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Maxwell's equations

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Maxwell's equations are the set of four equations, attributed to James Clerk Maxwell,that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.

Maxwell's four equations express, respectively, how electric charges produce electricfields (Gauss's law), the experimental absence of  magnetic charges, how currents producemagnetic fields (Ampère's law), and how changing magnetic fields produce electric fields(Faraday's law of induction). Maxwell, in 1864, was the first to put all four equationstogether and to notice that a correction was required to Ampere's law: changing electricfields act like currents, likewise producing magnetic fields.

Furthermore, Maxwell showed that the four equations, with his correction, predict wavesof oscillating electric and magnetic fields that travel through empty space at a speed thatcould be predicted from simple electrical experiments—using the data available at thetime, Maxwell obtained a velocity of 310,740,000 m/s. Maxwell (1865) wrote:

This velocity is so nearly that of light , that it seems we have strong reason to

conclude that light itself (including radiant heat, and other radiations if any) isan electromagnetic disturbance in the form of waves propagated through the

electromagnetic field according to electromagnetic laws. 

Maxwell was correct in this conjecture, though he did not live to see its vindication byHeinrich Hertz in 1888. Maxwell's quantitative explanation of light as an electromagneticwave is considered one of the great triumphs of 19th-century physics. (Actually, MichaelFaraday had postulated a similar picture of light in 1846, but had not been able to give aquantitative description or predict the velocity.) Moreover, it laid the foundation for manyfuture developments in physics, such as special relativity and its unification of electricand magnetic fields as a single tensor  quantity, and Kaluza and Klein's unification of electromagnetism with gravity and general relativity.

Historical developments of Maxwell's equations and

relativity

Maxwell's 1865 formulation was in terms of 20 equations in 20 variables, which includedseveral equations now considered to be auxiliary to what are now called "Maxwell'sequations" — the corrected Ampere's law (three component equations), Gauss's law for 

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charge (one equation), the relationship between total and displacement current densities(three component equations), the relationship between magnetic field and the vector potential (three component equations, which imply the absence of magnetic charge), therelationship between electric field and the scalar and vector potentials (three componentequations, which imply Faraday's law), the relationship between the electric and

displacement fields (three component equations), Ohm's law relating current density andelectric field (three component equations), and the continuity equation relating currentdensity and charge density (one equation).

The modern mathematical formulation of Maxwell's equations is due to Oliver Heaviside and Willard Gibbs, who in 1884 reformulated Maxwell's original system of equations to afar simpler representation using vector calculus. (In 1873 Maxwell also published aquaternion-based notation that ultimately proved unpopular.) The change to the vector notation produced a symmetric mathematical representation that reinforced theperception of physical symmetries between the various fields. This highly symmetricalformulation would directly inspire later developments in fundamental physics.

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

in the equations, Maxwell's equations were only thought to express electromagnetism inthe rest frame of the luminiferous aether  (the postulated medium for light, whoseinterpretation was considerably debated). When the Michelson-Morley experiment, conducted by Edward Morley and Albert Abraham Michelson, produced a null result for the change of the velocity of light due to the Earth's hypothesized motion through theaether, however, alternative explanations were sought by Lorentz and others. Thisculminated in Einstein's theory of  special relativity, which postulated the absence of anyabsolute rest frame (or aether) and the invariance of Maxwell's equations in all frames of reference.

The electromagnetic field equations have an intimate link with special relativity: themagnetic field equations can be derived from consideration of the transformation of theelectric field equations under relativistic transformations at low velocities. (In relativity,the equations are written in an even more compact, "manifestly covariant" form, in termsof the rank-2 antisymmetric field-strength 4-tensor that unifies the electric and magneticfields into a single object.)

Kaluza and Klein showed in the 1920s that Maxwell's equations can be derived byextending general relativity into five dimensions. This strategy of using higher dimensions to unify different forces is an active area of research in particle physics.

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Summary of the equations

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General case

NamePartial differential 

formIntegral form

Gauss's law:

Gauss's law for magnetism:

Faraday's law of induction:

Ampere's law + Maxwell'sextension:

where:

• ρ is the free electric charge density (SI unit: coulomb / cubic meter ), not includingdipole charges bound in a material

• is the magnetic flux density (SI unit: tesla), also called the magnetic induction.• is the electric displacement field (SI unit: coulomb per square meter).• is the electric field (SI unit: volt per meter),• is the magnetic field strength (SI unit: ampere per meter)• is the current density (SI unit: ampere per square meter)• is the divergence operator  (SI unit: 1 per meter),• is the curl operator (SI unit: 1 per meter).

Note that although SI units are given here for the various symbols, Maxwell's equationswill hold unchanged in many different unit systems (and with only minor modificationsin all others). The most commonly used systems of units are SI units, used for 

engineering, electronics and most practical physics experiments, and Planck units (alsoknown as "natural units"), used in theoretical physics, quantum physics and cosmology.An older system of units, the cgs system, is sometimes also used.

The second equation is equivalent to the statement that magnetic monopoles do not exist.The force exerted upon a charged particle by the electric field and magnetic field is givenby the Lorentz force equation:

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where is the charge on the particle and is the particle velocity. Note that this isslightly different when expressed in the cgs system of units below.

It is important to note that Maxwell's equations are generally applied to macroscopicaverages 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 inthis averaged sense that one can define quantities such as the permittivity andpermeability of a material, below. (The microscopic Maxwell's equations, ignoringquantum effects, are simply those of a vacuum — but one must include all atomiccharges and so on, which is normally an intractable problem.)

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In linear materials

In linear materials, the D and H fields are related to E and B by:

where:

ε is the electrical permittivity

μ is the magnetic permeability

(This can actually be extended to handle nonlinear materials as well, by making ε and μ

depend upon the field strength; see e.g. the Kerr and Pockels effects.)

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

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

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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-dependenceof 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 toobey the Kramers-Kronig relations).

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In vacuum, without charges or currents

The vacuum is a linear, homogeneous, isotropic, dispersionless medium, and theproportionality constants in the vacuum are denoted by ε0 and μ0 (neglecting very slightnonlinearities due to quantum effects). If there is no current or electric charge present inthe vacuum, we obtain the Maxwell's equations in free space:

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

Maxwell discovered that this quantity c is simply the speed of light in vacuum, and thusthat light is a form of electromagnetic radiation.

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Detail

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Charge density and the electric field

,

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where ρ is the free electric charge density (in units of C/m3), not including dipole chargesbound in a material, and is the electric displacement field (in units of C/m2). Thisequation corresponds to Coulomb's law for stationary charges in vacuum.

The equivalent integral form (by the divergence theorem), also known as Gauss's law, is:

where is the area of a differential square on the closed surface A with an outwardfacing surface normal defining its direction, and Qenclosed is the free charge enclosed by thesurface.

In a linear material , is directly related to the electric field via a material-dependentconstant called the permittivity, ε:

.

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:

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 , where εr  is the material's relative permittivity or its dielectric constant .

Compare Poisson's equation.

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The structure of the magnetic field

is the magnetic flux density (in units of teslas, T), also called the magnetic induction.

Equivalent integral form:

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is the area of a differential square on the surface A with an outward facing surfacenormal defining its direction.

Note: like the electric field's integral form, this equation only works if the integral is doneover a closed surface.

This equation is related to the magnetic field's structure because it states that given anyvolume element, the net magnitude of the vector components that point outward from thesurface 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 wayof putting it is that the field lines cannot originate from somewhere; attempting to followthe lines backwards to their source or forward to their terminus ultimately leads back tothe starting position. Hence, this is the mathematical formulation of the assumption thatthere are no magnetic monopoles.

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A changing magnetic field and the electric field

Equivalent integral Form:

where

where

ΦB is the magnetic flux through the area A described by the second equation

E is the electric field generated by the magnetic flux

s is a closed path in which current is induced, such as a wire.

The electromotive force (sometimes denoted , not to be confused with the permittivityabove) is equal to the value of this integral.

This law corresponds to the Faraday's law of  electromagnetic induction.

Note: some textbooks show the right hand sign of the Integral form with an N 

(representing the number of coils of wire that are around the edge of A) in front of theflux derivative. The N can be taken care of in calculating A (multiple wire coils meansmultiple surfaces for the flux to go through), and it is an engineering detail so it has beenomitted here.

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Note the negative sign; it is necessary to maintain conservation of energy. It is soimportant that it even has its own name, Lenz's law.

This equation relates the electric and magnetic fields, but it also has a lot of practicalapplications, too. This equation describes how electric motors and electric generators 

work. Specifically, it demonstrates that a voltage can be generated by varying themagnetic flux passing through a given area over time, such as by uniformly rotating aloop of wire through a fixed magnetic field. In a motor or generator, the fixed excitationis provided by the field circuit and the varying voltage is measured across the armaturecircuit. In some types of motors/generators, the field circuit is mounted on the rotor andthe armature circuit is mounted on the stator, but other types of motors/generators employthe reverse configuration.

Note: Maxwell's equations apply to a right-handed coordinate system. To apply themunmodified to a left handed system would mean a reversal of polarity of magnetic fields(not inconsistent, but confusingly against convention).

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The source of the magnetic field

where H is the magnetic field strength (in units of A/m), related to the magnetic flux B bya constant called the permeability, μ (B = μH), and J is the current density, defined by:J = ∫ρqvdV where v is a vector field called the drift velocity that describes the velocities

of that charge carriers which have a density described by the scalar function ρq.

In free space, the permeability μ is the permeability of free space, μ0, which is defined tobe exactly 4π×10-7 W/A·m. Also, the permittivity becomes the permittivity of free spaceε0. Thus, in free space, the equation becomes:

Equivalent integral form:

s is the edge of the open surface A (any surface with the curve s as its edge will do), andI encircled is the current encircled by the curve s (the current through any surface is definedby the equation: I through A = ∫AJ ·d A).

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Note: if the electric flux density does not vary rapidly, the second term on the right handside (the displacement flux) is negligible, and the equation reduces to Ampere's Law.

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Maxwell's equations in CGS units

The above equations are given in the International System of Units, or SI for short. In arelated unit system, called cgs (short for centimetre, gram, second), the equations take ona more symmetrical form, as follows:

Where c is the speed of light in a vacuum. The symmetry is more apparent when theelectromagnetic field is considered in a vacuum. The equations take on the following,highly symmetric form:

The force exerted upon a charged particle by the electric field and magnetic field is givenby the Lorentz force equation:

where is the charge on the particle and is the particle velocity. Note that this is

slightly different from the SI-unit expression above. For example, here the magnetic fieldhas the same units as the electric field .

Note: All variables that are in bold represent vector quantities.

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Formulation of Maxwell's equations in special relativity

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In special relativity, in order to more clearly express the fact that Maxwell's equations (invacuum) take the same form in any inertial coordinate system, the vacuum Maxwell'sequations are written in terms of  four-vectors and tensors in the "manifestly covariant"form:

,

and

where J is the 4-current density, F is the field strength tensor (Faraday tensor) (written

as a 4 × 4 matrix), and is the 4-gradient (so that is thed'Alembertian operator). (The α in the first equation is implicitly summed over, accordingto Einstein notation.) The first tensor equation expresses the two inhomogenous

Maxwell's equations: Gauss' law and Ampere's law with Maxwell's correction. Thesecond equation expresses the other two, homogenous equations: Faraday's law of induction and the absence of magnetic monopoles.

More explicitly, J = (cρ, J) (as a contravariant vector ), in terms of the charge density ρand the current density J. In terms of the 4-potential (as a contravariant vector ) A = (φ,A), where φ is the electric potential and A is the magnetic vector potential in the Lorenz

gauge ( ), F can be expressed as:

which leads to the 4 × 4 matrix (rank-2 tensor):

The fact that both electric and magnetic fields are combined into a single tensor expressesthe fact that, according to relativity, both of these are different aspects of the same thing—by changing frames of reference, what seemed to be an electric field in one frame canappear as a magnetic field in another frame, and vice versa.

Note that different authors sometimes employ different sign conventions for the abovetensors and 4-vectors (which does not affect the physical interpretation). Note also thatF αβ and F αβ are not the same: they are the contravariant and covariant forms of the tensor,related by the metric tensor  g . In special relativity the metric tensor introduces sign

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changes in some of F' s components; more complex metric dualities are encountered ingeneral relativity.

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Maxwell's equations in terms of differential forms

In a vacuum, where ε and μ are constant everywhere, Maxwell's equations simplifyconsiderably once you use the language of differential geometry and differential forms. Now, the electric and magnetic fields are jointly described by a 2-form in a 4-dimensional spacetime manifold which is usually called F. Maxwell's equations thenreduce to the Bianchi identity

where d is the exterior derivative, and the source equation

where these are represented in natural units where ε0 is 1. Here, J is a 1-form called the"electric current" satisfying the continuity equation

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See also

• gauge theory  for more details• vector calculus  . • natural units • Lorentz-Heaviside  units.

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References

• 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 theRoyal Society.)

• James Clerk Maxwell, A Treatise on Electricity and Magnetism, 3rd ed., vols. 1-2(1891) (reprinted: Dover, New York NY, 1954; ISBN 0-486-60636-8 and ISBN0-486-60637-6).

• John David Jackson, Classical Electrodynamics (Wiley, New York, 1998).• Edward M. Purcell, Electricity and Magnetism (McGraw-Hill, New York, 1985).

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• Banesh Hoffman, Relativity and Its Roots (Freeman, New York, 1983).• Charles F. Stevens, The Six Core Theories of Modern Physics, (MIT Press, 1995)

ISBN 0-262-69188-4.• Landau, L. D.  , The Classical Theory of Fields (Course of Theoretical Physics:

Volume 2), (Butterworth-Heinemann: Oxford, 1987).•

Fitzpatrick, Richard, "Lecture series: Relativity and electromagnetism (http://farside.ph.utexas.edu/~rfitzp/teaching/jk1/lectures/node

6.html)". Advanced Classical Electromagnetism, PHY387K. University of Texasat Austin, Fall 1996.