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> In Electromagnetism , Maxwell's equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell . They describe the interrelationship between electric fields, magnetic fields, electric charge, and electric current. Although Maxwell himself was not the originator of the individual equations, he derived them again independently in conjunction with his molecular vortex model of Faraday's lines of force, and he was the person who first grouped these equations all together into a coherent set. Most importantly, he introduced an extra term to Ampère's Circuital Law . This extra term is the time derivative of Electric Field and is known as Maxwell's Displacement Current . Maxwell's modified version of Ampère's Circuital Law enables the set of equations to be combined together to derive the Electromagnetic Wave Equation . Although Maxwell's equations were known before special relativity, they can be derived from Coulomb's Law and Special Relativity if one assumes Invariance Of Electric Charge . L. D. Landau , E. M. Lifshitz , ''The Classical Theory of Fields''http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics". ''Phys. Scr.'' 74 702-717 This in turn leads to a very interesting parallel with gravity in that the same reasoning can be applied to Newton's law of gravitation leading to a gravitational equivalent of Maxwell's equations. See '' Gravitomagnetism '' for more information. HISTORY OF MAXWELL'S EQUATIONS Maxwell's equations are a set of four equations that can all be found at various places in Maxwell's 1861 paper On Physical Lines of Force . They express (i) how Electric Charge s produce Electric Field s ( Gauss's Law ), (ii) the experimental absence of Magnetic Monopole s, (iii) how Electric Current s and changing electric fields produce Magnetic Field s ( Ampère's Circuital Law ), and (iv) how changing magnetic fields produce electric fields ( Faraday's Law Of Induction ). Apart from Maxwell's amendment to Ampère's Circuital Law , none of these equations are original. However, Maxwell uniquely re-derived them hydrodynamically and mechanically using his vortex model of Faraday's lines of force. In the year 1884 Oliver Heaviside selected these four equations, and in conjunction with Willard Gibbs , he put them into modern vector notation. This gives rise to the claim by some scientists that Maxwell's equations are in actual fact Heaviside's equations. This matter is further confused by the fact that the term 'Maxwell's Equations' is also used to describe a set of eight equations labelled (A) to '''(H)''' in Maxwell's 1865 paper '' A Dynamical Theory Of The Electromagnetic Field ''. It therefore helps when referring to 'Maxwell's Equations' to specify whether we are talking about the original eight equations or the modified 'Heaviside Four'. The two sets of equations are physically equivalent to all intents and purposes although Gauss's Law is the only actual equation that appears in both sets. The Maxwell/Ampère equation in the 'Heaviside Four' is an amalgamation of two equations in the original eight. SUMMARY OF THE MODERN HEAVISIDE VERSIONS Symbols in bold represent Vector quantities, whereas symbols in ''italics'' represent Scalar quantities. General case The Equations are given in SI Units . See Below for CGS Units . The following table provides the meaning of each symbol and the SI unit of measure: The equations are given here in SI units. Unlike the equations of mechanics (for example), Maxwell's equations are not unchanged in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places. For example, the electric field and the magnetic field have the same unit (gauss) in the Gaussian system. Other than SI (used in engineering), the units commonly used are Gaussian Units (based on the cgs system and considered to have some theoretical advantages over SIIntroduction to Electrodynamics by Griffiths), Lorentz-Heaviside Units (used mainly in particle physics) and Planck Units (used in theoretical physics). In order to connect the theory of classical electrodynamics to mechanics we need to add another equation to the four Maxwell's Equations. The force exerted upon a charged particle by the Electric Field and Magnetic Field is given by the Lorentz Force equation: : where is the charge on the particle and is the particle velocity. This is slightly different when expressed in the cgs system of units below. This extra equation appeared in cartesian format as equation (D) of the original eight 'Maxwell's Equations'. Maxwell's equations are generally applied to ''macroscopic averages'' of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo Quantum Mechanical effects as well). It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material. Below the microscopic, Maxwell's equations, ignoring quantum effects, are simply those of a vacuum — but one must include all atomic charges and so on, which is generally an intractable problem. In linear materials In linear materials, the polarization density (in coulombs per square meter) and magnetization density (in amperes per meter) are given by: : : and the and fields are related to and by: : : where: is the Electrical Susceptibility of the material, is the Magnetic Susceptibility of the material, is the electrical Permittivity of the material, and is the magnetic Permeability of the material (This can actually be extended to handle nonlinear materials as well, by making ε and '''μ''' depend upon the field strength; see e.g. the Kerr and Pockels Effect s.) In non-dispersive, isotropic media, ε and μ are time-independent scalars, and Maxwell's equations reduce to : : : : In a uniform (homogeneous) medium, ε and μ are constants independent of position, and can thus be furthermore interchanged with the spatial derivatives. More generally, ε and μ can be rank-2 Tensor s (3×3 Matrices ) describing Birefringent (anisotropic) materials. Also, although for many purposes the time/frequency-dependence of these constants can be neglected, every real material exhibits some Material Dispersion by which ε and/or μ depend upon Frequency (and causality constrains this dependence to obey the Kramers-Kronig Relations ). In vacuum, without charges or currents The vacuum is a linear, homogeneous, isotropic, dispersionless medium, and the proportionality constants in the vacuum are denoted by ε0 and μ0 . : : Since there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space: : : : : These equations have a solution in terms of travelling sinusoidal plane waves, with the electric and magnetic field directions orthogonal to one another and the direction of travel, and with the two fields in phase, travelling at the speed : Maxwell Discovered that this quantity ''c'' is simply the Speed Of Light In Vacuum , and thus that light is a form of Electromagnetic Radiation . The current SI values for the speed of light, the electric and the magnetic constant are summarized in the following table: THE HEAVISIDE VERSIONS IN DETAIL (1) Gauss's Law Gauss's law yields the sources (and sinks) of electric charge. : where is the ''free'' electric charge density (in units of C/m3), not including dipole charges bound in a material, and is the Electric Displacement Field (in units of C/m2). The solution to Gauss's Law is Coulomb's Law for stationary charges in vacuum. The equivalent integral form (by the Divergence Theorem ), also known as Gauss' Law , is: : where is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and is the free charge enclosed by the surface. In a ''linear material'', is directly related to the electric field via a material-dependent constant called the Permittivity , : :. Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as , and appears in: : where, again, is the electric field (in units of V/m), is the total charge density (including bound charges), and (approximately 8.854 pF/m) is the permittivity of free space. can also be written as , where is the material's relative permittivity or its '' Dielectric Constant ''. Compare Poisson's Equation . (2) The Divergence of the Magnetic Field The divergence of a magnetic field is always zero and hence magnetic field lines are solenoidal. : is the magnetic flux density (in units of teslas, T), also called the magnetic induction. Equivalent integral form: : is the area of a differential square on the surface with an outward facing surface normal defining its direction. Like the electric field's integral form, this equation only works if the integral is done over a closed surface. This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, this is a mathematical formulation of the statement that there are no Magnetic Monopole s. (3) Faraday's Law of Electromagnetic Induction : The equivalent integral form is (according to Stoke's Theorem ): : where is the electric field, is the boundary of the surface . If a conducting wire, following the contour , is introduced into the field, the so-called Electromotive Force in this wire is equal to the value of these integrals (over the fields in absence of the wire!). The negative sign was established experimentally by Faraday in 1831, a common modern textbook interpretation is that it is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's Law . This equation relates the electric and magnetic fields, but it also has a number of practical applications. In circuit theory it takes the form of the relationship of induced voltage due to a changing current in an inductance, sometimes called a reverse or back emf. This equation describes how Electric Motor s and Electric Generator s work. Specifically, it demonstrates that a voltage can be generated by varying the magnetic flux passing through a given area over time, such as by uniformly rotating a loop of wire through a fixed magnetic field. In a motor or generator, the fixed excitation is provided by the Field circuit and the varying voltage is measured across the Armature circuit. In some types of motors/generators, the field circuit is mounted on the rotor and the armature circuit is mounted on the stator, but other types of motors/generators reverse the configuration. Maxwell's equations apply to a right-handed coordinate system. To apply them unmodified to a left handed system would reverse the polarity of magnetic fields (not inconsistent, but confusingly against convention). (4) Ampère's Circuital Law Ampère's Circuital Law describes the source of the magnetic field, : where is the is known as the Displacement Current . It was Maxwell who added the displacement current term to Ampère's Circuital Law at equation (112) in his 1861 paper On Physical Lines of Force . This addition means that either Maxwell's original eight equations, or the modified Heaviside four equations can be combined together to obtain the electromagnetic wave equation. Maxwell used the displacement current in conjunction with the original eight equations in his 1864 paper '' A Dynamical Theory Of The Electromagnetic Field '' to derive the Electromagnetic Wave Equation in a much more cumbersome fashion than that which is employed when using the 'Heaviside Four'. Most modern textbooks derive the electromagnetic wave equation using the 'Heaviside Four'. In free space, the permeability μ is the permeability of free space, μ0, which is defined to be ''exactly'' 4π×10-7 Wb/A•m. Also, the permittivity becomes the permittivity of free space ε0. Thus, in free space, the equation becomes: : Equivalent integral form: : '''' is the edge of the open surface ''A'' (any surface with the curve '''' as its edge will do), and ''I''encircled is the current encircled by the curve '''' (the current through any surface is defined by the equation: ). In some situations, this integral form of Ampere-Maxwell Law appears in: : for : is sometimes called Displacement Current . The displacement current concept was Maxwell's greatest innovation in electromagnetic theory. It states that a magnetic field appears during the charge or discharge of a capacitor. With this concept, and the Faraday law equation, Maxwell was able to derive the wave equations, and by showing that the prediced wave velocity was the same as the measured velocity of light, Maxwell asserted that light waves are electromagnetic waves. If the Electric Flux Density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's Law . MAXWELL'S EQUATIONS IN CGS UNITS The above equations are given in the International System Of Units , or SI for short. In a related unit system, called cgs (short for Centimeter-gram-second ), the equations take the following form: : : : : Where ''c'' is the speed of light in a vacuum. For the electromagnetic field in a vacuum, the equations become: : : : : In this system of units the relation between magnetic induction, Magnetic Field and total Magnetization take the form: : With the linear approximation: : for vacuum is zero and therefore: : and in the ferro or ferri magnetic materials where is much bigger than 1: : The force exerted upon a charged particle by the Electric Field and Magnetic Field is given by the Lorentz Force equation: : where is the charge on the particle and is the particle velocity. This is slightly different from the SI -unit expression above. For example, here the magnetic field has the same units as the electric field . FORMULATION OF MAXWELL'S EQUATIONS IN SPECIAL RELATIVITY See Also: Formulation of Maxwell's equations in special relativity In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwell's equations are written in terms of Four-vector s and tensors in the "manifestly covariant" form (cgs units): :, and : where is the 4-current , is the Field Strength Tensor , is the Levi-Civita Symbol , and : is the 4-gradient . Repeated indices are summed over according to Einstein Summation Convention . We have displayed the results in several common notations. The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and Ampere's law with Maxwell's correction. The second equation is an expression of the homogenous equations, Faraday's law of induction and the absence of Magnetic Monopole s. MAXWELL'S EQUATIONS IN TERMS OF DIFFERENTIAL FORMS In a Vacuum , where ε and μ are constant everywhere, Maxwell's equations simplify considerably once the language of Differential Geometry and Differential Form s is used. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional Spacetime manifold. Maxwell's equations then reduce to the Bianchi Identity : where d denotes the Exterior Derivative - a differential operator acting on forms - and the source equation
: As the exterior derivative is defined on any Manifold , this formulation of electromagnetism works for any 4-dimensional oriented manifold with a Lorentz metric, e.g. on the curved space-time of general relativity. In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call : the constitutive transformation. The role of this transformation is comparable to the Hodge duality transformation. The Maxwell equations in the presence of matter then become: : : where the current 3-form J still satisfies the continuity equation dJ= 0. When the fields are expressed as linear combinations (of Exterior Product s) of basis forms , :. the constitutive relation takes the form : where the field coefficient functions are antisymmetric in the indices and the constitutive coefficients are antisymmetric in the corresponding pairs. The Hodge duality transformation leading to the vacuum equations discussed above are obtained by taking : which up to scaling is the only invariant tensor of this type that can be defined with the metric. In this formulation, electromagnetism generalises immediately to any 4 dimensional oriented manifold or with small adaptations any manifold, requiring not even a metric. Thus the expression of Maxwell's equations in terms of differential forms leads to a further notational simplification. Whereas Maxwell's Equations could be written as two tensor equations instead of eight scalar equations, from which the propagation of electromagnetic disturbances and the continuity equation could be derived with a little effort, using differential forms leads to an even simpler derivation of these results. The price one pays for this simplification, however, is a need for knowledge of more technical mathematics. Conceptual insight from this formulation On the conceptual side, from a point of view of physics, this shows that the second and third Maxwell equations should be grouped together, be called the homogeneous ones, and be seen as geometric ''identities'' expressing nothing else that the ''field'' F derives from a more "fundamental" ''potential'' '''A''', while the first and last one should be seen as the dynamical ''equations of motion'', obtained via the Lagrangian principle of Least Action , from the "interaction term" '''A J''' (introduced through Gauge Covariant Derivative s), coupling the field to matter. Often, the time derivative in the third law motivates calling this equation "dynamical", which is somewhat misleading; in the sense of the preceding analysis, this is rather an artifact of breaking and Fadeev-Popov Ghosts . THE ORIGINAL EIGHT MAXWELL'S EQUATIONS In Part III of . The eight original Maxwell's equations will now be listed in modern vector notation, (A) The Law of Total Currents : (B) Definition of the Magnetic Vector Potential : (C) Ampère's Circuital Law : (D) The Lorentz Force. Electric fields created by convection, induction, and by charges. : (E) The Electric Elasticity Equation : (F) Ohm's Law : (G) Gauss's Law : (H) Equation of Continuity of Charge : Notation : is the Magnetic Field , which Maxwell called the "magnetic intensity". : is the electric current density (with being the total current including displacement current). : is the Displacement Field (called the "electric displacement" by Maxwell). : is the free charge density (called the "quantity of free electricity" by Maxwell). : is the magnetic Vector Potential (called the "angular impulse" by Maxwell). : is the Electric Field (called the "electromotive force" by Maxwell, not to be confused with the scalar quantity that is now called Electromotive Force ). : is the Electric Potential (which Maxwell also called "electric potential"). : is the Electrical Conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the Resistivity ). Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive Permittivity ε and Permeability μ, although he also discussed the possibility of anisotropic materials. It is of particular interest to note that Maxwell includes a term in his expression for the "electromotive force" at equation D , which corresponds to the magnetic force per unit charge on a moving conductor with as an additional electromagnetic equation that is not included as part of the set. When Maxwell derives the Electromagnetic Wave Equation in his 1864 paper, he uses equation D as opposed to using Faraday's law of electromagnetic induction as in modern textbooks. Maxwell however drops the term from equation D when he is deriving the Electromagnetic Wave Equation , and he considers the situation only from the rest frame. CLASSICAL ELECTRODYNAMICS AS THE CURVATURE OF A LINE BUNDLE An elegant and intuitive way to formulate Maxwell's equations is to use complex Line Bundle s or Principal Bundle s with fibre U(1) . The Connection on the line bundle has a Curvature which is a two form that automatically satisfies and can be interpreted as a field strength. If the line bundle is trivial with flat reference connection ''d'' we can write and F = ''d'' '''A''' with '''A''' the 1-form composed of the Electric Potential and the Magnetic Vector Potential . In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov-Bohm Effect . In this experiment, a static magnetic field runs through a long super conducting tube. Because of the Meissner Effect the superconductor perfectly shields off the magnetic field so the magnetic field strength is zero outside of the tube. Since there is no electric field either, the Maxwell tensor F = 0 in the space time region outside the tube, during the experiment. This means by definition that the connection is flat there. However the connection depends on the magnetic field through the tube since the Holonomy along a non contractible curve encircling the super conducting tube is the magnetic flux through the tube in the proper units. This can be detected quantum mechanically with a double split electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern. ''(See ''Michael Murray'', Line Bundles , ''2002 (PDF web link)'' for a simple mathematical review of this formulation. See also ''R. Bott'', On some recent interactions between mathematics and physics, ''Canadian Mathematical Bulletin, 28 (1985) )no. 2 pp 129-164.)'' LINKS TO RELATIVITY In the late 19th century, because of the appearance of a velocity, : in the equations, Maxwell's equations were only thought to express electromagnetism in the rest frame of the Luminiferous Aether (the postulated medium for light, whose interpretation was considerably debated). The symbols represent the Permittivity and Permeability of free space. The prevailing theory of the aether was that it was a medium that supported electromagnetic waves and that it was at rest relative to the sun, in accordance with the Copernican hypothesis. Maxwell's work suggested to the American scientist A.A. Michelson that the velocity of the earth through the stationary aether could be detected by a light wave interferometer that he had invented. When the Michelson-Morley Experiment , was conducted by Edward Morley and Albert Abraham Michelson in 1887, it produced a Null Result for the change of the velocity of light due to the Earth's motion through the hypothesized aether. Two alternative explanations for this result were investigated. Michelson conducted experiments which sought to prove that the aether was dragged by the earth according to the Stokes aether theory. Another solution was suggested by George FitzGerald , Joseph Larmor and Hendrik Lorentz . Both Larmor (1897) and Lorentz (1899, 1904) derived the Lorentz Transformation (so named by Henri Poincaré ) as one under which Maxwell's equations were invariant. Poincaré (1900) analyzed the coordination of moving clocks by exchanging light signals. He also established the group property of the Lorentz transformation (Poincaré 1905). This culminated in Einstein's theory of Special Relativity , which postulated the absence of any absolute rest frame, dismissed the aether as unnecessary, and established the invariance of Maxwell's equations in all inertial frames of reference. The electromagnetic field equations have an intimate link with special relativity, because the equations of special relativity are derived from Maxwell's equations by the Lorentz invariance requirement. The magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities, and the same may be done with the electric field equations. Einstein motivated the special theory by noting that a description of a Conductor Moving With Respect To A Magnet must generate a consistent set of fields irrespective of whether the frame is the magnet frame or the conductor frame. {Link without Title} In relativity, the equations Are Written in an even more compact, "manifestly Covariant " form, in terms of the rank-2 antisymmetric field-strength 4- Tensor that unifies the electric and magnetic fields into a single object. Kaluza And Klein showed in the 1920s that Maxwell's equations can be derived by extending General Relativity into five dimensions. This strategy of using higher dimensions to unify different forces is an active area of research in Particle Physics . MAXWELL'S EQUATIONS IN CURVED SPACETIME See Also: Maxwell's equations in curved spacetime Traditional formulation Matter and energy generate curvature in Spacetime . This is the subject of General Relativity . Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum will also generate curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with Covariant Derivative s. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs units): :, and :. Here, : is a Christoffel Symbol that characterizes the curvature of spacetime and is the covariant derivative. Formulation in terms of differential forms The above formulation is related to the Differential Form formulation of the Maxwell equations as follows. We have implicitly chosen local coordinates and therefore have a basis of 1-forms in every point of the open set where the coordinates are defined. Using this basis we have:
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: Here ''g'' is as usual the determinant of the metric tensor . SEE ALSO
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