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Maxwell's equations

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For thermodynamic relations, see Maxwell relations.
Maxwell's equations (sometimes called the Maxwell 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 electric fields (Gauss' law), the experimental absence of magnetic monopoles, how currents and changing electric fields produce magnetic fields (the Ampere-Maxwell law), and how changing magnetic fields produce electric fields (Faraday's law of induction).

Contents

Historical development of Maxwell's equations

Maxwell, in 1864, was the first to put all four equations together and to notice that a correction was required to Ampere's law: changing electric fields act like currents, likewise producing magnetic fields. (This additional term is called the displacement current.) The most common modern notation for these equations was developed by Oliver Heaviside.

Furthermore, Maxwell showed that waves of oscillating electric and magnetic fields travel through empty space at a speed that could be predicted from simple electrical experiments—using the data available at the time, 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) is an 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 the first experimental confirmation by Heinrich Hertz in 1888. Maxwell's quantitative explanation of light as an electromagnetic wave is considered one of the great triumphs of 19th-century physics. (Actually, Michael Faraday had postulated a similar picture of light in 1846, but had not been able to give a quantitative description or predict the velocity.) Moreover, it laid the foundation for many future developments in physics, such as special relativity and its unification of electric and magnetic fields as a single tensor quantity, and Kaluza and Klein's unification of electromagnetism with gravity and general relativity.

Maxwell's 1865 formulation was in terms of 20 equations in 20 variables, which included several equations now considered to be auxiliary to what are now called "Maxwell's equations":

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

Links to relativity

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

[c=\frac}]
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. 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 motion through the hypothesized aether, however, alternative explanations were sought 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) analysed 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 ether as unnecessary, and established the invariance of Maxwell's equations in all inertial frames of reference.

The electromagnetic field equations have an intimate link with special relativity: the magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities. Einstein motivated the special theory by noting that a description of a conductor moving with respect to a magnet must generate a consistent set of fields irrespective of whether the frame is the magnet frame or the conductor frame..[link]

In relativity, the equations are written in an even more compact, "manifestly covariant" form, in terms of the rank-2 antisymmetric field-strength 4-tensor that unifies the electric and magnetic fields into a single object.

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

Summary of the equations

Symbols in bold represent vector quantities, whereas symbols in italics represent scalar quantities.

General case

Name Differential form Integral form
Gauss's law: [\nabla \cdot \mathbf = \rho] [\oint_S \mathbf \cdot d\mathbf = \int_V \rho dV]
Gauss' law for magnetism
(absence of magnetic monopoles): 		[\nabla \cdot \mathbf = 0] [\oint_S \mathbf \cdot d\mathbf = 0]
Faraday's law of induction: [\nabla \times \mathbf = -\frac} ] [\oint_C \mathbf \cdot d\mathbf = - \ \int_S \mathbf \cdot d\mathbf]
Ampère's law
(with Maxwell's extension): 		[\nabla \times \mathbf = \mathbf + \frac} ] [\oint_C \mathbf \cdot d\mathbf = \int_S \mathbf \cdot d \mathbf + \int_S \mathbf \cdot d \mathbf]

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

Symbol Meaning SI Unit of Measure
[\mathbf] electric field volt per meter
[\mathbf] magnetic field
also called the auxiliary field 		ampere per meter
[\mathbf] electric displacement field
also called the electric flux density 		coulomb per square meter
[\mathbf] magnetic flux density
also called the magnetic induction
also called the magnetic field 		tesla, or equivalently,
weber per square meter
[\ \rho \ ] free electric charge density,
not including dipole charges bound in a material 		coulomb per cubic meter
[\mathbf] free current density,
not including polarization or magnetization currents bound in a material 		ampere per square meter
[d\mathbf] differential vector element of surface area A, with infinitesimally
small magnitude and direction normal to surface S 		square meters
[ dV \ ] differential element of volume V enclosed by surface S cubic meters
[ d \mathbf ] differential vector element of path length tangential to contour C enclosing surface S meters
[\nabla \cdot] the divergence operator per meter
[\nabla \times] the curl operator per meter

Although SI units are given here for the various symbols, Maxwell's equations will hold unchanged in many different unit systems (and with only minor modifications in all others). The most commonly used systems of units are SI units, used for engineering, electronics and most practical physics experiments, and Planck units (also known 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 given by the Lorentz force equation:

[\mathbf = q (\mathbf + \mathbf \times \mathbf),]
where [ q \ ] is the charge on the particle and [ \mathbf \ ] is the particle velocity. This is slightly different when expressed in the cgs system of units below.

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 normally an intractable problem).

In linear materials

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

[ \mathbf = \chi_e \varepsilon_0 \mathbf ]
[ \mathbf = \chi_m \mathbf ]
and the D and B fields are related to E and H by:

[\mathbf \ \ = \ \ \varepsilon_0 \mathbf + \mathbf \ \ = \ \ (1 + \chi_e) \varepsilon_0 \mathbf \ \ = \ \ \varepsilon \mathbf ]
[\mathbf \ \ = \ \ \mu_0 ( \mathbf + \mathbf ) \ \ = \ \ (1 + \chi_m) \mu_0 \mathbf \ \ = \ \ \mu \mathbf ]
where:

[ \chi_e ] is the electrical susceptibility of the material,

[ \chi_m ] is the magnetic susceptibility of the material,

ε is the electrical permittivity of the material, and

μ is the magnetic permeability of the material

(This can actually be extended to handle nonlinear materials as well, by making ε and μ depend upon the field strength; see e.g. the Kerr and Pockels effects.)

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

[\nabla \cdot \varepsilon \mathbf = \rho ]
[\nabla \cdot \mu \mathbf = 0]
[\nabla \times \mathbf = - \mu \frac} ]
[\nabla \times \mathbf = \mathbf + \varepsilon \frac} ]
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 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 (neglecting very slight nonlinearities due to quantum effects).

[\mathbf = \varepsilon_0 \mathbf ]
[\mathbf = \mu_0 \mathbf ]
Since there is no current or electric charge present in the vacuum, we obtain the Maxwell equations in free space:

[\nabla \cdot \mathbf = 0]
[\nabla \cdot \mathbf = 0]
[\nabla \times \mathbf = - \mu_0 \frac} ]
[\nabla \times \mathbf = \ \ \varepsilon_0 \frac} ]
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 direction of travel, and with the two fields in phase, travelling at the speed

[c = \frac} ]
Maxwell discovered that this quantity c is simply the speed of light in vacuum, and thus that light is a form of electromagnetic radiation. The currently accepted values for the speed of light, the permittivity, and the permeability are summarized in the following table:

Symbol Name Numerical Value SI Unit of Measure Type
[ c \ ] Speed of light [ 2.998 \times 10^ ] meters per second defined
[ \ \varepsilon_0 ] Permittivity [ 8.854 \times 10^ ] farads per meter derived
[\ \mu_0 \ ] Permeability [ 4 \pi \times 10^ ] henries per meter defined

Detail

Charge density and the electric field

[\nabla \cdot \mathbf = \rho],
where [] is the free electric charge density (in units of C/m3), not including dipole charges bound in a material, and [\mathbf] is the electric displacement field (in units of C/m2). This equation corresponds to Coulomb's law for stationary charges in vacuum.

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

[\oint_S \mathbf \cdot d\mathbf = Q_\mathrm]
where [d\mathbf] is the area of a differential square on the closed surface A with an outward facing surface normal defining its direction, and [Q_\mathrm] is the free charge enclosed by the surface.

In a linear material, [\mathbf] is directly related to the electric field [\mathbf] via a material-dependent constant called the permittivity, [\epsilon]:

[\mathbf = \varepsilon \mathbf].
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 [\epsilon_0], and appears in:

[\nabla \cdot \mathbf = \frac]
where, again, [\mathbf] is the electric field (in units of V/m), [\rho_t] is the total charge density (including bound charges), and [\epsilon_0] (approximately 8.854 pF/m) is the permittivity of free space. [\epsilon] can also be written as [\varepsilon_0 \cdot \varepsilon_r], where [\epsilon_r] is the material's relative permittivity or its dielectric constant.

Compare Poisson's equation.

The structure of the magnetic field

[\nabla \cdot \mathbf = 0]
[\mathbf] is the magnetic flux density (in units of teslas, T), also called the magnetic induction.

Equivalent integral form:

[\oint_S \mathbf \cdot d\mathbf = 0]
[d\mathbf] 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 the mathematical formulation of the asumption that there are no magnetic monopoles.

A changing magnetic flux and the electric field

[\nabla \times \mathbf = -\frac }]
Equivalent integral Form:

[ \oint_ \mathbf \cdot d\mathbf = - \frac }} ] where [ \Phi_} = \int_ \mathbf \cdot d\mathbf]
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

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

The electromotive force (sometimes denoted [\mathcal], not to be confused with the permittivity above) is equal to the value of this integral.

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

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 the flux derivative. The N can be taken care of in calculating A (multiple wire coils means multiple surfaces for the flux to go through), and it is an engineering detail so it has been omitted here.

The negative sign is necessary to maintain conservation of energy. It is so important that it even has its own name, Lenz's law.

This equation relates the electric and magnetic fields, but it also has a lot of practical applications, too. 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 employ the reverse configuration.

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

The source of the magnetic field

[ \nabla \times \mathbf = \mathbf + \frac } ]
where H is the magnetic field strength (in units of A/m), related to the magnetic flux B by a constant called the permeability, μ (B = μH), and J is the current density, defined by: [\begin\mathbf = \rho_q\mathbf\end] where v is a vector field called the drift velocity that describes the velocities of the charge carriers which have a density described by the scalar function ρq.

In free space, the permeability μ is the permeability of free space, μ0, which is defined to be exactly 4π×10-7 W/A·m. Also, the permittivity becomes the permittivity of free space ε0. Thus, in free space, the equation becomes:

[\nabla \times \mathbf = \mu_0 \mathbf + \mu_0\varepsilon_0 \frac}]
Equivalent integral form:

[\oint_C \mathbf \cdot d\mathbf = \mu_0 I_\mathrm + \mu_0\varepsilon_0 \int_S \frac} \cdot d \mathbf]
[\mathbf] is the edge of the open surface A (any surface with the curve [\mathbf] as its edge will do), and Iencircled is the current encircled by the curve [\mathbf] (the current through any surface is defined by the equation: [\beginI_\ A} = \int_S \mathbf\cdot d\mathbf\end]).In some situations, this integral form of Ampere-Maxwell Law appears in:

[\oint_C \mathbf \cdot d\mathbf = \mu_0 (I_\mathrm + I_\mathrm)]
for

[\varepsilon_0 \int_S \frac} \cdot d \mathbf]
is sometimes called displacement current

If the electric flux density does not vary rapidly, the second term on the right hand side (the displacement flux) is negligible, and the equation reduces to Ampere's law.

Maxwell's equations in CGS units

The above equations are given in the International System of Units, or SI for short. In a related unit system, called cgs (short for centimeter-gram-second), the equations take the following form:

[ \nabla \cdot \mathbf = 4\pi\rho]
[ \nabla \cdot \mathbf = 0]
[ \nabla \times \mathbf = -\frac \frac} ]
[ \nabla \times \mathbf = \frac \frac} + \frac \mathbf]
Where c is the speed of light in a vacuum. For the electromagnetic field in a vacuum, the equations become:

[\nabla \cdot \mathbf = 0]
[\nabla \cdot \mathbf = 0]
[\nabla \times \mathbf = -\frac \frac} ]
[\nabla \times \mathbf = \frac \frac} ]
The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:

[\mathbf = q (\mathbf + \frac} \times \mathbf),]
where [ q \ ] is the charge on the particle and [ \mathbf \ ] is the particle velocity. This is slightly different from the SI-unit expression above. For example, here the magnetic field [ \mathbf \ ] has the same units as the electric field [ \mathbf \ ].

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):

[ J^ b = \over } \equiv \partial_a F^ \equiv }_ \,\!],
and

[0 = \partial_c F_ + \partial_b F_ + \partial_a F_ \equiv }_ + }_ +}_ \equiv \epsilon_ }_ ]
where [\, J^a] is the 4-current, [\, F^] is the field strength tensor, [\, \epsilon_] is the Levi-Civita symbol, and

[ } \equiv \partial_a \equiv _ \equiv (\partial/\partial ct, \nabla)]
is the 4-gradient. Repeated indices are summed over according to Einstein summation convention. We have displayed the results in several common notations.

The first tensor equation is an expression of the two inhomogeneous Maxwell's equations, Gauss' law and Ampere's law with Maxwell's correction. The second equation is an expression of the homogenous equations, Faraday's law of induction and the absence of magnetic monopoles.

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

[d\bold=0]
where d denotes the exterior derivative - a differential operator acting on forms - and the source equation
[d * }=\bold]
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 [1/4\pi\epsilon_0=1]. Here, the 3-form J is called the "electric current" or "current (3-)form" satisfying the continuity equation
[d}=0]
As the exterior derivative is defined on any manifold, this formulation of electromagnetism works for any 4-dimensional oriented manifold with a Lorentz metric, e.g. on the curved space-time of general relativity.

In a linear, macroscopic theory, the influence of matter on the electromagnetic field is described through more general linear transformation in the space of 2-forms. We call

[ C:\Lambda^2\ni\bold\mapsto \bold\in\Lambda^]
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:
[ d\bold = 0]
[ d\bold = \bold]
where the tree current form J still satisfies the continuity equation dJ= 0.

When the fields are expressed as linear combinations (of exterior products) of basis forms [\bold^p],

[ \bold = F_\bold^p\wedge\bold^q].
the constitutive relation takes the form
[ G_ = C_^F_]
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
[ C_^ = g^g^ \epsilon_ \sqrt ]
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.

Often, the time derivative in the third law motivates calling this equation "dynamical", which is somehow 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 A→A' = A-dα: see also gauge fixing and Fadeev-Popov ghosts.

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 [\nabla] on the line bundle has a curvature [\bold = \nabla^2] which is a two form that automatically satisfies [ d\bold = 0 ] and can be interpreted as a field strength. If the line bundle is trivial with flat reference connection d we can write [\nabla = d+\bold] and F = d A with A the 1-form comprised of the electric potential and the magnetic vector potential.

In quantum mechanics, the connection itself is used to define the dynamics of the system. Some feel that this formulation allows a more natural description of the Aharonov-Bohm effect. In this system a magnetic field shielded by a long super conducting tube defines a flat (F = 0) but non trivial connection outside of the tube. The connection has a non trivial holonomy along a curve encircling the tube which corresponds to a phase shift for electrons waves travelling either side of the tube. This can be detected by a double split electron diffraction experiment by changing the magnetic field. The magnetic field remains constant zero outside of the tube so is undectable classically. (See Michael Murray, [Line Bundles], 2002 (PDF web link) for a simple mathematical review of this formulation. See also R. Bott, On some recent interactions between mathematics and physics, Canadian Mathematical Bulletin, 28 (1985) )no. 2 pp 129-164.)''

Maxwell's equations in curved spacetime

Traditional formulation

Matter and energy generate curvature in spacetime. This is the subject of general relativity. Curvature of spacetime affects electrodynamics. Electrodynamics also generates 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. The sourced and source-free equations become (cgs units):

[ J^ b = \partial_a F^ + _ F^ + _ F^ \equiv D_a F^ \equiv }_ \,\!],
and

[0 = \partial_c F_ + \partial_b F_ + \partial_a F_ = D_c F_ + D_b F_ + D_a F_].
Here,

[_ \!]
is a Christoffel symbol that characterizes the curvature of spacetime and [ D_c ] is the covariant derivative.

Formulation in terms of differential forms

The above formulation is related to the differential form formulation of the Maxwell equations as follows. We have implicitly chosen local coordinates xa and therefore have a basis of 1-forms d xa in every point of the open set where the coordinates are defined. Using this basis we have:

[ \bold = F_ d\,x^a \wedge d\,x^b]
  • The current form
    • [ \bold = J^a \sqrt \, \epsilon_ d\,x^b \wedge d\,x^c \wedge d\,x^d]
  • the Bianchi identity
    • [ d\bold = 2(\partial_c F_ + \partial_b F_ + \partial_a F_)d\,x^a\wedge d\,x^b \wedge d\,x^c = 0]
  • the source equation
    • [ d * \bold = }_\sqrt \, \epsilon_d\,x^c \wedge d\,x^d \wedge d\,x^e = \bold]
  • the continuity equation
    • [ d\bold = _ \sqrt \, \epsilon_d\,x^a\wedge d\,x^b \wedge d\,x^c \wedge d\,x^d = 0]
    Here g is as usual the determinant of the metric tensor gab.

    See also

    References

    Journal articles

    The developments before relativity see

    University level textbooks

    Undergraduate

    Graduate

    External links


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