FACTOID # 7: The top five best educated states are all in the Northeast.

 Home Encyclopedia Statistics States A-Z Flags Maps FAQ About

 WHAT'S NEW

SEARCH ALL

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Maxwell's equations

In electromagnetism, Maxwell's equations are a set of four equations that were first presented as a distinct group in 1884 by Oliver Heaviside in conjunction with Willard Gibbs. These equations had appeared throughout James Clerk Maxwell's 1861 paper entitled On Physical Lines of Force. Maxwells relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Oliver Heaviside (May 18, 1850 â€“ February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and... Josiah Willard Gibbs (February 11, 1839 New Haven â€“ April 28, 1903 New Haven) was one of the very first American theoretical physicists and chemists. ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance â€” eponymously named Maxwells equations â€” including an important modification (extension) of the AmpÃ¨res...

Those equations describe the interrelationship between electric field, magnetic field, electric charge, and electric current. Although Maxwell himself was the originator of only one of these equations (by virtue of modifying an already existing equation), he derived them all again independently in conjunction with his molecular vortex model of Faraday's "lines of force". 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. ... Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... Electric current is the flow (movement) of electric charge. ... Michael Faraday, FRS (September 22, 1791 â€“ August 25, 1867) was an English chemist and physicist (or natural philosopher, in the terminology of that time) who contributed to the fields of electromagnetism and electrochemistry. ...

Although Maxwell's equations were known before special relativity, they can be derived from Coulomb's law and special relativity if one assumes invariance of electric charge.[1][2] For more information, see links to relativity section. Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Charge invariance refers to the fixed electric charge of a particle, regardless of speed. ... For thermodynamic relations, see Maxwell relations. ...

Maxwell's equations are a set of four equations originally appearing separately in Maxwell's 1861 paper On Physical Lines of Force as equation (54) Faraday's law, equation (56) div B = 0, equation (112) Ampère's law with Maxwell's correction, and equation (113) Gauss's law. They express respectively how changing magnetic fields produce electric fields, the experimental absence of magnetic monopoles, how electric currents and changing electric fields produce magnetic fields (Ampère's circuital law with Maxwell's correction), and how electric charges produce electric fields. Faradays law can mean: Faradays law of induction (electromagnetic fields) Faradays law of electrolysis Category: ... An electric current produces a magnetic field. ... In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... In physics, a magnetic monopole is a hypothetical particle that may be loosely described as a magnet with only one pole (see electromagnetic theory for more on magnetic poles). ... Electric current is the flow (movement) of electric charge. ... Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... 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. ...

Maxwell introduced an extra term to Ampère's circuital law which is the time derivative of electric field and known as Maxwell's displacement current. This modification is the most significant aspect of Maxwell's work in electromagnetism. In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ... 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. ... Displacement current is a quantity related to changing electric field. ... Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...

In Maxwell's 1865 paper, A Dynamical Theory of the Electromagnetic Field Maxwell's modified version of Ampère's circuital law enabled him to derive the electromagnetic wave equation, hence demonstrating that light is an electromagnetic wave. A Dynamical Theory of the Electromagnetic Field was the third of James Clerk Maxwells papers concerned with electromagnetism. ... In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ... Lasers used for visual effects during a musical performance. ...

Apart from Maxwell's amendment to Ampère's circuital law, none of these equations were original. Maxwell however uniquely re-derived them hydrodynamically and mechanically using his vortex model of Faraday's lines of force. In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ...

In 1884 Oliver Heaviside, in conjunction with Willard Gibbs, grouped these equations together and restated them in modern vector notation. It is important however to note that in doing so, Heaviside used partial time derivative notation as opposed to the total time derivative notation used by Maxwell at equation (54). The consequence of this is that we lose the vXB term that appeared in Maxwell's follow up equation (77). Nowadays, the vXB term sits beside the group known as Maxwell's equations and bears the name Lorentz Force. Oliver Heaviside (May 18, 1850 â€“ February 3, 1925) was a self-taught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, developed techniques for applying Laplace transforms to the solution of differential equations, reformulated Maxwells field equations in terms of electric and... Josiah Willard Gibbs (February 11, 1839 â€“ April 28, 1903) was an American mathematical physicist who contributed much of the theoretical foundation that led to the development of chemical thermodynamics and was one of the founders of vector analysis. ... Lorentz force. ...

This whole matter is confused because the term Maxwell's equations is also used for a set of eight equations in Maxwell's 1865 paper, A Dynamical Theory of the Electromagnetic Field, and this confusion is yet further confused by virtue of the fact that six of those eight equations are each written as three separate equations for the x, y, and z, axes, hence allowing even Maxwell to refer to them as twenty equations in twenty unknowns. A Dynamical Theory of the Electromagnetic Field was the third of James Clerk Maxwells papers concerned with electromagnetism. ...

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

## Summary of the modern Heaviside versions

Symbols in bold represent vector quantities, whereas symbols in italics represent scalar quantities. A vector going from A to B. In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ...

### General case

The Equations are given in SI units. See below for CGS units. The International System of Units (symbol: SI) (for the French phrase Système International dUnités) is the most widely used system of units. ... CGS is an acronym for centimetre-gram-second. ...

Name Differential form Integral form
Gauss's law: $nabla cdot mathbf{E} = frac {rho} {epsilon_0}$ $oint_S mathbf{E} cdot mathrm{d}mathbf{A} = frac {mathbf{Q}_S}{epsilon_0}$
Gauss' law for magnetism
(absence of magnetic monopoles):
$oint_S mathbf{B} cdot mathrm{d}mathbf{A} = 0$
Faraday's law of induction: $nabla times mathbf{E} = -frac{partial mathbf{B}} {partial t}$ $oint_{partial S} mathbf{E} cdot mathrm{d}mathbf{l} = - frac {d mathbf{Phi}_{B,S}}{dt}$
Ampère's Circuital Law
(with Maxwell's correction):
$nabla times mathbf{B} = mu_0 mathbf{J} + mu_0 epsilon_0 frac{partial mathbf{E}} {partial t}$ $oint_{partial S} mathbf{B} cdot mathrm{d}mathbf{l} = mu_0 mathbf{I}_S + mu_0 epsilon_0 frac {d mathbf{Phi}_{E,S}}{dt}$

The following table provides the meaning of each symbol and the SI unit of measure: In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... This article is about the concept of integrals in calculus. ... In physics and mathematical analysis, Gausss law is the electrostatic application of the generalized Gausss theorem giving the equivalence relation between any flux, e. ... In physics, a magnetic monopole is a hypothetical particle that may be loosely described as a magnet with only one pole (see electromagnetic theory for more on magnetic poles). ... Faradays law of induction (more generally, the law of electromagnetic induction) states that the induced emf (electromotive force) in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. ... In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ... â€œSIâ€ redirects here. ...

Symbol Meaning (first term is the most common) SI Unit of Measure
$nabla cdot$ the divergence operator, also known as del per meter (factor contributed by applying either operator)
$nabla times$ the curl operator
$frac {partial}{partial t}$ partial derivative with respect to time per second (factor contributed by applying the operator)
electric field
also called the electric flux density
volt per meter or, equivalently,
newton per coulomb
$mathbf{B}$ Magnetic field
also called the magnetic induction
also called the magnetic field density
also called the magnetic flux density
tesla, or equivalently,
weber per square meter
$rho$ electric charge density coulomb per cubic meter
ε0 Permittivity of free space, a universal constant farads per meter
$oint_S mathbf{E} cdot mathrm{d}mathbf{A}$ The flux of the electric field over any closed Gaussian surface S joule-meter per coulomb
$mathbf{Q}_S$ net unbalanced electric charge enclosed by the Gaussian surface S, including so-called Bound charges coulombs
$oint_S mathbf{B} cdot mathrm{d}mathbf{A}$ The flux of the magnetic field over any closed surface S Tesla meter-squared or webber
$oint_{partial S} mathbf{E} cdot mathrm{d}mathbf{l}$ line integral of the electric field along the boundary (therefore necessarily a closed curve) of the surface S Joule per coulomb
$mathbf{Phi}_{B,S} = int_S mathbf{B} cdot mathrm{d} mathbf{A}$ magnetic flux over any surface S (not necessarily closed) webber
μ0 magnetic permeability of free space, a universal constant henries per meter, or newtons per ampere squared
$mathbf{J}$ current density ampere per square meter
$oint_{partial S} mathbf{B} cdot mathrm{d}mathbf{l}$ line integral of the magnetic field over the closed boundary of the surface S tesla-meter
$mathbf{I}_S = int_S mathbf{J} cdot mathrm{d} mathbf{A}$ net electrical current passing through the surface S amperes
$mathbf{Phi}_{E,S} = int_S mathbf{E} cdot mathrm{d} mathbf{A}$ Electric flux over any surface S, not necessarily closed
$mathrm{d}mathbf{A}$ differential vector element of surface area A, with infinitesimally

small magnitude and direction normal to surface S In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ... For other uses, see Curl (disambiguation). ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or 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 (as opposed to the total derivative, in which all variables are allowed to vary). ... 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. ... Josephson junction array chip developed by NIST as a standard volt. ... The metre, or meter (symbol: m) is the SI base unit of length. ... For other uses, see Newton (disambiguation). ... The coulomb (symbol: C) is the SI unit of electric charge. ... Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... SI unit. ... In physics, the weber (symbol: Wb) is the SI unit of magnetic flux. ... A square metre (US spelling: square meter) is by definition the area enclosed by a square with sides each 1 metre long. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... The coulomb (symbol: C) is the SI unit of electric charge. ... The cubic metre (symbol m³) is the SI derived unit of volume. ... Permittivity is an intensive physical quantity that describes how an electric field affects and is affected by a medium. ... The farad (symbol F) is the SI unit of capacitance (named after Michael Faraday). ... In physics, Gausss law gives the relation between the electric flux flowing out a closed surface and the charge enclosed in the surface. ... A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, ideal wire. ... Electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. ... Magnetic flux, represented by the Greek letter Î¦ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... In mathematics a closed surface (2-manifold) is a space like the sphere, the torus, the Klein bottle. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... Magnetic flux, represented by the Greek letter Î¦ (phi), is a measure of quantity of magnetism, taking account of the strength and the extent of a magnetic field. ... In electromagnetism, permeability is the degree of magnetization of a material that responds linearly to an applied magnetic field. ... The henry (symbol H) is the SI unit of inductance. ... In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ... This article is about path integrals in the general mathematical sense, and not the path integral formulation of physics which was studied by Richard Feynman. ... In electricity, current is the rate of flow of charges, usually through a metal wire or some other electrical conductor. ... In physics, Gausss law gives the relation between the electric flux flowing out a closed surface and the charge enclosed in the surface. ... The differential dy In calculus, a differential is an infinitesimally small change in a variable. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

square meters
$mathrm{d} mathbf{l}$ differential vector element of path length tangential to contour meters

The equations are given here in SI units. Unlike the equations of mechanics (for example), Maxwell's equations are not unchanged in other unit systems. Though the general form remains the same, various definitions get changed and different constants appear at different places. For example, the electric field and the magnetic field have the same unit (gauss) in the Gaussian system. Other than SI (used in engineering), the units commonly used are Gaussian units (based on the cgs system and considered to have some theoretical advantages over SI[3]), Lorentz-Heaviside units (used mainly in particle physics) and Planck units (used in theoretical physics). In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. ... The Comet Nucleus Tour (CONTOUR) was a Discovery-class space mission. ... CGS is an acronym for centimetre-gram-second. ... Lorentz-Heaviside units (or Heaviside-Lorentz units) for Maxwells equations are often used in relativistic calculations. ... In physics, Planck units are physical units of measurement defined exclusively in terms of the five universal physical constants shown in the table below in such a manner that all of these physical constants take on the numerical value of one when expressed in terms of these units. ...

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: 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. ... Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... Lorentz force. ...

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

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. For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ...

### In linear materials

In linear materials, the polarization density, $mathbf{P}$ (in coulombs per square meter), and magnetization density, $mathbf{M}$ (in amperes per meter), are given by:

$mathbf{P} = chi_e varepsilon_0 mathbf{E}$
$mathbf{M} = chi_m mathbf{H}$

and the and $mathbf{B}$ fields are related to and $mathbf{H}$ by:

$mathbf{D} = varepsilon_0 mathbf{E} + mathbf{P} = (1 + chi_e) varepsilon_0 mathbf{E} = varepsilon mathbf{E}$
$mathbf{B} = mu_0 (mathbf{H} + mathbf{M}) = (1 + chi_m) mu_0 mathbf{H} = mu mathbf{H}$

where:

χe is the electrical susceptibility of the material, The electric susceptibility Ï‡e of a dielectric material is a measure of how easily it polarizes in response to an electric field. ...

χm is the magnetic susceptibility of the material, In physics and electrical engineering, the magnetic susceptibility is the degree of magnetization of a material in response to an applied magnetic field. ...

$varepsilon$ is the electrical permittivity of the material, and 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 of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

μ is the magnetic permeability of the material In electromagnetism, permeability is the degree of magnetization of a material that responds linearly to an applied magnetic field. ...

(This can actually be extended to handle nonlinear materials as well, by making ε and μ depend upon the field strength; see e.g. the Kerr and Pockels effects.) The Kerr effect or the quadratic electro-optic effect is a change in the refractive index of a material in response to the intensity of an external electric field. ... The Pockels effect, or Pockels electro-optic effect, produces birefringence in an optical medium induced by a constant or varying electric field. ...

In non-dispersive, isotropic media, ε and μ are time-independent scalars, and Maxwell's equations reduce to

$nabla cdot varepsilon mathbf{E} = rho$
$nabla cdot mu mathbf{H} = 0$
$nabla times mathbf{E} = - mu frac{partial mathbf{H}} {partial t}$
$nabla times mathbf{H} = mathbf{J} + varepsilon frac{partial mathbf{E}} {partial t}$

In a uniform (homogeneous) medium, ε and μ are constants independent of position, and can thus be furthermore interchanged with the spatial derivatives.

More generally, ε and μ can be rank-2 tensors (3×3 matrices) describing birefringent (anisotropic) materials. Also, although for many purposes the time/frequency-dependence of these constants can be neglected, every real material exhibits some material dispersion by which ε and/or μ depend upon frequency (and causality constrains this dependence to obey the Kramers-Kronig relations). In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... For the square matrix section, see square matrix. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on... Dispersion of a light beam in a prism. ... For other uses, see Frequency (disambiguation). ... In mathematics and physics, the Kramers-Kronig relations describe the relation between the real and imaginary part of a certain class of complex-valued functions. ...

### 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. Vacuum permittivity is the electric constant Îµ0 (also known as the permittivity of free space, or by the term dielectric constant of vacuum), which is a fundamental physical constant. ... The magnetic constant Î¼0 (equal to the vacuum permeability, also known as the permeability of free space) is a universal physical constant, relating mechanical and electromagnetic units of measurement. ...

$mathbf{D} = varepsilon_0 mathbf{E}$
$mathbf{B} = mu_0 mathbf{H}$

Since there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:

$nabla times mathbf{E} = - frac{partialmathbf{B}} {partial t}$
$nabla times mathbf{B} = mu_0varepsilon_0 frac{partial mathbf{E}} {partial t}$

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

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

Maxwell discovered that this quantity c is 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: Lasers used for visual effects during a musical performance. ... â€œLightspeedâ€ redirects here. ... Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... â€œSIâ€ redirects here. ...

Symbol Name Numerical Value SI Unit of Measure Type
$c$ Speed of light in vacuum $2.99792458 times 10^8$ meters per second defined
$varepsilon_0$ electric constant $8.85419 times 10^{-12}$ farads per meter derived
$mu_0$ magnetic constant $4 pi times 10^{-7}$ henries per meter defined

â€œLightspeedâ€ redirects here. ... The electric constant () is the permittivity of vacuum, a physical constant, defined by: where: - magnetic constant - speed of light In SI units, the value is exactly expressed by: = 2. ... The farad (symbol F) is the SI unit of capacitance (named after Michael Faraday). ... The magnetic constant () is the permeability of vacuum. ... An inductor. ...

## The Heaviside versions in detail

### Gauss's law

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

$nabla cdot mathbf{D} = rho$

where ρ is the free electric charge density (in units of C/m³), not including dipole charges bound in a material, and 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. In physics, the electric displacement field or electric flux density or electric induction is a vector field that appears in Maxwells equations. ... Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ...

The equivalent integral form (by the divergence theorem), also known as Gauss' law, is: In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ... In physics, Gausss law gives the relation between the electric flux flowing out a closed surface and the charge enclosed in the surface. ...

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

where $mathrm{d}mathbf{A}$ is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and Qenclosed is the free charge enclosed by the surface.

In a linear material, is directly related to the electric field via a material-dependent constant called the permittivity, ε: 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 of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

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

Any material can be treated as linear, as long as the electric field is not extremely strong. The permittivity of free space is referred to as ε0, and appears in:

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

where, again, is the electric field (in units of V/m), ρt is the total charge density (including bound charges), and ε0 (approximately 8.854 pF/m) is the permittivity of free space. ε can also be written as $varepsilon_0 varepsilon_r$, where εr is the material's relative permittivity or its dielectric constant. The relative dielectric constant of a material under given conditions is a measure of the extent to which it concentrates electrostatic lines of flux. ...

Compare Poisson's equation. In mathematics, Poissons equation is a partial differential equation with broad utility in electrostatics, mechanical engineering and theoretical physics. ...

### The divergence of the magnetic field

The divergence of a magnetic field is always zero and hence magnetic field lines are solenoidal.

$mathbf{B}$ is the magnetic flux density (in units of teslas, T), also called the magnetic induction.

Equivalent integral form:

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

$mathrm{d}mathbf{A}$ is the area of a differential square on the surface A with an outward facing surface normal defining its direction.

Like the electric field's integral form, this equation only works if the integral is done over a closed surface.

This equation is related to the magnetic field's structure because it states that given any volume element, the net magnitude of the vector components that point outward from the surface must be equal to the net magnitude of the vector components that point inward. Structurally, this means that the magnetic field lines must be closed loops. Another way of putting it is that the field lines cannot originate from somewhere; attempting to follow the lines backwards to their source or forward to their terminus ultimately leads back to the starting position. Hence, this is a mathematical formulation of the statement that there are no magnetic monopoles. In physics, a magnetic monopole is a hypothetical particle that may be loosely described as a magnet with only one pole (see electromagnetic theory for more on magnetic poles). ...

### Faraday's law of electromagnetic induction

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

The equivalent integral form is (according to Stoke's Theorem): Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ...

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

where

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

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

If a conducting wire, following the contour C, is introduced into the field, the so-called electromotive force in this wire is equal to the value of these integrals (over the fields in absence of the wire!). Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ...

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. Lenzs law (pronounced (IPA) ) gives the direction of the induced electromotive force (emf) and current resulting from electromagnetic induction. ...

This equation relates the electric and magnetic fields, but it also has a number of practical applications. In circuit theory it takes the form of the relationship of induced voltage due to a changing current in an inductance, sometimes called a reverse or back emf. This equation describes how electric motors and electric generators work. Specifically, it demonstrates that a voltage can be generated by varying the magnetic flux passing through a given area over time, such as by uniformly rotating a loop of wire through a fixed magnetic field. In a motor or generator, the fixed excitation is provided by the field circuit and the varying voltage is measured across the armature circuit. In some types of motors/generators, the field circuit is mounted on the rotor and the armature circuit is mounted on the stator, but other types of motors/generators reverse the configuration. For other kinds of motors, see motor. ... Generator redirects here. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... In electrical engineering, an armature is usually the rotating part of an electric motor or dynamo. ...

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).

### Ampère's circuital law

Ampère's circuital law describes the source of the magnetic field, In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ...

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

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: $mathbf{J} = rho_qmathbf{v}$ where is a vector field called the drift velocity that describes the velocities of the charge carriers which have a density described by the scalar function ρq. The second term on the right hand side of Ampère's Circuital Law is known as the displacement current. In physics, a magnetic field is an entity produced by moving electric charges (electric currents) that exerts a force on other moving charges. ... In electromagnetism, permeability is the degree of magnetization of a material that responds linearly to an applied magnetic field. ... In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ... Displacement current is a quantity related to 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. 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. In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ...

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'. A Dynamical Theory of the Electromagnetic Field was the third of James Clerk Maxwells papers concerned with electromagnetism. ... Lasers used for visual effects during a musical performance. ...

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:

$nabla times mathbf{B} = mu_0 mathbf{J} + mu_0varepsilon_0 frac{partial mathbf{E}}{partial t}$

Equivalent integral form:

$oint_C mathbf{B} cdot mathrm{d}mathbf{l} = mu_0 I_mathrm{encircled} + mu_0varepsilon_0 int_S frac{partial mathbf{E}}{partial t} cdot mathrm{d} mathbf{A}$

$mathbf{l}$ is the edge of the open surface A (any surface with the curve $mathbf{l}$ as its edge will do), and Iencircled is the current encircled by the curve $mathbf{l}$ (the current through any surface is defined by the equation: $begin{matrix}I_{mathrm{through} A} = int_S mathbf{J}cdot mathrm{d}mathbf{A}end{matrix}$). In some situations, this integral form of Ampere-Maxwell Law appears in:

$oint_C mathbf{B} cdot mathrm{d}mathbf{l} = mu_0 (I_mathrm{enc} + I_mathrm{d,enc})$

for

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

is sometimes called displacement current. The displacement current concept was Maxwell's greatest innovation in electromagnetic theory. It states that a magnetic field appears during the charge or discharge of a capacitor. With this concept, and the Faraday law equation, Maxwell was able to derive the wave equations, and by showing that the prediced wave velocity was the same as the measured velocity of light, Maxwell asserted that light waves are electromagnetic waves. Displacement current is a quantity related to changing electric field. ...

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 electric displacement field or electric flux density is a vector-valued field that appears in Maxwells equations and that generalizes the electric field. ... In physics, AmpÃ¨res Circuital law, discovered by AndrÃ©-Marie AmpÃ¨re, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. ...

## Maxwell's original eight equations

In Part III of 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), Maxwell formulated eight equations labelled A to H. These eight equations were to become known as Maxwell's equations. Today, however, references to Maxwell's equations usually refer to the Heaviside restatements. Heaviside's versions of Maxwell's equations actually contain only one of the original eight, Gauss's law (Maxwell's equation G). Another of Heaviside's four equations is an amalgamation of Maxwell's law of total currents (equation A) with Ampère's circuital law (equation C). This amalgamation, which Maxwell himself originally made at equation (112) in his 1861 paper "On Physical Lines of Force", is the one that modifies Ampère's circuital law to include Maxwell's displacement current. A Dynamical Theory of the Electromagnetic Field was the third of James Clerk Maxwells papers concerned with electromagnetism. ... Displacement current is a quantity related to changing electric field. ...

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

(A) The law of total currents
$mathbf{J}_{tot} = mathbf{J} + frac{partialmathbf{D}}{partial t}$
(B) Definition of the magnetic vector potential
$mu mathbf{H} = nabla times mathbf{A}$
(C) Ampère's circuital law
$nabla times mathbf{H} = mathbf{J}_{tot}$
(D) The Lorentz force (electric fields created by convection, induction, and by charges)
$mathbf{E} = mu mathbf{v} times mathbf{H} - frac{partialmathbf{A}}{partial t}-nabla phi$
(E) The electric elasticity equation
$mathbf{E} = frac{1}{epsilon} mathbf{D}$
(F) Ohm's law
$mathbf{E} = frac{1}{sigma} mathbf{J}$
(G) Gauss's law
$nabla cdot mathbf{D} = rho$
(H) Equation of continuity of charge
$nabla cdot mathbf{J} = -frac{partialrho}{partial t}$
Notation
$mathbf{H}$ is the magnetic field, which Maxwell called the "magnetic intensity".
$mathbf{J}$ is the electric current density (with $mathbf{J}_{tot}$ being the total current including displacement current).
is the displacement field (called the "electric displacement" by Maxwell).
ρ 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).
φ is the electric potential (which Maxwell also called "electric potential").
σ is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive permittivity ε and permeability μ, although he also discussed the possibility of anisotropic materials. Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... In physics, the electric displacement field or electric flux density or electric induction is a vector field that appears in Maxwells equations. ... In vector calculus, a vector potential is a vector field whose curl is a given vector field. ... 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. ... Electromotive force (emf) is the amount of energy gained per unit charge that passes through a device in the opposite direction to the electric field existing across that device. ... This article does not cite any references or sources. ... Electrical conductivity or specific conductivity is a measure of a materials 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. ... 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 of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... Permeability has several meanings: In electromagnetism, permeability is the degree of magnetisation of a material in response to a magnetic field. ...

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 velocity . This means that equation D is effectively the Lorentz force. This equation first appeared at equation (77) in Maxwell's 1861 paper "On Physical Lines of Force" quite some time before Lorentz thought of it. Nowadays, the Lorentz force sits alongside Maxwell's equations as an additional electromagnetic equation that is not included in Maxwell's set. This article is about velocity in physics. ...

When Maxwell derives the electromagnetic wave equation in his 1864 paper, he uses equation D rather than Faraday's law of electromagnetic induction, which is used in modern textbooks. However, Maxwell drops the term from equation D when he is deriving the electromagnetic wave equation, and he considers the situation only from the rest frame. Lasers used for visual effects during a musical performance. ... Lasers used for visual effects during a musical performance. ...

## 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: Look up si, Si, SI in Wiktionary, the free dictionary. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... This article or section is in need of attention from an expert on the subject. ...

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: Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... Magnetization is a property of some materials (e. ...

With the linear approximation:

χm for vacuum is zero and therefore:

and in the ferro or ferri magnetic materials where χm 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: 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. ... Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ... Lorentz force. ...

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 . Look up si, Si, SI in Wiktionary, the free dictionary. ...

## Formulation of Maxwell's equations in special relativity

In special relativity, in order to more clearly express the fact that Maxwell's equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwell's equations are written in terms of four-vectors and tensors in the "manifestly covariant" form (cgs units): In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ...

,

and

where is the 4-current, is the field strength tensor, is the Levi-Civita symbol, and In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, &#961; the charge density, and j the conventional current density. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations. The four-gradient is the four-vector generalization of the gradient: and is sometimes also represented as D. The square of D is the four-Laplacian, which is called the dAlembertian operator: . As it is the dot product of two four-vectors, the dAlembertian is a Lorentz invariant... This article or section does not adequately cite its references or sources. ...

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and Ampere's law with Maxwell's correction. The second equation is an expression of the homogenous equations, Faraday's law of induction and the absence of magnetic monopoles. In physics, a magnetic monopole is a hypothetical particle that may be loosely described as a magnet with only one pole (see electromagnetic theory for more on magnetic poles). ...

It has been suggested that the vXB component of the Lorentz force can be derived from Coulomb's law and special relativity if one assumes invariance of electric charge.[4][5] Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Charge invariance refers to the fixed electric charge of a particle, regardless of speed. ...

## Maxwell's equations in terms of differential forms

In a vacuum, where ε and μ are constant everywhere, Maxwell's equations simplify considerably once the language of differential geometry and differential forms is used. The electric and magnetic fields are now jointly described by a 2-form F in a 4-dimensional spacetime manifold. Maxwell's equations then reduce to the Bianchi identity Look up Vacuum in Wiktionary, the free dictionary. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... For other uses of this term, see Spacetime (disambiguation). ... In differential geometry, the curvature form describes curvature of principal bundle with connection. ...

where d denotes the exterior derivative - a differential operator acting on forms - and the source equation In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

where the (dual) Hodge star operator * is a linear transformation from the space of 2 forms to the space of 4-2 forms defined by the metric in Minkowski space (or in four dimensions by its conformal class), and the fields are in natural units where 1 / 4πε0 = 1. Here, the 3-form J is called the "electric current" or "current (3-)form" satisfying the continuity equation In mathematics, the Hodge star operator on an oriented inner product space V is a linear operator on the exterior algebra of V, interchanging the subspaces of k-vectors and n&#8722;k-vectors where n = dim V, for 0 &#8804; k &#8804; n. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In physics, natural units are physical units of measurement defined in terms of universal physical constants in such a manner that some chosen physical constants take on the numerical value of one when expressed in terms of a particular set of natural units. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

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. On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ...

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 products) of basis forms , In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

.

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 derivatives), coupling the field to matter. A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ... In physics, gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... 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. 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 AA' = A-dα: see also gauge fixing and Fadeev-Popov ghosts. Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ... In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ... 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 models, they are not even bilinear), and usually contains two derivatives with respect to time (or space); in the case of fermions... In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. ... In physics, a Faddeev-Popov ghost ci is a field that violates the spin-statistics relation. ...

## Classical electrodynamics as the curvature of a line bundle

An elegant and intuitive way to formulate Maxwell's equations is to use complex line bundles or principal bundles with fibre U(1). The connection 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 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 Ã— G of a space X with a group G. Analogous to the Cartesian product, a principal bundle P is equipped with An action of G on P, analogous to... In mathematics, the unitary group of degree n, denoted U(n), is the group of nÃ—n unitary matrices, with the group operation that of matrix multiplication. ... In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. ... In mathematics, curvature refers to a number of loosely related concepts in different areas of geometry. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... This article does not cite any references or sources. ... In physics, the magnetic potential is a method of representing the magnetic field by using a potential value instead of the actual vector field. ...

In quantum mechanics, the connection itself is used to define the dynamics of the system. This formulation allows a natural description of the Aharonov-Bohm effect. In this experiment, a static magnetic field runs through a long superconducting tube. Because of the Meissner effect the superconductor perfectly shields off the magnetic field so the magnetic field strength is zero outside of the tube. Since there is no electric field either, the Maxwell tensor F = 0 in the space time region outside the tube, during the experiment. This means by definition that the connection is flat there. However the connection depends on the magnetic field through the tube since the holonomy along a non-contractible curve encircling the superconducting tube is the magnetic flux through the tube in the proper units. This can be detected quantum mechanically with a double-slit electron diffraction experiment on an electron wave traveling around the tube. The holonomy corresponds to an extra phase shift, which leads to a shift in the diffraction pattern. (See Michael Murray, Line Bundles, 2002 (PDF web link) for a simple mathematical review of this formulation. See also R. Bott, On some recent interactions between mathematics and physics, Canadian Mathematical Bulletin, 28 (1985) )no. 2 pp 129-164.) 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 electromagnetic fields in regions from which the particle is excluded. ... Diagram of the Meissner effect. ... In differential geometry, the holonomy group of a connection on a vector bundle over a smooth manifold M is the group of linear transformations induced by parallel transport around closed loops in M. There is an analogous notion for connections on principal bundles over M. The holonomy group of a...

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 luminiferous aether: it was hypothesised that the Earth moves through a medium of aether that carries light In the late 19th century luminiferous aether (light-bearing aether) was the term used to describe a medium for the propagation of light. ... 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 of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ... In electromagnetism, permeability is the degree of magnetization of a material that responds linearly to an applied magnetic field. ... 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 Morley at what is now Case Western Reserve University, and is considered by some to be the first strong evidence against the theory of... Edward Morley (1887). ... His signature. ... Generally, a null result is a result which is null (nothing): that is, the absence of an observable result. ... George FitzGerald George Francis FitzGerald, or Fitzgerald, (3 August 1851 â€“ 22 February 1901) was a professor of natural and experimental philosophy (i. ... Sir Joseph Larmor (11 July 1857 â€“ 19 May 1942), an Northern Irish physicist, mathematician and politician, researched electricity, dynamics, and thermodynamics. ... Hendrik Antoon Lorentz (July 18, 1853, Arnhem â€“ February 4, 1928, Haarlem) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and elucidation of the Zeeman effect. ... A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed (=rotation of Minkowski space). ... Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... â€œEinsteinâ€ redirects here. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...

The electromagnetic field equations have an intimate link with special relativity, because the equations of special relativity are derived from Maxwell's equations by the Lorentz invariance requirement. The magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities, and the same may be done with the electric field equations. Einstein motivated the special theory by noting that a description of a conductor moving with respect to a magnet must generate a consistent set of fields irrespective of whether the frame is the magnet frame or the conductor frame.[2] Conductor moving in a magnetic field. ...

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: In special relativity, in order to more clearly express the fact that Maxwells equations (in vacuum) take the same form in any inertial coordinate system, the vacuum Maxwells equations are written in terms of four-vectors and tensors in the manifestly covariant form (cgs units): , and where is... In category theory, see covariant functor. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

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. In physics, Kaluza-Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... Thousands of particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...

## Maxwell's equations in curved spacetime

To meet Wikipedias quality standards, this article or section may require cleanup. ...

Matter and energy generate curvature in spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. An electromagnetic field having energy and momentum will also generate curvature in spacetime. Maxwell's equations in curved spacetime can be obtained by replacing the derivatives in the equations in flat spacetime with covariant derivatives. (Whether this is the appropriate generalization requires separate investigation.) The sourced and source-free equations become (cgs units): For other uses of this term, see Spacetime (disambiguation). ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ...

,

and

.

Here,

is a Christoffel symbol that characterizes the curvature of spacetime and Dγ 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 connection derived from the metric tensor. ...

### Formulation in terms of differential forms

The above formulation is related to the differential form formulation of the Maxwell equations as follows. We have implicitly chosen local coordinates xα and therefore have a basis of 1-forms dxα in every point of the open set where the coordinates are defined. Using this basis we have: A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

• The field form
• The current form
• the Bianchi identity
• the source equation
• the continuity equation

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

Lasers used for visual effects during a musical performance. ... Perhaps the most useful solutions to the electromagnetic wave equation are sinusoidal plane-wave solutions. ... Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. ... Maxwells equations describe the behavior of electromagnetic fields; electric field, magnetic field, electric flux density and magnetic flux density. ... The Dynamics of photons in the double-slit experiment describes the relationship between classical electromagnetic waves and photons, the quantum counterpart of classical electromagnetic waves, in the context of the double-slit experiment. ... Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. ... Computational electromagnetics, computational electrodynamics or electromagnetic modeling refers to the process of modeling the interaction of electromagnetic fields with physical objects and the environment. ... Finite-difference time-domain (FDTD) is a popular computational electrodynamics modeling technique. ... Jefimenkos equations describe the behavior of the electric and magnetic fields in terms of the sources at retarded times. ... Conductor moving in a magnetic field. ... The Abraham-Lorentz force is the average force on an accelerating charged particle caused by the particle emitting electromagnetic radiation. ... The theoretical and experimental justification for the SchrÃ¶dinger equation motivates the discovery of the SchrÃ¶dinger equation, the equation that describes the dynamics of nonrelativistic particles. ...

## References

### Journal articles

The developments before relativity James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance â€” eponymously named Maxwells equations â€” including an important modification (extension) of the AmpÃ¨res... A Dynamical Theory of the Electromagnetic Field was the third of James Clerk Maxwells papers concerned with electromagnetism. ... is the 342nd day of the year (343rd in leap years) in the Gregorian calendar. ... 1864 (MDCCCLXIV) was a leap year starting on Friday (see link for calendar) of the Gregorian calendar or a leap year starting on Sunday of the 12-day-slower Julian calendar. ...

• Joseph Larmor (1897) "On a dynamical theory of the electric and luminiferous medium", Phil. Trans. Roy. Soc. 190, 205-300 (third and last in a series of papers with the same name).
• Hendrik Lorentz (1899) "Simplified theory of electrical and optical phenomena in moving systems", Proc. Acad. Science Amsterdam, I, 427-43.
• Hendrik Lorentz (1904) "Electromagnetic phenomena in a system moving with any velocity less than that of light", Proc. Acad. Science Amsterdam, IV, 669-78.
• Henri Poincaré (1900) "La theorie de Lorentz et la Principe de Reaction", Archives Neerlandaies, V, 253-78.
• Henri Poincaré (1901) Science and Hypothesis
• Henri Poincaré (1905) "Sur la dynamique de l'electron", Comptes Rendues, 140, 1504-8.

see Sir Joseph Larmor (11 July 1857 â€“ 19 May 1942), an Northern Irish physicist, mathematician and politician, researched electricity, dynamics, and thermodynamics. ... Hendrik Antoon Lorentz (July 18, 1853, Arnhem â€“ February 4, 1928, Haarlem) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and elucidation of the Zeeman effect. ... Hendrik Antoon Lorentz (July 18, 1853, Arnhem â€“ February 4, 1928, Haarlem) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and elucidation of the Zeeman effect. ... Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... Jules TuPac Henri PoincarÃ© (April 29, 1854 â€“ July 17, 1912) (IPA: [][1]) was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ...

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

### University level textbooks

• Sadiku, Matthew N. O. (2006). Elements of Electromagnetics (4th ed.). Oxford University Press. ISBN 0-19-5300483.
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Hoffman, Banesh, 1983. Relativity and Its Roots. W. H. Freeman.
• Lounesto, Pertti, 1997. Clifford Algebras and Spinors. Cambridge Univ. Press. Chpt. 8 sets out several variants of the equations, using exterior algebra and differential forms.
• Edward Mills Purcell (1985). Electricity and Magnetism. McGraw-Hill. ISBN 0-07-004908-4.
• Stevens, Charles F., 1995. The Six Core Theories of Modern Physics. MIT Press. ISBN 0-262-69188-4.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.
• Schwarz, Melvin (1987). Principles of Electrodynamics. Dover Publications. ISBN 0-486-65493-1.
• Ulaby, Fawwaz T. (2007). Fundamentals of Applied Electromagnetics (5th ed.). Pearson Education, Inc.. ISBN 0-13-241326-4.

In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Edward Mills Purcell (August 30, 1912 â€“ March 7, 1997) was an American physicist who shared the 1952 Nobel Prize for Physics for his independent discovery (published 1946) of nuclear magnetic resonance in liquids and in solids. ...

John David Jackson (born 1925) is a physics professor emeritus at the University of California, Berkeley. ... Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist, who made fundamental contributions to many areas of theoretical physics. ... James Clerk Maxwell (13 June 1831 â€“ 5 November 1879) was a Scottish mathematician and theoretical physicist from Edinburgh, Scotland, UK. His most significant achievement was aggregating a set of equations in electricity, magnetism and inductance â€” eponymously named Maxwells equations â€” including an important modification (extension) of the AmpÃ¨res... Charles W. Misner is one of the authors of Gravitation. He has also invented Misner space, a topology and relativity-related mathematical structure. ... Kip S. Thorne Professor Kip Stephen Thorne, Ph. ... John Archibald Wheeler (born July 9, 1911) is an eminent American theoretical physicist. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ...

#### Computational techniques

• R. F. Harrington (1993). Field Computation by Moment Methods. Wiley-IEEE Press. ISBN 0-78031-014-4.
• W. C. Chew, J.-M. Jin, E. Michielssen, and J. Song (2001). Fast and Efficient Algorithms in Computational Electromagnetics. Artech House Publishers. ISBN 1-58053-152-0.
• J. Jin (2002). The Finite Element Method in Electromagnetics, 2nd. ed.. Wiley-IEEE Press. ISBN 0-47143-818-9.
• Allen Taflove and Susan C. Hagness (2005). Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed.. Artech House Publishers. ISBN 1-58053-832-0.

Allen Taflove is a professor in the Department of Electrical Engineering and Computer Science of Northwestern Universitys McCormick School of Engineering. ...

## Footnotes

1. ^ L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields
2. ^ http://www.cse.secs.oakland.edu/haskell/SpecialRelativity.htm J H Field (2006) "Classical electromagnetism as a consequence of Coulomb's law, special relativity and Hamilton's principle and its relationship to quantum electrodynamics". Phys. Scr. 74 702-717
3. ^ Introduction to Electrodynamics by Griffiths
4. ^ L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields
5. ^ 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

Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics. ... Evgeny Mikhailovich Lifshitz (&#1045;&#1074;&#1075;&#1077;&#1085;&#1080;&#1081; &#1052;&#1080;&#1093;&#1072;&#1081;&#1083;&#1086;&#1074;&#1080;&#1095; &#1051;&#1080;&#1092;&#1096;&#1080;&#1094;) (February 21, 1915 &#8211; October 29, 1985) was a Russian physicist. ... Lev Davidovich Landau Lev Davidovich Landau (Russian language: Ð›ÐµÌÐ² Ð”Ð°Ð²Ð¸ÌÐ´Ð¾Ð²Ð¸Ñ‡ Ð›Ð°Ð½Ð´Ð°ÌÑƒ) (January 22, 1908 â€“ April 1, 1968) was a prominent Soviet physicist who made fundamental contributions to many areas of theoretical physics. ... Evgeny Mikhailovich Lifshitz (&#1045;&#1074;&#1075;&#1077;&#1085;&#1080;&#1081; &#1052;&#1080;&#1093;&#1072;&#1081;&#1083;&#1086;&#1074;&#1080;&#1095; &#1051;&#1080;&#1092;&#1096;&#1080;&#1094;) (February 21, 1915 &#8211; October 29, 1985) was a Russian physicist. ...

Results from FactBites:

 Maxwell's equations (1488 words) Maxwell's equations are the set of four equations by James Clerk Maxwell that describe the behavior of both the electric and magnetic fields. Maxwell's equations provided the basis for the unification of electric field and magnetic field, the electromagnetic description of light, and ultimately, Albert Einstein's theory of relativity. Maxwell discovered that this quantity c is simply the speed of light in vacuum, and thus that light is a form of electromagnetic radiation.
More results at FactBites »

Share your thoughts, questions and commentary here