Maxwell's equations
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Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, electric and magnetic circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar, etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.[note 1] The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulation is credited to Oliver Heaviside.[1]
Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum, c (299792458 m/s).[2] Known as electromagnetic radiation, these waves occur at various wavelengths to produce a spectrum of radiation from radio waves to gamma rays.
In
The equations have two major variants. The microscopic equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the
The publication of the equations marked the
History of the equations
Conceptual descriptions
Gauss's law
Gauss's law describes the relationship between an electric field and electric charges: an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of free space.
Gauss's law for magnetism
Gauss's law for magnetism states that electric charges have no magnetic analogues, called magnetic monopoles; no north or south magnetic poles exist in isolation.[3] Instead, the magnetic field of a material is attributed to a dipole, and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field.[note 3]
Faraday's law
The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds to curl of an electric field.[3] In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface.
The
Ampère's law with Maxwell's addition
The original law of Ampère states that magnetic fields relate to electric current. Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.
Maxwell's addition to Ampère's law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields.
The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents,[note 4] matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.
Formulation in terms of electric and magnetic fields (microscopic or in vacuum version)
In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate
The differential and integral formulations are mathematically equivalent; both are useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely local and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using
Key to the notation
Symbols in bold represent
- the total electric charge density (total charge per unit volume), ρ, and
- the total electric current density (total current per unit area), J.
The
- the permittivity of free space, ε0, and
- the permeability of free space, μ0, and
- the speed of light,
Differential equations
In the differential equations,
- the nabla symbol, ∇, denotes the three-dimensional gradient operator, del,
- the ∇⋅ symbol (pronounced "del dot") denotes the divergence operator,
- the ∇× symbol (pronounced "del cross") denotes the curl operator.
Integral equations
In the integral equations,
- Ω is any volume with closed boundary surface ∂Ω, and
- Σ is any surface with closed boundary curve ∂Σ,
The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the
- is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed
- is a volume integral over the volume Ω,
- is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed.
- is a surface integral over the surface Σ,
- The total electric charge Q enclosed in Ω is the volume integral over Ω of the charge density ρ (see the "macroscopic formulation" section below): where dV is the volume element.
- The net magnetic flux ΦB is the surface integral of the magnetic field B passing through a fixed surface, Σ:
- The net electric flux ΦE is the surface integral of the electric field E passing through Σ:
- The net electric current densityJ passing through Σ:where dS denotes the differential vector element of surface area S, normal to surface Σ. (Vector area is sometimes denoted by A rather than S, but this conflicts with the notation for magnetic vector potential).
Formulation in SI units convention
Name | Integral equations | Differential equations |
---|---|---|
Gauss's law | ||
Gauss's law for magnetism | ||
Maxwell–Faraday equation (Faraday's law of induction) | ||
Ampère's circuital law (with Maxwell's addition) |
Formulation in Gaussian units convention
The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing
Name | Integral equations | Differential equations |
---|---|---|
Gauss's law | ||
Gauss's law for magnetism | ||
Maxwell–Faraday equation (Faraday's law of induction) | ||
Ampère's circuital law (with Maxwell's addition) |
The equations simplify slightly when a system of quantities is chosen in the speed of light, c, is used for nondimensionalization, so that, for example, seconds and lightseconds are interchangeable, and c = 1.
Further changes are possible by absorbing factors of 4π. This process, called rationalization, affects whether Coulomb's law or Gauss's law includes such a factor (see Heaviside–Lorentz units, used mainly in particle physics).
Relationship between differential and integral formulations
The equivalence of the differential and integral formulations are a consequence of the
Flux and divergence
According to the (purely mathematical) Gauss divergence theorem, the electric flux through the boundary surface ∂Ω can be rewritten as
The integral version of Gauss's equation can thus be rewritten as
Similarly rewriting the magnetic flux in Gauss's law for magnetism in integral form gives
which is satisfied for all Ω if and only if everywhere.
Circulation and curl
By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls) over a surface it bounds, i.e.
The line integrals and curls are analogous to quantities in classical
Charge conservation
The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the modified Ampere's law has zero divergence by the div–curl identity. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives:
In particular, in an isolated system the total charge is conserved.
Vacuum equations, electromagnetic waves and speed of light
In a region with no charges (ρ = 0) and no currents (J = 0), such as in a vacuum, Maxwell's equations reduce to:
Taking the curl (∇×) of the curl equations, and using the curl of the curl identity we obtain
The quantity has the dimension of (time/length)2. Defining , the equations above have the form of the standard wave equations
Already during Maxwell's lifetime, it was found that the known values for and give , then already known to be the
In materials with relative permittivity, εr, and relative permeability, μr, the phase velocity of light becomes
which is usually[note 5] less than c.
In addition, E and B are perpendicular to each other and to the direction of wave propagation, and are in
Macroscopic formulation
The above equations are the microscopic version of Maxwell's equations, expressing the electric and the magnetic fields in terms of the (possibly atomic-level) charges and currents present. This is sometimes called the "general" form, but the macroscopic version below is equally general, the difference being one of bookkeeping.
The microscopic version is sometimes called "Maxwell's equations in a vacuum": this refers to the fact that the material medium is not built into the structure of the equations, but appears only in the charge and current terms. The microscopic version was introduced by Lorentz, who tried to use it to derive the macroscopic properties of bulk matter from its microscopic constituents.[12]: 5
"Maxwell's macroscopic equations", also known as Maxwell's equations in matter, are more similar to those that Maxwell introduced himself.
Name | Integral equations (SI convention) |
Differential equations (SI convention) |
Differential equations (Gaussian convention) |
---|---|---|---|
Gauss's law | |||
Ampère's circuital law (with Maxwell's addition) | |||
Gauss's law for magnetism | |||
Maxwell–Faraday equation (Faraday's law of induction) |
In the macroscopic equations, the influence of bound charge Qb and bound current Ib is incorporated into the
The cost of this splitting is that the additional fields D and H need to be determined through phenomenological constituent equations relating these fields to the electric field E and the magnetic field B, together with the bound charge and current.
See below for a detailed description of the differences between the microscopic equations, dealing with total charge and current including material contributions, useful in air/vacuum;[note 6] and the macroscopic equations, dealing with free charge and current, practical to use within materials.
Bound charge and current
When an electric field is applied to a
Somewhat similarly, in all materials the constituent atoms exhibit
The very complicated and granular bound charges and bound currents, therefore, can be represented on the macroscopic scale in terms of P and M, which average these charges and currents on a sufficiently large scale so as not to see the granularity of individual atoms, but also sufficiently small that they vary with location in the material. As such, Maxwell's macroscopic equations ignore many details on a fine scale that can be unimportant to understanding matters on a gross scale by calculating fields that are averaged over some suitable volume.
Auxiliary fields, polarization and magnetization
The definitions of the auxiliary fields are:
where P is the polarization field and M is the magnetization field, which are defined in terms of microscopic bound charges and bound currents respectively. The macroscopic bound charge density ρb and bound current density Jb in terms of polarization P and magnetization M are then defined as
If we define the total, bound, and free charge and current density by
Constitutive relations
In order to apply 'Maxwell's macroscopic equations', it is necessary to specify the relations between
For materials without polarization and magnetization, the constitutive relations are (by definition)[9]: 2
An alternative viewpoint on the microscopic equations is that they are the macroscopic equations together with the statement that vacuum behaves like a perfect linear "material" without additional polarization and magnetization. More generally, for linear materials the constitutive relations are[15]: 44–45
- For homogeneous materials, ε and μ are constant throughout the material, while for inhomogeneous materials they depend on location within the material (and perhaps time).[16]: 463
- For isotropic materials, ε and μ are scalars, while for anisotropic materials (e.g. due to crystal structure) they are tensors.[15]: 421 [16]: 463
- Materials are generally dispersive, so ε and μ depend on the frequency of any incident EM waves.[15]: 625 [16]: 397
Even more generally, in the case of non-linear materials (see for example nonlinear optics), D and P are not necessarily proportional to E, similarly H or M is not necessarily proportional to B. In general D and H depend on both E and B, on location and time, and possibly other physical quantities.
In applications one also has to describe how the free currents and charge density behave in terms of E and B possibly coupled to other physical quantities like pressure, and the mass, number density, and velocity of charge-carrying particles. E.g., the original equations given by Maxwell (see History of Maxwell's equations) included Ohm's law in the form
Alternative formulations
Following is a summary of some of the numerous other mathematical formalisms to write the microscopic Maxwell's equations, with the columns separating the two homogeneous Maxwell equations from the two inhomogeneous ones involving charge and current. Each formulation has versions directly in terms of the electric and magnetic fields, and indirectly in terms of the
Each table describes one formalism. See the main article for details of each formulation. SI units are used throughout.
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields
3D Euclidean space + time |
|
|
Potentials (any gauge)
3D Euclidean space + time |
|
|
Potentials ( Lorenz gauge )
3D Euclidean space + time |
|
|
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields
space + time spatial metric independent of time |
||
Potentials
space (with § topological restrictions) + time spatial metric independent of time |
|
|
Potentials (Lorenz gauge)
space (with topological restrictions) + time spatial metric independent of time |
|
|
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields
any space + time |
|
|
Potentials (any gauge)
any space (with § topological restrictions) + time |
|
|
Potential (Lorenz Gauge)
any space (with topological restrictions) + time spatial metric independent of time |
|
|
Relativistic formulations
The Maxwell equations can also be formulated on a spacetime-like
Each table below describes one formalism.
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields Minkowski space |
||
Potentials (any gauge) Minkowski space |
||
Potentials (Lorenz gauge) Minkowski space |
|
|
Fields any spacetime |
||
Potentials (any gauge) any spacetime (with §topological restrictions) |
||
Potentials (Lorenz gauge) any spacetime (with topological restrictions) |
|
Formulation | Homogeneous equations | Inhomogeneous equations |
---|---|---|
Fields any spacetime |
||
Potentials (any gauge) any spacetime (with topological restrictions) |
||
Potentials (Lorenz gauge) any spacetime (with topological restrictions) |
|
- In the tensor calculus formulation, the Ricci tensor, Rαβ and raising and lowering of indices are defined by the Lorentzian metric, gαβ and the d'Alembert operator is defined as ◻ = ∇α∇α. The topological restriction is that the second real cohomologygroup of the space vanishes (see the differential form formulation for an explanation). This is violated for Minkowski space with a line removed, which can model a (flat) spacetime with a point-like monopole on the complement of the line.
- In the differential form formulation on arbitrary space times, F = 1/2Fαβdxα ∧ dxβ is the electromagnetic tensor considered as a 2-form, A = Aαdxα is the potential 1-form, is the current 3-form, d is the exterior derivative, and is the Hodge staron forms defined (up to its orientation, i.e. its sign) by the Lorentzian metric of spacetime. In the special case of 2-forms such as F, the Hodge star depends on the metric tensor only for its local scale. This means that, as formulated, the differential form field equations are conformally invariant, but the Lorenz gauge condition breaks conformal invariance. The operator is the d'Alembert–Laplace–Beltrami operator on 1-forms on an arbitrary Lorentzian spacetime. The topological condition is again that the second real cohomology group is 'trivial' (meaning that its form follows from a definition). By the isomorphism with the second de Rham cohomology this condition means that every closed 2-form is exact.
Other formalisms include the geometric algebra formulation and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation[17][18] was used.
Solutions
Maxwell's equations are
As for any differential equation,
Jefimenko's equations (or the closely related Liénard–Wiechert potentials) are the explicit solution to Maxwell's equations for the electric and magnetic fields created by any given distribution of charges and currents. It assumes specific initial conditions to obtain the so-called "retarded solution", where the only fields present are the ones created by the charges. However, Jefimenko's equations are unhelpful in situations when the charges and currents are themselves affected by the fields they create.
Overdetermination of Maxwell's equations
Maxwell's equations seem overdetermined, in that they involve six unknowns (the three components of E and B) but eight equations (one for each of the two Gauss's laws, three vector components each for Faraday's and Ampere's laws). (The currents and charges are not unknowns, being freely specifiable subject to charge conservation.) This is related to a certain limited kind of redundancy in Maxwell's equations: It can be proven that any system satisfying Faraday's law and Ampere's law automatically also satisfies the two Gauss's laws, as long as the system's initial condition does, and assuming conservation of charge and the nonexistence of magnetic monopoles.[30][31] This explanation was first introduced by Julius Adams Stratton in 1941.[32]
Although it is possible to simply ignore the two Gauss's laws in a numerical algorithm (apart from the initial conditions), the imperfect precision of the calculations can lead to ever-increasing violations of those laws. By introducing dummy variables characterizing these violations, the four equations become not overdetermined after all. The resulting formulation can lead to more accurate algorithms that take all four laws into account.[33]
Both identities , which reduce eight equations to six independent ones, are the true reason of overdetermination.[34][35]
Equivalently, the overdetermination can be viewed as implying conservation of electric and magnetic charge, as they are required in the derivation described above but implied by the two Gauss's laws.
For linear algebraic equations, one can make 'nice' rules to rewrite the equations and unknowns. The equations can be linearly dependent. But in differential equations, and especially partial differential equations (PDEs), one needs appropriate boundary conditions, which depend in not so obvious ways on the equations. Even more, if one rewrites them in terms of vector and scalar potential, then the equations are underdetermined because of gauge fixing.
Maxwell's equations and quantum mechanics
Maxwell's equations are valid in both the classical and the quantum realm. In the Heisenberg representation of Quantum Mechanics, the equations of the E and B operators are precisely Maxwell's equations. Of course since the fields are quantum operators, there are many aspects which differ from the classical fields. For example, the E field acts like the momentum conjugate to the spatial components of the vector potential A. This of course leads to many aspects of the quantum electromagnetic field with differ from them as classical fields but they still obey the same evolution equations as the classical field does.
Of course once one examines the effects of the electromagnetic fields on charged matter, and those effects then change the electromagnetic field are examined, the field equations become non-linear, and the quantum behaviour of non-linear field can be very different from the classical behaviour of the non-linear fields. That however does not alter the fact that if one remains in the linear regime, the fields obey Maxwell's equations.
Variations
Popular variations on the Maxwell equations as a classical theory of electromagnetic fields are relatively scarce because the standard equations have stood the test of time remarkably well.
Magnetic monopoles
Maxwell's equations posit that there is
See also
Explanatory notes
- ^ Electric and magnetic fields, according to the theory of relativity, are the components of a single electromagnetic field.
- ^ In general relativity, however, they must enter, through its stress–energy tensor, into Einstein field equations that include the spacetime curvature.
- ^ The absence of sinks/sources of the field does not imply that the field lines must be closed or escape to infinity. They can also wrap around indefinitely, without self-intersections. Moreover, around points where the field is zero (that cannot be intersected by field lines, because their direction would not be defined), there can be the simultaneous begin of some lines and end of other lines. This happens, for instance, in the middle between two identical cylindrical magnets, whose north poles face each other. In the middle between those magnets, the field is zero and the axial field lines coming from the magnets end. At the same time, an infinite number of divergent lines emanate radially from this point. The simultaneous presence of lines which end and begin around the point preserves the divergence-free character of the field. For a detailed discussion of non-closed field lines, see L. Zilberti "The Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.
- ^ The quantity we would now call 1/√ε0μ0, with units of velocity, was directly measured before Maxwell's equations, in an 1855 experiment by Wilhelm Eduard Weber and Rudolf Kohlrausch. They charged a leyden jar (a kind of capacitor), and measured the electrostatic force associated with the potential; then, they discharged it while measuring the magnetic force from the current in the discharge wire. Their result was 3.107×108 m/s, remarkably close to the speed of light. See Joseph F. Keithley, The story of electrical and magnetic measurements: from 500 B.C. to the 1940s, p. 115.
- anomalous dispersion) where the phase velocity can exceed c, but the "signal velocity" will still be < c
- ^ In some books—e.g., in U. Krey and A. Owen's Basic Theoretical Physics (Springer 2007)—the term effective charge is used instead of total charge, while free charge is simply called charge.
- ^ See magnetic monopole for a discussion of monopole searches. Recently, scientists have discovered that some types of condensed matter, including spin ice and topological insulators, which display emergent behavior resembling magnetic monopoles. (See sciencemag.org and nature.com.) Although these were described in the popular press as the long-awaited discovery of magnetic monopoles, they are only superficially related. A "true" magnetic monopole is something where ∇ ⋅ B ≠ 0, whereas in these condensed-matter systems, ∇ ⋅ B = 0 while only ∇ ⋅ H ≠ 0.
References
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- ^ "The NIST Reference on Constants, Units, and Uncertainty".
- ^ a b c Jackson, John. "Maxwell's equations". Science Video Glossary. Berkeley Lab. Archived from the original on 2019-01-29. Retrieved 2016-06-04.
- ^ J. D. Jackson, Classical Electrodynamics, section 6.3
- ^ Principles of physics: a calculus-based text, by R. A. Serway, J. W. Jewett, page 809.
- ^ Bruce J. Hunt (1991) The Maxwellians, chapter 5 and appendix, Cornell University Press
- ^ "Maxwell's Equations". Engineering and Technology History Wiki. 29 October 2019. Retrieved 2021-12-04.
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- ^ ISBN 978-0-471-43132-9.
- ^ Littlejohn, Robert (Fall 2007). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory" (PDF). Physics 221A, University of California, Berkeley lecture notes. Retrieved 2008-05-06.
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Peter Monk (2003). Finite Element Methods for Maxwell's Equations. Oxford UK: Oxford University Press. p. 1 ff. ISBN 978-0-19-850888-5.
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Thomas B. A. Senior & John Leonidas Volakis (1995-03-01). Approximate Boundary Conditions in Electromagnetics. London UK: Institution of Electrical Engineers. p. 261 ff. ISBN 978-0-85296-849-9.
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Further reading
- Imaeda, K. (1995), "Biquaternionic Formulation of Maxwell's Equations and their Solutions", in Ablamowicz, Rafał; Lounesto, Pertti (eds.), Clifford Algebras and Spinor Structures, Springer, pp. 265–280, ISBN 978-90-481-4525-6
Historical publications
- On Faraday's Lines of Force – 1855/56. Maxwell's first paper (Part 1 & 2) – Compiled by Blaze Labs Research (PDF).
- On Physical Lines of Force – 1861. Maxwell's 1861 paper describing magnetic lines of force – Predecessor to 1873 Treatise.
- James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459–512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
- A Dynamical Theory Of The Electromagnetic Field – 1865. Maxwell's 1865 paper describing his 20 equations, link from Google Books.
- J. Clerk Maxwell (1873), "A Treatise on Electricity and Magnetism":
- The developments before relativity:
- Larmor Joseph (1897). . Phil. Trans. R. Soc. 190: 205–300.
- Lorentz Hendrik (1899). . Proc. Acad. Science Amsterdam. I: 427–443.
- Lorentz Hendrik (1904). . Proc. Acad. Science Amsterdam. IV: 669–678.
- Henri Poincaré (1900) "La théorie de Lorentz et le Principe de Réaction" (in French), Archives Néerlandaises, V, 253–278.
- La Science et l'Hypothèse" (in French).
- Henri Poincaré (1905) "Sur la dynamique de l'électron" (in French), Comptes Rendus de l'Académie des Sciences, 140, 1504–1508.
- Catt, Walton and Davidson. "The History of Displacement Current" Archived 2008-05-06 at the Wayback Machine. Wireless World, March 1979.
External links
- "Maxwell equations", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- maxwells-equations.com — An intuitive tutorial of Maxwell's equations.
- The Feynman Lectures on Physics Vol. II Ch. 18: The Maxwell Equations
- Wikiversity Page on Maxwell's Equations
Modern treatments
- Electromagnetism (ch. 11), B. Crowell, Fullerton College
- Lecture series: Relativity and electromagnetism, R. Fitzpatrick, University of Texas at Austin
- Electromagnetic waves from Maxwell's equations on Project PHYSNET.
- MIT Video Lecture Series (36 × 50 minute lectures) (in .mp4 format) – Electricity and Magnetism Taught by Professor Walter Lewin.
Other
- Silagadze, Z. K. (2002). "Feynman's derivation of Maxwell equations and extra dimensions". Annales de la Fondation Louis de Broglie. 27: 241–256. Bibcode:2001hep.ph....6235S.
- Nature Milestones: Photons – Milestone 2 (1861) Maxwell's equations