In Physics, the electric displacement field (also called electrical field/flux density is a Vector field \mathbf{D} that appears in Maxwell's equations In differential calculus, a differential is traditionally an Infinitesimally small change in a Variable. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. A contour line (also Level set, isopleth, isoline, isogram or isarithm) of a function of two Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material. Permittivity is a Physical quantity that describes how an Electric field affects and is affected by a Dielectric medium and is determined by the ability In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field. At the microscopic level, Maxwell's equations, ignoring quantum effects, describe fields, charges and currents in free space — but at this level of detail one must include all charges, even those at an atomic level, generally an intractable problem. In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes

History

Although James Clerk Maxwell was not the originator of these equations, he nevertheless derived three of them again independently in conjunction with his molecular vortex model of Faraday's "lines of force", along with the full version of Faraday's law of induction. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. Michael Faraday, FRS ( September 22 1791 – August 25 1867) was an English Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In doing so he made an important addition to Ampère's circuital law. In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed

Maxwell also developed Faraday's law of induction into another equation, which used to be listed as a 'Maxwell's equation' but is nowadays known as the Lorentz force law. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation ^[3]

The term Maxwell's equations

Controversy has always surrounded the term Maxwell's equations concerning the extent to which Maxwell himself was involved in these equations. The term Maxwell's equations nowadays applies to a set of four equations that were grouped together as a distinct set in 1884 by Oliver Heaviside, in conjunction with Willard Gibbs. Josiah Willard Gibbs ( February 11, 1839 &ndash April 28, 1903) was an American theoretical Physicist, Chemist

The importance of Maxwell's role in these equations lies in the correction he made to Ampère's circuital law in his 1861 paper On Physical Lines of Force. In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed He added the displacement current term to Ampère's circuital law and this enabled him to derive the electromagnetic wave equation in his later 1865 paper A Dynamical Theory of the Electromagnetic Field and demonstrate the fact that light is an electromagnetic wave. Displacement current is a quantity that arises in a changing electric field In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium A Dynamical Theory of the Electromagnetic Field which was written in the year 1864 is the third of James Clerk Maxwell 's papers concerned with Electromagnetism Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. This fact was then later confirmed experimentally by Heinrich Hertz in 1887. Heinrich Rudolf Hertz ( February 22, 1857 – January 1, 1894) was a German physicist who clarified and expanded the electromagnetic theory

The reason that these equations are called Maxwell's equations is disputed. Some say that these equations were originally called the Heaviside-Hertz equations but that Einstein for whatever reason later referred to them as the Maxwell-Hertz equations. see pages 110-112 of Nahin's book^[4]^[5]

These equations are based on the works of James Clerk Maxwell, and Heaviside made no secret of the fact that he was working from Maxwell's papers. Heaviside aimed to produce a symmetrical set of equations that were crucial as regards deriving the telegrapher's equations. The telegrapher's equations (or just telegraph equations) are a pair of linear Differential equations which describe the Voltage and current on The net result was a set of four equations, three of which had appeared in substance throughout Maxwell's previous papers, in particular Maxwell's 1861 paper On Physical Lines of Force and 1865 paper A Dynamical Theory of the Electromagnetic Field. A Dynamical Theory of the Electromagnetic Field which was written in the year 1864 is the third of James Clerk Maxwell 's papers concerned with Electromagnetism The fourth was a partial time derivative version of Faraday's law of induction that doesn't include motionally induced EMF. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of ^[6]

Of Heaviside's equations, the most important in deriving the telegrapher's equations was the version of Ampère's circuital law that had been amended by Maxwell in this 1861 paper to include what is termed the displacement current. The telegrapher's equations (or just telegraph equations) are a pair of linear Differential equations which describe the Voltage and current on In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed Displacement current is a quantity that arises in a changing electric field

Maxwell's On Physical Lines of Force (1861)

See also: Image:On Physical Lines of Force.pdf (Alternate source. )

Three of Heaviside's four equations appeared throughout Maxwell's 1861 paper On Physical Lines of Force:

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

(ii) At equation (112) we see Ampère's circuital law with Maxwell's correction. In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed It is this correction called displacement current which is the most significant aspect of Maxwell's work in electromagnetism as it enabled him to later derive the electromagnetic wave equation in his 1865 paper A Dynamical Theory of the Electromagnetic Field, and hence show that light is an electromagnetic wave. Displacement current is a quantity that arises in a changing electric field Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium A Dynamical Theory of the Electromagnetic Field which was written in the year 1864 is the third of James Clerk Maxwell 's papers concerned with Electromagnetism It is therefore this aspect of Maxwell's work which gives Heaviside's equations their full significance. (Interestingly, Kirchhoff derived the telegrapher's equations in 1857 without using displacement current. The telegrapher's equations (or just telegraph equations) are a pair of linear Differential equations which describe the Voltage and current on Displacement current is a quantity that arises in a changing electric field But he did use Poisson's equation and the equation of continuity which are the mathematical ingredients of the displacement current. Displacement current is a quantity that arises in a changing electric field Nevertheless, Kirchhoff believed his equations to be applicable only inside an electric wire and so he is not credited with having discovered that light is an electromagnetic wave).

(iii) At equation (113) we see Gauss's law.

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

Maxwell's A Dynamical Theory of the Electromagnetic Field (1865)

Main article: A Dynamical Theory of the Electromagnetic Field

Confusion over the term "Maxwell's equations" is further increased because it is also sometimes used for a set of eight equations that appeared in Part III of Maxwell's 1865 paper A Dynamical Theory of the Electromagnetic Field, entitled "General Equations of the Electromagnetic Field" [1] (page 480 of the article and page 2 of the pdf link), a confusion compounded by the writing of six of those eight equations as three separate equations (one for each of the Cartesian axes), resulting in twenty equations in twenty unknowns. A Dynamical Theory of the Electromagnetic Field which was written in the year 1864 is the third of James Clerk Maxwell 's papers concerned with Electromagnetism A Dynamical Theory of the Electromagnetic Field which was written in the year 1864 is the third of James Clerk Maxwell 's papers concerned with Electromagnetism (As noted above, this terminology is not common: Modern references to the term "Maxwell's equations" usually refer to the Heaviside restatements. )

These original eight equations are nearly identical to the Heaviside versions in substance, but they have some superficial differences. In fact, only one of the Heaviside versions is completely unchanged from these original equations, and that is Gauss's law (Maxwell's equation G below). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation A below) with Ampère's circuital law (equation C below). This amalgamation, which Maxwell himself originally made at equation (112) in his 1861 paper "On Physical Lines of Force" (see above), is the one that modifies Ampère's circuital law to include Maxwell's displacement current. Displacement current is a quantity that arises in a changing electric field

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

(A) The law of total currents

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

(B) The equation of magnetic force

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

(C) Ampère's circuital law

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

(D) Electromotive force created by convection, induction, and by static electricity. (This is in effect the Lorentz force)

$\mathbf{E} = \mu \mathbf{v} \times \mathbf{H} - \frac{\partial\mathbf{A}}{\partial t}-\nabla \phi$

(E) The electric elasticity equation

$\mathbf{E} = \frac{1}{\epsilon} \mathbf{D}$

(F) Ohm's law

$\mathbf{E} = \frac{1}{\sigma} \mathbf{J}$

(G) Gauss's law

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

(H) Equation of continuity

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

Notation: $\mathbf{H}$ is the magnetizing field, which Maxwell called the "magnetic intensity". In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges; $\mathbf{J}$ 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). In Physics, the electric displacement field (also called electrical field/flux density is a Vector field \mathbf{D} that appears in Maxwell's equations; $ρ$ 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). In Vector calculus, a vector potential is a Vector field whose curl is a given vector field; $\mathbf{E}$ is called the "electromotive force" by Maxwell. The term electromotive force is nowadays used for voltage, but it is clear from the context that Maxwell's meaning corresponded more to the modern term electric field. Electromotive force ( emf, \mathcal{E} is a term used to characterize electrical devices such as Voltaic cells thermoelectric devices electrical In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can; $Φ$ is the electric potential (which Maxwell also called "electric potential"). At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field; $σ$ is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity). Electrical conductivity or specific conductivity is a measure of a material's ability to conduct an Electric current. Electrical resistivity (also known as specific electrical resistance) is a measure of how strongly a material opposes the flow of Electric current.

It is interesting to note the $\mu \mathbf{v} \times \mathbf{H}$ term that appears in equation D. Equation D is therefore effectively the Lorentz force, similarly to equation (77) of his 1861 paper (see above). In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation

When Maxwell derives the electromagnetic wave equation in his 1865 paper, he uses equation D to cater for electromagnetic induction rather than Faraday's law of induction which is used in modern textbooks. The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of (Faraday's law itself does not appear among his equations. ) However, Maxwell drops the $\mu \mathbf{v} \times \mathbf{H}$ term from equation D when he is deriving the electromagnetic wave equation, as he considers the situation only from the rest frame. The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium

Details and special cases of Maxwell's equations

Bound charge and current

Main articles: Bound charge and Bound current

If an electric field is applied to a dielectric material, each of the molecules responds by forming a microscopic dipole -- its atomic nucleus will move a tiny distance in the direction of the field, while its electrons will move a tiny distance in the opposite direction. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Magnetization is defined as the quantity of Magnetic moment per unit volume A dielectric is a nonconducting substance ie an insulator. The term was coined by William Whewell in response to a request from Michael Faraday. The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J This is called polarization of the material. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses The distribution of charge that results from these tiny movements turn out to be identical to having a layer of positive charge on one side of the material, and a layer of negative charge on the other side -- a macroscopic separation of charge, even though all of the charges involved are "bound" to a single molecule. This is called bound charge. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Likewise, in a magnetized material, there is effectively a "bound current" circulating around the material, despite the fact that no individual charge is travelling a distance larger than a single molecule. Magnetization is defined as the quantity of Magnetic moment per unit volume In Physics, Astronomy, Chemistry, and Electrical engineering, the term magnetic moment of a system (such as a loop of Electric current

Proof that the two general formulations are equivalent

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

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

$\rho_b = -\nabla\cdot\mathbf{P}$

$\mathbf{J}_b = \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}$

$\mathbf{D} = \epsilon_0\mathbf{E} + \mathbf{P}$

$\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$

$\rho = \rho_b + \rho_f \$

$\mathbf{J} = \mathbf{J}_b + \mathbf{J}_f$

where P and M are polarization and magnetization, and ρ_b and J_b are bound charge and current, respectively. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Magnetization is defined as the quantity of Magnetic moment per unit volume Plugging in these relations, it can be easily demonstrated that the two formulations of Maxwell's equations given in Section 1 are precisely equivalent.

Constitutive relations

In order to apply Maxwell's equations (the formulation in terms of free charge and current, and D and H), it is necessary to specify the relations between D and E, and H and B. These are called constitutive relations, and correspond physically to specifying the response of bound charge and current to the field, or equivalently, how much polarization and magnetization a material acquires in the presence of electromagnetic fields. In Structural analysis, constitutive relations connect applied stresses or Forces to strains or Deformations The constitutive relation In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses Magnetization is defined as the quantity of Magnetic moment per unit volume

Case without magnetic or dielectric materials

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

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

where ε₀ and μ₀ are two universal constants, called the permittivity of free space and permeability of free space, respectively. Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also

Case of linear materials

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

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

where ε and μ are constants (which depend on the material), called the permittivity and permeability, respectively, of the material. Isotropy is uniformity in all directions Precise definitions depend on the subject area In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency Permittivity is a Physical quantity that describes how an Electric field affects and is affected by a Dielectric medium and is determined by the ability In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field.

General case

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

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

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

The strength of the fields (the case of nonlinearity, which occurs when ε and μ are functions of E and B; see, for example, Kerr and Pockels effects),
The direction of the fields (the case of anisotropy, birefringence, or dichroism; which occurs when ε and μ are second-rank tensors),
The frequency with which the fields vary (the case of dispersion, which occurs when ε and μ are functions of frequency; see, for example, Kramers–Kronig relations),
The position inside the material (the case of a nonuniform material, which occurs when ε and μ vary from point to point within the material; for example in a domained structure, heterostructure or a liquid crystal),
The history of the fields (the case of hysteresis, which occurs when ε and μ are functions of both present and past values of the fields). Nonlinear optics (NLO is the branch of Optics that describes the behaviour of Light in nonlinear media, that is media in which the dielectric polarization The Kerr effect or the quadratic electro-optic effect ( QEO effect) is a change in the Refractive index of a material in response to an Electric field The Pockels effect, or Pockels electro-optic effect produces Birefringence in an optical medium induced by a constant or varying Electric field. Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity Birefringence, or double refraction, is the decomposition of a ray of Light into two rays (the ordinary ray and the extraordinary ray Dichroic redirects here For the filter see Dichroic filter. For the glass see Dichroic glass. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency The Kramers–Kronig relations are mathematical properties connecting the real and imaginary parts of any complex function which is analytic A magnetic domain describes a region within a material which has uniform Magnetization. The Heterojunction bipolar transistor ( HBT) is an improvement of the Bipolar junction transistor (BJT that can handle signals of very high frequencies Liquid crystals are substances that exhibit a phase of matter that has properties between those of a conventional Liquid, and those of a Solid A system with hysteresis can be summarised as a system that may be in any number of states independent of the inputs to the system

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

Substituting in the constitutive relations above, Maxwell's equations in a linear material (differential form only) are:

$\nabla \cdot \mathbf{E} = \frac {\rho_f} {\epsilon}$

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

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

$\nabla \times \mathbf{B} = \mu \mathbf{J}_f + \mu \epsilon \frac{\partial \mathbf{E}} {\partial t}.$

These are formally identical to the general formulation in terms of E and B (given above), except that the permittivity of free space was replaced with the permittivity of the material (see also displacement field, electric susceptibility and polarization density), the permeability of free space was replaced with the permeability of the material (see also magnetization, magnetic susceptibility and magnetic field), and only free charges and currents are included (instead of all charges and currents). Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes Permittivity is a Physical quantity that describes how an Electric field affects and is affected by a Dielectric medium and is determined by the ability Displacement field may refer to Displacement field (mechanics Electric displacement field The electric susceptibility χe of a Dielectric material is a measure of how easily it polarizes in response to an Electric field. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field. Magnetization is defined as the quantity of Magnetic moment per unit volume In Electromagnetism the magnetic susceptibility ( Latin: susceptibilis “receptiveness” is the degree of Magnetization of a material in response In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges

Maxwell's equations in vacuum

See also: Electromagnetic wave equation and Sinusoidal plane-wave solutions of the electromagnetic wave equation

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

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

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

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

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

These equations have a solution in terms of traveling sinusoidal plane waves, with the electric and magnetic field directions orthogonal to one another and the direction of travel, and with the two fields in phase, traveling at the speed^[7]

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

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

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

which reduces to the electromagnetic wave equation due to an identity in vector calculus. The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium Sinusoidal plane-wave solutions are particular solutions to the Electromagnetic wave equation. In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which 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 electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. The current SI values for the speed of light, the electric and the magnetic constant are summarized in the following table (values from NIST:Latest value of the constants):

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	$8.854187817\ldots \times 10^{-12}$	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

Nondimensionalization and unobservability of the speed of light

Because c₀ and μ₀ have defined values (they are properties of the ideal reference state of free space), they are not subject to alteration due to experimental observation. Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical This is about the capacitance unit of measure For the charge unit see Faraday (unit. The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also The henry (symbol H is the SI unit of Inductance. It is named after Joseph Henry (1797-1878 the American scientist who discovered electromagnetic In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes 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. Nondimensionalization is the partial or full removal of units from a Mathematical equation by a suitable substitution of Variables. Experiment could in principle, however, alter the standard meter, for example, as a result of greater measurement accuracy. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International

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. In Physics, a magnetic monopole is a hypothetical particle that is a Magnet with only one pole (see Maxwell's equations for more on magnetic Except for this, the equations are symmetric under interchange of electric and magnetic field. In fact, symmetric equations can be written when all charges are zero, and this is how the wave equation is derived (see immediately above). The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium

Fully symmetric equations can also be written if one allows for the possibility of "magnetic charges" analogous to electric charges. [2] 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 (hypothetical)
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:	$\vec{\nabla} \cdot \vec{B} = 0$	$\vec{\nabla} \cdot \vec{B} = 4 \pi \rho_m$
Maxwell-Faraday equation (Faraday's law of induction):	$-\vec{\nabla} \times \vec{E} = \frac{\partial \vec{B}} {\partial t}$	$-\vec{\nabla} \times \vec{E} = \frac{\partial \vec{B}}{\partial t} + 4 \pi\vec{j}_m$
Ampère's law (with Maxwell's extension):	$\vec{\nabla} \times \vec{B} = \frac{\partial \vec{E}} {\partial t} + 4 \pi \vec{j}_e$ $\$	$\vec{\nabla} \times \vec{B} = \frac{\partial \vec{E}} {\partial t} + 4 \pi \vec{j}_e$
Note: the Bivector notation embodies the sign swap, and these four equations can be written as only one equation. Planck units are Units of measurement named after the German physicist Max Planck, who first proposed them in 1899 Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Differential geometry, a p -vector is the Tensor obtained by taking Linear combinations of the Wedge product of p

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?"

Materials and dynamics

See also: Computational electromagnetics

The fields in Maxwell's equations are generated by charges and currents. Computational electromagnetics, computational electrodynamics or electromagnetic modeling refers to the process of modeling the interaction of electromagnetic fields Conversely, the charges and currents are affected by the fields through the Lorentz force equation:

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

where q is the charge on the particle and v is the particle velocity. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation (It also should be remembered that the Lorentz force is not the only force exerted upon charged bodies, which also may be subject to gravitational, nuclear, etc. forces. ) Therefore, in both classical and quantum physics, the precise dynamics of a system form a set of coupled differential equations, which are almost always too complicated to be solved exactly, even at the level of statistical mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Mathematics simultaneous equations are a set of Equations containing multiple variables A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics ^[8] This remark applies to not only the dynamics of free charges and currents (which enter Maxwell's equations directly), but also the dynamics of bound charges and currents, which enter Maxwell's equations through the constitutive equations, as described next. In Structural analysis, constitutive relations connect applied stresses or Forces to strains or Deformations The constitutive relation

Commonly, real materials are approximated as "continuum" media with bulk properties such as the refractive index, permittivity, permeability, conductivity, and/or various susceptibilities. Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium Permittivity is a Physical quantity that describes how an Electric field affects and is affected by a Dielectric medium and is determined by the ability In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field. Electrical conductivity or specific conductivity is a measure of a material's ability to conduct an Electric current. These lead to the macroscopic Maxwell's equations, which are written (as given above) in terms of free charge/current densities and D, H, E, and B ( rather than E and B alone ) along with the constitutive equations relating these fields. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric For example, although a real material consists of atoms whose electronic charge densities can be individually polarized by an applied field, for most purposes behavior at the atomic scale is not relevant and the material is approximated by an overall polarization density related to the applied field by an electric susceptibility. History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses The electric susceptibility χe of a Dielectric material is a measure of how easily it polarizes in response to an Electric field.

Continuum approximations of atomic-scale inhomogeneities cannot be determined from Maxwell's equations alone. but require some type of quantum mechanical analysis such as quantum field theory as applied to condensed matter physics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In quantum field theory (QFT the forces between particles are mediated by other particles Condensed matter physics is the field of Physics that deals with the macroscopic physical properties of Matter. See, for example, density functional theory, Green–Kubo relations and Green's function (many-body theory). Density functional theory (DFT is a quantum mechanical theory used in Physics and Chemistry to investigate the Electronic structure (principally Green–Kubo relations give exact mathematical expression for transport coefficients in terms of integrals of time correlation functions In Many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with Correlation function, but refers specifically Various approximate transport equations have evolved, for example, the Boltzmann equation or the Fokker–Planck equation or the Navier-Stokes equations. The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of The Fokker–Planck equation describes the Time evolution of the Probability density function of the position of a particle and can be generalized to other observables The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of viscous Fluid substances such Some examples where these equations are applied are magnetohydrodynamics, fluid dynamics, electrohydrodynamics, superconductivity, plasma modeling. Magnetohydrodynamics (MHD ( magnetofluiddynamics or hydromagnetics) is the Academic discipline which studies the dynamics of electrically Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion Electrohydrodynamics (EHD, also known as electro-fluid-dynamics (EFD or electrokinetics, is the study of the dynamics of electrically conducting Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Plasma Modeling refers to solving Equations of motion that describe the state of a plasma. An entire physical apparatus for dealing with these matters has developed. A different set of homogenization methods (evolving from a tradition in treating materials such as conglomerates and laminates) are based upon approximation of an inhomogeneous material by a homogeneous effective medium^[9]^[10] (valid for excitations with wavelengths much larger than the scale of the inhomogeneity). A conglomerate (kɒnˈglɒmərət is a rock consisting of individual stones that have become cemented together A laminate is a material constructed by uniting two or more layers of material together In Physics wavelength is the distance between repeating units of a propagating Wave of a given Frequency. ^[11]^[12]^[13]^[14]

Theoretical results have their place, but often require fitting to experiment. Continuum-approximation properties of many real materials rely upon measurement,^[15] for example, ellipsometry measurements. Ellipsometry is a versatile and powerful Optical technique for the investigation of the Dielectric properties (complex Refractive index or Dielectric

In practice, some materials properties have a negligible impact in particular circumstances, permitting neglect of small effects. For example: optical nonlinearities can be neglected for low field strengths; material dispersion is unimportant where frequency is limited to a narrow bandwidth; material absorption can be neglected for wavelengths where a material is transparent; and metals with finite conductivity often are approximated at microwave or longer wavelengths as perfect metals with infinite conductivity (forming hard barriers with zero skin depth of field penetration). Nonlinear optics (NLO is the branch of Optics that describes the behaviour of Light in nonlinear media, that is media in which the dielectric polarization In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency Bandwidth is the difference between the upper and lower Cutoff frequencies of for example a filter, a Communication channel, or a Signal spectrum In Physics, absorption of electromagnetic radiation is the process by which the Energy of a Photon is taken up by matter typically the electrons of an The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across Microwaves are electromagnetic waves with Wavelengths ranging from 1 mm to 1 m or frequencies between 0 A perfect conductor is an Electrical conductor with no Resistivity. Skin depth is a measure of the distance an Alternating current can penetrate beneath the surface of a conductor.

And, of course, some situations demand that Maxwell's equations and the Lorentz force be combined with other forces that are not electromagnetic. An obvious example is gravity. Gravitation is a natural Phenomenon by which objects with Mass attract one another A more subtle example, which applies where electrical forces are weakened due to charge balance in a solid or a molecule, is the Casimir force from quantum electrodynamics. In Physics, the Casimir effect and the Casimir-Polder force are physical forces arising from a quantized field. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. ^[16]

The connection of Maxwell's equations to the rest of the physical world is via the fundamental sources of charges and currents and the forces on them, and also by the properties of physical materials.

Boundary conditions

Although Maxwell's equations apply throughout space and time, practical problems are finite and solutions to Maxwell's equations inside the solution region are joined to the remainder of the universe through boundary conditions^[17]^[18] ^[19] and started in time using initial conditions. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints In Mathematics, in the field of Differential equations an initial value problem is an Ordinary differential equation together with specified value called ^[20] In some cases, like waveguides or cavity resonators, the solution region is largely isolated from the universe, for example, by metallic walls, and boundary conditions at the walls define the fields with influence of the outside world confined to the input/output ends of the structure. A waveguide is a structure which guides waves such as Electromagnetic waves Light, or Sound waves A resonator is a device or system that exhibits Resonance or resonant behavior that is it naturally oscillates at some frequencies, called its resonance ^[21] In other cases, the universe at large sometimes is approximated by an artificial absorbing boundary,^[22]^[23]^[24] or, for example for radiating antennas or communication satellites, these boundary conditions can take the form of asymptotic limits imposed upon the solution. A perfectly matched layer ( PML) is an artificial absorbing layer for Wave equations commonly used to truncate computational regions in Numerical methods An antenna is a Transducer designed to transmit or Receive electromagnetic waves In other words antennas convert electromagnetic waves into A communications satellite (sometimes abbreviated to comsat) is an artificial Satellite stationed in space for the purposes of Telecommunications. ^[25] In addition, for example in an optical fiber or thin-film optics, the solution region often is broken up into subregions with their own simplified properties, and the solutions in each subregion must be joined to each other across the subregion interfaces using boundary conditions. An optical fiber (or fibre) is a Glass or Plastic fiber that carries Light along its length Thin-film optics is the branch of Optics that deals with very thin structured layers of different materials ^[26]^[27]^[28] Following are some links of a general nature concerning boundary value problems: Examples of boundary value problems, Sturm-Liouville theory, Dirichlet boundary condition, Neumann boundary condition, mixed boundary condition, Cauchy boundary condition, Sommerfeld radiation condition. We will use k to denote the Square root of the absolute value of \lambda In Mathematics and its applications a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855 and Joseph Liouville In Mathematics, the Dirichlet (or first type) boundary condition is a type of Boundary condition, named after Johann Peter Gustav Lejeune In Mathematics, the Neumann (or second type) boundary condition is a type of Boundary condition, named after Carl Neumann. In Mathematics, a mixed boundary condition for a Partial differential equation indicates that different Boundary conditions are used on different parts of In Mathematics, a Cauchy (pronounced "koe-she" Boundary condition imposed on an Ordinary differential equation or a Partial differential Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation as "the sources must be sources not sinks of Needless to say, one must choose the boundary conditions appropriate to the problem being solved. See also Kempel^[29] and the extraordinary book by Friedman. ^[30]

The Heaviside versions in detail

Gauss's law

Main article: Gauss's law

Gauss's law describes the relation between the electric field and the distribution of electric charge, as follows:

$\nabla \cdot \mathbf{D} = \rho_f \ .$

The formulation of Table 1 is assumed; that is, ρ_f is the "free" electric charge density (in units of C/m³), not including bound charge from the polarization of a material, and $\mathbf{D}$ is the electric displacement field (in units of C/m²). In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric The coulomb (symbol C) is the SI unit of Electric charge. It is named after Charles-Augustin de Coulomb. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses In Physics, the electric displacement field (also called electrical field/flux density is a Vector field \mathbf{D} that appears in Maxwell's equations For stationary charges in vacuum, the force exerted upon one point charge by another as found from Gauss's law is Coulomb's law. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form

The equivalent integral form (by the divergence theorem) of Gauss' law is:

$\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A} = Q_\mathrm{enclosed}\$

where:

S is any fixed, closed surface,

The integral is a surface integral, i. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought e. $\mathrm{d}\mathbf{A}$ is a vector whose magnitude is the area of a differential square on the closed surface A, and whose direction is an outward-facing normal vector, and

Q enclosed

is the free charge enclosed within the surface S. (If the surface itself is charged, that gives an extra contribution weighted by a factor 1/2. )

In a linear, isotropic, homogeneous material, with instantaneous response to field changes, $\mathbf{D}$ is directly related to the electric field $\mathbf{E}$ via a material-dependent constant called the permittivity, $ε$ :

$\mathbf{D} = \varepsilon \mathbf{E}$ . Isotropy is uniformity in all directions Precise definitions depend on the subject area Permittivity is a Physical quantity that describes how an Electric field affects and is affected by a Dielectric medium and is determined by the ability

The material permittivity $ε$ can also be written as ε₀ ε_r where ε_r is the material's relative permittivity or its dielectric constant. Measurement The relative static permittivity εr can be measured for static Electric fields as follows first the Capacitance of a test No material (except free space) is precisely linear and isotropic, but many materials are approximately so. In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes The permittivity of free space, or electric constant, is denoted as $ε 0$ (approximately 8. Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical 854 p F/m), 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). pico- (symbol p) is a prefix denoting a factor of 10-12 in the International System of Units (SI This is about the capacitance unit of measure For the charge unit see Faraday (unit. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International The volt (symbol V) is the SI derived unit of electric Potential difference or Electromotive force. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International The formulation of Table 2 is assumed. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric

Some insight into Gauss' law is found using the Maxwell-Faraday equation:

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

which shows the solenoidal portion of E is determined by the time variation of the magnetic field. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric In Vector calculus a solenoidal vector field (also known as an incompressible vector field) is a Vector field v with Divergence zero Thus, in electrostatics (that is, when the system is unchanging in time), by Helmholtz decomposition the E-field can be expressed in terms of a scalar field as:

$\mathbf{E} (\mathbf{r} ) = -\nabla \phi(\mathbf{r}) \ .$

Time independence not only allows E to be expressed as a gradient, but also removes any time-delay in material response (ε independent of time), so the equation determining the electrostatic potential ɸ (r ) is:

$\nabla \cdot \mathbf{D} (\mathbf{r}) = -\nabla \cdot \left( \epsilon ( \mathbf{r} ) \nabla \phi ( \mathbf{r} ) \right) = \rho_f ( \mathbf{r} ) \ ,$

which is Poisson's equation in the case where ε is independent of position (that is, when the material is homogeneous). Electrostatics is the branch of Science that deals with the Phenomena arising from what seems to be stationary Electric charges Since Classical In Mathematics, in the area of Vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field In Mathematics, Poisson's equation is a Partial differential equation with broad utility in Electrostatics, Mechanical engineering and Theoretical The formulation of Table 1 is assumed. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric That is, the bound charge is subsumed under the permittivity, and only the free charge is explicit on the right side of the equation.

Gauss's law for magnetism

Main article: Gauss's law for magnetism

"Gauss's law for magnetism" states that the divergence of the magnetic field is always zero (in other words, the magnetic field is a solenoidal vector field):

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

where $\mathbf{B}$ is the magnetic B-field (in units of tesla, denoted "T"), also called "magnetic flux density", "magnetic induction", or simply "magnetic field". In Vector calculus a solenoidal vector field (also known as an incompressible vector field) is a Vector field v with Divergence zero In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges The tesla (symbol T) is the SI derived unit of Magnetic field B (which is also known as "magnetic flux density" and "magnetic It is interpreted as saying there is no "magnetic" charge that is the analog of the electric charge, and often this equation is referred to as "the absence of magnetic monopoles". In Physics, a magnetic monopole is a hypothetical particle that is a Magnet with only one pole (see Maxwell's equations for more on magnetic Differently put, the basic entity for magnetism is the magnetic dipole, which orients itself in a magnetic field. In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a

By the divergence theorem, the above divergence equation has an equivalent integral form:

$\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0 \ ,$

where $\mathrm{d}\mathbf{A}$ is an infinitesimal vector corresponding to the area of a differential square on the surface S with an outward facing surface normal defining its direction. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

Like the electric field's integral form, this equation works only 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, the above reference to this law as saying there are no magnetic monopoles.

The Maxwell-Faraday equation

Main article: Faraday's law of induction

The Maxwell-Faraday equation states:^[31]

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

This equation is usually referred to as "Faraday's law of induction", but in fact it is only a restricted form of Faraday's law; for example, it doesn't apply to situations involving motionally induced EMF. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of

The Maxwell-Ampère equation

Main articles: Ampère's circuital law and Displacement current

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 ρ_q. In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed Displacement current is a quantity that arises in a changing electric field In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges The ampere, in practice often shortened to amp, (symbol A is a unit of Electric current, or amount of Electric charge per second The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International In Electromagnetism, permeability is the degree of Magnetization of a material that responds linearly to an applied Magnetic field. The second term on the right hand side of Ampère's Circuital Law is known as the displacement current. In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed Displacement current is a quantity that arises in a changing electric field

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. In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed

Maxwell used the displacement current in conjunction with the original eight equations in his 1865 paper A Dynamical Theory of the Electromagnetic Field to derive a wave equation that has the velocity of light. A Dynamical Theory of the Electromagnetic Field which was written in the year 1864 is the third of James Clerk Maxwell 's papers concerned with Electromagnetism Most modern textbooks derive this electromagnetic wave equation using the 'Heaviside Four'. The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of Electromagnetic waves through a medium

In free space, the permeability μ is the magnetic constant, μ₀, which is defined to be exactly 4π×10^-7 Wb/A•m. The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also In Physics, the weber (symbol Wb ˈveɪbɚ ˈwiːbɚ is the SI unit of Magnetic flux. The ampere, in practice often shortened to amp, (symbol A is a unit of Electric current, or amount of Electric charge per second The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International Also, the permittivity becomes the electric constant ε₀, also a defined quantity. Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical Thus, in free space, the equation becomes:

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

Using Stokes theorem the equivalent integral form can be found:

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

C is the edge of the open surface A (any surface with the curve C as its edge will do), and I_encircled is the current encircled by the curve C (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 Differential geometry, Stokes' theorem is a statement about the integration of Differential forms which generalizes several Theorems from Sometimes this integral form of Ampere-Maxwell Law is written as:

$\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 (I_\mathrm{enc} + I_\mathrm{d,enc})$ because the term $\varepsilon_0 \int_S \frac{\partial \mathbf{E}}{\partial t} \cdot \mathrm{d} \mathbf{A}$

is displacement current. Displacement current is a quantity that arises in a changing electric field The displacement current concept was Maxwell's greatest innovation in electromagnetic theory. It implies that a magnetic field appears during the charge or discharge of a capacitor. 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. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed

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:

$\nabla \cdot \mathbf{D} = 4\pi\rho_f$

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

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

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

Where c is the speed of light in a vacuum. The centimetre-gram-second system ( CGS) is a system of physical units. 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. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges Magnetization is defined as the quantity of Magnetic moment per unit volume In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation 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} \$ .

Maxwell's equations and special relativity

Main article: Classical electromagnetism and special relativity

Maxwell's equations have a close relation to special relativity: Not only were Maxwell's equations a crucial part of the historical development of special relativity, but also, special relativity has motivated a compact mathematical formulation Maxwell's equations, in terms of covariant tensors. The theory of Special relativity plays an important role in the modern theory of Classical electromagnetism. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial For other uses of "covariant" or "contravariant" see Covariance and contravariance.

Historical developments

Main article: History of special relativity

Maxwell's electromagnetic wave equation only applied in what he believed to be the rest frame of the luminiferous medium because he didn't use the vXB term of his equation (D) when he derived it. The History of special relativity consists of many theoretical and empirical results of physicists like Hendrik Lorentz and Henri Poincaré, which culminated in the Maxwell's idea of the luminiferous medium was that it comprised of aethereal vortices aligned solenoidally along their rotation axes.

The American scientist A. A. Michelson set out to determine the velocity of the earth through the luminiferous medium aether using a light wave interferometer that he had invented. When the Michelson-Morley experiment was conducted by Edward Morley and Albert Abraham Michelson in 1887, it produced a null result for the change of the velocity of light due to the Earth's motion through the hypothesized aether. The Michelson–Morley experiment, one of the most important and famous experiments in the History of physics, was performed in 1887 by Albert Michelson and Edward Williams Morley ( January 29, 1838 - February 24 1923) was an American Scientist famous for the Michelson-Morley Albert Abraham Michelson ( December 19, 1852 &ndash May 9, 1931) was a Polish - American Physicist known Generally a null result is a result which is Null (nothing that is the proposed result is absent This null result was in line with the theory that was proposed in 1845 by George Stokes which suggested that the aether was entrained with the Earth's orbital motion. Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist

Hendrik Lorentz objected to Stokes' aether drag model and in along with George FitzGerald and Joseph Larmor, he suggested another approach. Hendrik Antoon Lorentz ( July 18, 1853 &ndash February 4, 1928) was a Dutch Physicist who shared the 1902 Nobel George Francis FitzGerald ( 3 August 1851 &ndash 21 February 1901) was an Irish professor of "natural and experimental philosophy" Sir Joseph Larmor ( 11 July 1857 Magheragall, County Antrim, Northern Ireland – 19 May 1942 Holywood 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. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Poincaré (1900) analyzed the coordination of moving clocks by exchanging light signals. He also established mathematically the group property of the Lorentz transformation (Poincaré 1905).

This culminated in Albert Einstein's revolutionary theory of special relativity, which postulated the absence of any absolute rest frame, dismissed the aether as unnecessary (a bold idea, which did not come to Lorentz nor to Poincaré), and established the invariance of Maxwell's equations in all inertial frames of reference, in contrast to the famous Newtonian equations for classical mechanics. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects But the transformations between two different inertial frames had to correspond to Lorentz' equations and not - as former believed - to those of Galileo (called Galilean transformations). Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher The Galilean transformation is used to transform between the coordinates of two Reference frames which differ only by constant relative motion within the constructs of Newtonian ^[32] Indeed, Maxwell's equations played a key role in Einstein's famous paper on special relativity; for example, in the opening paragraph of the paper, he motivated his theory by noting that a description of a conductor moving with respect to a magnet must generate a consistent set of fields irrespective of whether the force is calculated in the rest frame of the magnet or that of the conductor. The moving magnet and conductor problem is a famous Thought experiment, originating in the 19th century concerning the intersection of Classical electromagnetism and [3]

General relativity has also had a close relationship with Maxwell's equations. For example, Kaluza and Klein showed in the 1920s that Maxwell's equations can be derived by extending general relativity into five dimensions. In Physics, Kaluza–Klein theory (or KK theory, for short is a model that seeks to unify the two fundamental forces of Gravitation and General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 This strategy of using higher dimensions to unify different forces continues to be an active area of research in particle physics. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them

Covariant formulation of Maxwell's equations

Main article: Covariant formulation of classical electromagnetism

In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form (cgs units). The covariant formulation of Classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular Maxwell's equations In relativity, a four-vector is a vector in a four-dimensional real Vector space, called Minkowski space.

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

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

(Here SI units are used; in cgs units, one would have to replace c by 1. The electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually )

The other ingredient is the four-current: $J^\alpha = (c\rho,\vec{J})$ where $ρ$ is the charge density and J is the current density. In special and General relativity, the four-current is the Lorentz covariant Four-vector that replaces the Electromagnetic Current The linear surface or volume charge density is the amount of Electric charge in a line, Surface, or Volume. Current density is a measure of the Density of flow of a conserved charge.

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

${4 \pi \over c }j^{\beta} = {\partial F^{\alpha\beta} \over {\partial x^{\alpha}} } \ \stackrel{\mathrm{def}}{=}\ \partial_{\alpha} F^{\alpha\beta} \ \stackrel{\mathrm{def}}{=}\ {F^{\alpha\beta}}_{,\alpha} \,\!$ ,

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 $\, \epsilon_{\alpha\beta\gamma\delta}$ 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. The Levi-Civita symbol, also called the Permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in Tensor The four-gradient is the Four-vector generalization of the Gradient: \partial_\alpha \ \stackrel{\mathrm{def}}{=}\ \left(\frac{1}{c} \frac{\partial}{\partial Repeated indices are summed over according to Einstein summation convention. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational We have displayed the results in several common notations. Upper and lower components of a vector, $v α$ and $v α$ 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. In Classical electromagnetism, Ampère's circuital law, discovered by André-Marie Ampère, relates the integrated Magnetic field around a closed The second equation is an expression of the two homogeneous equations, Faraday's law of induction and Gauss's law for magnetism. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of

Alternative covariant presentations of Maxwell's equations also exist, for example in terms of the four-potential; see Covariant formulation of classical electromagnetism for details. The electromagnetic four-potential is a covariant Four-vector defined in volt·seconds/meter (and in maxwell/centimeter in parentheses The covariant formulation of Classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular Maxwell's equations

Maxwell's equations in terms of differential forms

In free space, where ε = ε₀ and μ = μ₀ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. In Classical physics, free space is a concept of Electromagnetic theory, corresponding to a theoretically "perfect" Vacuum, and sometimes Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical The vacuum permeability, referred to by international standards organizations as the magnetic constant, and denoted by the symbol μ 0 (also Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In what follows, cgs units, not SI units are used, however. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS 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 $1 / 4πε 0 = 1$ . In Differential geometry, the curvature form describes Curvature of a connection on a Principal bundle. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity In Mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a Riemannian manifold or Pseudo-Riemannian In Physics, natural units are Physical units of Measurement defined in terms of universal Physical constants, such that some chosen physical Here, the 3-form J is called the "electric current form" or "current 3-form" satisfying the continuity equation

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

The current 3-form can be integrated over a 3-dimensional space-time region. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is A continuity equation is a Differential equation that describes the conservative transport of some kind of quantity The physical interpretation of this integral is the charge in that region if it is spacelike, or the amount of charge that flows through a surface in a certain amount of time if that region is a spacelike surface cross a timelike interval. As the exterior derivative is defined on any manifold, the differential form version of the Bianchi identity makes sense for any 4-dimensional manifold, whereas the source equation is defined if the manifold is oriented and has a Lorentz metric. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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.

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. The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system This article discusses the history of the principle of least action Gauge theory is a peculiar Quantum field theory where the Lagrangian is invariant under certain transformations In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold.

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. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Probability theory and Statistics, covariance is a measure of how much two variables change together (the Variance is a special case of the covariance 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. In Physics, a kinetic term is the part of the Lagrangian that is Bilinear in the fields (this does not include the mass term! (and for nonlinear sigma In the Physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees

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). In Mathematics, a line bundle expresses the concept of a line that varies from point to point of a space In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G In Mathematics, the circle group, denoted by T (or in Blackboard bold by \mathbb T is the multiplicative group of all Complex 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. In Geometry, the notion of a connection (also connexion) makes precise the idea of transporting data along a curve or family of curves in a parallel and In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry If the line bundle is trivial with flat reference connection d we can write $\nabla = \mathrm{d}+\bold{A}$ and F = dA with A the 1-form composed of the electric potential and the magnetic vector potential. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field The magnetic potential provides a mathematical way to define a Magnetic field in Classical electromagnetism.

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. The Aharonov-Bohm effect, sometimes called the Ehrenberg-Siday-Aharonov-Bohm effect is a quantum mechanical phenomenon by which a charged particle is affected by In this experiment, a static magnetic field runs through a long magnetic wire (e. g. an Fe wire magnetized longitudinally). Outside of this wire the magnetic induction is zero, in contrast to the vector potential, which essentially depends on the magnetic flux through the cross-section of the wire and does not vanish outside. Since there is no electric field either, the Maxwell tensor F = 0 throughout the space-time region outside the tube, during the experiment. This means by definition that the connection $\nabla$ is flat there.

However, as mentioned, the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the tube is the magnetic flux through the tube in the proper units. In Differential geometry, the holonomy of a connection on a Smooth manifold is a general geometrical consequence of the Curvature of the connection 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. )

Maxwell's equations in curved spacetime

Main article: Maxwell's equations in curved spacetime

Traditional formulation

Matter and energy generate curvature of spacetime. In Physics, Maxwell's equations in curved spacetime govern the dynamics of the Electromagnetic field in curved Spacetime (where the metric may SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS This is the subject of general relativity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 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. In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. (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. In Mathematics and Physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900 are coordinate-space expressions for the Levi-Civita

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. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric The equivalence of the more traditional general relativistic formulation using the covariant derivative with the differential form formulation can be seen as follows. Choose local coordinates $x α$ which gives a basis of 1-forms $d x α$ in every point of the open set where the coordinates are defined. Using this basis and cgs units we define

The antisymmetric infinitesimal field tensor $F αβ$ , corresponding to the field 2-form F

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

The current-vector infinitesimal 3-form J

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

Here g is as usual the determinant of the metric tensor $g αβ$ . A small computation that uses the symmetry of the Christoffel symbols (i. In Mathematics and Physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900 are coordinate-space expressions for the Levi-Civita 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

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

Footnotes and references

^ The Lorentz force law was actually derived by Maxwell under the name of "Equation for Electromotive Force" and was one of an earlier set of eight Maxwell's equations. In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W See the "History" section.
^ David J Griffiths (1999). Introduction to electrodynamics, Third Edition, Prentice Hall, pp. 559-562. ISBN 013805326X.
^ Ironically it is an equation which Maxwell himself was absolutely responsible for even though it doesn't count as a "Maxwell's equation". This extra equation appeared in an original list of eight Maxwell's equations in his 1865 paper entitled A Dynamical Theory of the Electromagnetic Field. A Dynamical Theory of the Electromagnetic Field which was written in the year 1864 is the third of James Clerk Maxwell 's papers concerned with Electromagnetism Maxwell derived it from Faraday's law when Lorentz was still a young boy and he termed it the equation of electromotive force.
^ Oliver Heaviside ((2001) Facsimile of 1893 Edition). Electromagnetic theory. Adamant Media Corporation, Vol. 1. ISBN 1402172982.
^ Oliver Heaviside ((2007) Facsimile of 1912 Edition). Electromagnetic theory. Cosimo Classics, Vol. 3. ISBN 1602062625.
^ In this article, this version is termed the Maxwell-Faraday equation to keep clear the distinction from Faraday's law of induction. Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of
^ See NIST Special Publication 330, Appendix 2, p. 45 : "Current practice is to use c₀ to denote the speed of light in vacuum (ISO 31). "
^ These complications show there is merit in separating the Lorentz force from the main four Maxwell equations. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation The four Maxwell's equations express the fields' dependence upon current and charge, setting apart the calculation of these currents and charges. As noted in this subsection, these calculations may well involve the Lorentz force only implicitly. Separating these complicated considerations from the Maxwell's equations provides a useful framework.
^ D. E. Aspnes, "Local-field effects and effective-medium theory: A microscopic perspective," Am. J. Phys. 50, p. 704-709 (1982).
^ Habib Ammari & Hyeonbae Kang (2006). Inverse problems, multi-scale analysis and effective medium theory : workshop in Seoul, Inverse problems, multi-scale analysis, and homogenization, June 22-24, 2005, Seoul National University, Seoul, Korea. Providence RI: American Mathematical Society. ISBN 0821839683.
^ O. C. Zienkiewicz, Robert Leroy Taylor, J. Z. Zhu, Perumal Nithiarasu (2006). The Finite Element Method, Sixth Edition, Oxford UK: Butterworth-Heinemann, pp. 550 ff. ISBN 0750663219.
^ See, e. g. : N. Bakhvalov and G. Panasenko, Homogenization: Averaging Processes in Periodic Media (Kluwer: Dordrecht, 1989); V. V. Jikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer: Berlin, 1994).
^ Vitaliy Lomakin, Steinberg BZ, Heyman E, & Felsen LB (2003). "Multiresolution Homogenization of Field and Network Formulations for Multiscale Laminate Dielectric Slabs". IEEE Transactions on Antennas and Propagation 51 (10): 2761 ff.
^ AC Gilbert (Ronald R Coifman, Editor) (2000). Topics in Analysis and Its Applications: Selected Theses. Singapore: World Scientific Publishing Company, 155. ISBN 9810240945.
^ Edward D. Palik & Ghosh G (1998). Handbook of Optical Constants of Solids. London UK: Academic Press. ISBN 0125444222.
^ F Capasso, JN Munday, D. Iannuzzi & HB Chen Casimir forces and quantum electrodynamical torques: physics and nanomechanics
^ Peter Monk (2003). Finite Element Methods for Maxwell's Equations. Oxford UK: Oxford University Press, pp. 1 ff. ISBN 0198508883.
^ Thomas B. A. Senior & John Leonidas Volakis (1995). Approximate Boundary Conditions in Electromagnetics. London UK: Institution of Electrical Engineers, pp. 261 ff. ISBN 0852968493.
^ T Hagstrom (Björn Engquist & Gregory A. Kriegsmann, Eds. ) (1997). Computational Wave Propagation. Berlin: Springer, pp. 1 ff. ISBN 0387948740.
^ Henning F. Harmuth & Malek G. M. Hussain (1994). Propagation of Electromagnetic Signals. Singapore: World Scientific, p. 17. ISBN 9810216890.
^ S. F. Mahmoud (1991). Electromagnetic Waveguides: Theory and Applications applications. London UK: Institution of Electrical Engineers, Chapter 2. ISBN 0863412327.
^ Jean-Michel Lourtioz (2005). Photonic Crystals: Towards Nanoscale Photonic Devices. Berlin: Springer, pp. 84. ISBN 354024431X.
^ S. G. Johnson, Notes on Perfectly Matched Layers, online MIT course notes (Aug. 2007).
^ Taflove A & Hagness S C (2005). Computational Electrodynamics: The Finite-difference Time-domain Method. Boston MA: Artech House, Chapters 6 & 7. ISBN 1580538320.
^ David M Cook (2003). The Theory of the Electromagnetic Field. Mineola NY: Courier Dover Publications, pp. 335 ff. ISBN 0486425673.
^ Korada Umashankar (1989). Introduction to Engineering Electromagnetic Fields. Singapore: World Scientific, §10. 7; pp. 359ff. ISBN 9971509210.
^ Joseph V. Stewart (2001). Intermediate Electromagnetic Theory. Singapore: World Scientific, Chapter III, pp. 111 ff Chapter V, Chapter VI. ISBN 9810244703.
^ Tai L. Chow (2006). Electromagnetic theory. Sudbury MA: Jones and Bartlett, pp. 333ff and Chapter 3: pp. 89ff. ISBN 0-7637-3827-1.
^ John Leonidas Volakis, Arindam Chatterjee & Leo C. Kempel (1989). Finite element method for electromagnetics : antennas, microwave circuits, and scattering applications. New York: Wiley IEEE, pp. 79 ff. ISBN 0780334256.
^ Bernard Friedman (1990). Principles and Techniques of Applied Mathematics. Mineola NY: Dover Publications. ISBN 0486664449.
^ The term Maxwell-Faraday equation frequently is replaced by Faraday's law of induction or even Faraday's law. These last two terms have multiple meanings, so Maxwell-Faraday equation is used here to avoid confusion.
^ Details can be found in U. Krey, A. Owen, Basic Theoretical Physics - A Concise Overview, Springer, Berlin and elsewhere, 2007, ISBN 978-3-540-36804-5

External links

Modern treatments

Electromagnetism, B. Crowell, Fullerton College
Lecture series: Relativity and electromagnetism, R. Fitzpatrick, University of Texas at Austin
Electromagnetic waves from Maxwell's equations on Project PHYSNET.
MIT Video Lecture Series (30 x 50 minute lectures)- Electricity and Magnetism Taught by Professor Walter Lewin. Walter H G Lewin is currently a professor of Physics at the Massachusetts Institute of Technology (MIT

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 Faraday's Lines of Force - 1855/56 Maxwell's first paper (Part 1 & 2) - Compiled by Blaze Labs Research (PDF)
On Physical Lines of Force - 1861 Maxwell's 1861 paper describing magnetic lines of Force - Predecessor to 1873 Treatise
Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society. )
Catt, Walton and Davidson. "The History of Displacement Current". Wireless World, March 1979.

Feynman’s derivation of Maxwell equations

Feynman's derivation of Maxwell equations and extra dimensions

Other

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Contents

General formulations of Maxwell's equations

Table 1: Formulation in terms of free charge and current

Table 2: Formulation in terms of total charge and current

Table 3: Definitions and units

History

The term Maxwell's equations

Maxwell's On Physical Lines of Force (1861)

Maxwell's A Dynamical Theory of the Electromagnetic Field (1865)

Details and special cases of Maxwell's equations

Bound charge and current

Proof that the two general formulations are equivalent

Constitutive relations

Case without magnetic or dielectric materials

Case of linear materials

General case

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

Maxwell's equations in vacuum

Nondimensionalization and unobservability of the speed of light

With magnetic monopoles

Materials and dynamics

Boundary conditions

The Heaviside versions in detail

Gauss's law

Gauss's law for magnetism

The Maxwell-Faraday equation

The Maxwell-Ampère equation

Maxwell's equations in CGS units

Maxwell's equations and special relativity

Historical developments

Covariant formulation of Maxwell's equations

Maxwell's equations in terms of differential forms

Conceptual insight from this formulation

Classical electrodynamics as the curvature of a line bundle

Maxwell's equations in curved spacetime

Traditional formulation

Formulation in terms of differential forms

Footnotes and references

See also

Further reading

Journal articles

University level textbooks

Undergraduate

Graduate

Older classics

Computational techniques

External links

Modern treatments

Historical

Feynman’s derivation of Maxwell equations

Other