Categories: Electromagnetism | Partial differential equations | Equations

Maxwell's equations

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 electric fields (Gauss' law), the experimental absence of magnetic charges, how currents produce magnetic fields (Ampere'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 equations together and to notice that a correction was required to Ampere's law: changing electric fields act like currents, likewise producing magnetic fields.

Furthermore, Maxwell showed that the four equations, with his correction, predict waves of oscillating electric and magnetic fields that 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 its vindication 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.

Contents

1 Historical developments of Maxwell's equations and relativity

2 Summary of the equations

2.1 General case
2.2 In linear materials
2.3 In vacuum, without charges or currents

3 Detail

3.1 Charge density and the electric field
3.2 The structure of the magnetic field
3.3 A changing magnetic field and the electric field
3.4 The source of the magnetic field

4 Maxwell's equations in CGS units

5 Formulation of Maxwell's equations in special relativity

6 Maxwell's equations in terms of differential forms

7 See also

8 References

Historical developments of Maxwell's equations and 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 corrected Ampere's law (three component equations), Gauss' law for 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), the relationship between electric field and the scalar and vector potentials (three component equations, which imply Faraday's law), the relationship between the electric and displacement fields (three component equations), Ohm's law relating current density and electric field (three component equations), and the continuity equation relating current density 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 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.

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

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

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). 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 the aether, however, alternative explanations were sought by Lorentz and others. This culminated in Einstein's theory of special relativity, which postulated the absence of any absolute 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: the magnetic field equations can be derived from consideration of the transformation of the electric field equations under relativistic transformations at low velocities. (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

General case

Name	Partial differential form	Integral form
Gauss' law:	$\nabla \cdot \mathbf{D} = \rho$	$\oint_S \mathbf{D} \cdot d\mathbf{s} = Q_{\mathrm{encl}}$
Gauss' law for magnetism:	$\nabla \cdot \mathbf{B} = 0$	$\oint_S \mathbf{B} \cdot d\mathbf{s} = 0$
Faraday's law of induction:	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$	$\oint_C \mathbf{E} \cdot d\mathbf{l} = -\int_{\partial C} \ {d\mathbf{B}\over dt} \cdot d\mathbf{s}$
Ampere's law + Maxwell's extension:	$\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}$	$\oint_C \mathbf{H} \cdot d\mathbf{l} = I_{\mathrm{encl}} + \frac{d \mathbf{\Phi_D}}{dt}$

where:

ρ

is the free electric charge density (SI unit: coulomb per cubic meter), not including dipole charges bound in a material

$\mathbf{B}$ is the magnetic flux density (SI unit: tesla), also called the magnetic induction.

$\mathbf{D}$ is the electric displacement field (SI unit: coulomb per square meter).

$\mathbf{E}$ is the electric field (SI unit: volt per meter),

$\mathbf{H}$ is the magnetic field strength (SI unit: ampere per meter)

$\mathbf{J}$ is the current density (SI unit: ampere per square meter)

$\nabla \cdot$ is the divergence operator (SI unit: 1 per meter),

$\nabla \times$ is the curl operator (SI unit: 1 per meter).

Note that 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:

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

where $q \$ is the charge on the particle and $\mathbf{v} \$ is the particle velocity. Note that this is slightly different when expressed in the cgs system of units below.

It is important to note that 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 D and H fields are related to E and B by:

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

$\mathbf{B} = \mu \mathbf{H}$

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's equations reduce to

$\nabla \cdot \varepsilon \mathbf{E} = \rho$

$\nabla \cdot \mathbf{B} = 0$

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

$\nabla \times \mathbf{B / \mu} = \mathbf{J} + \varepsilon \frac{\partial \mathbf{E}} {\partial t}$

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 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). If there is no current or electric charge present in the vacuum, we obtain the Maxwell's equations in free space:

$\nabla \cdot \mathbf{E} = 0$

$\nabla \cdot \mathbf{B} = 0$

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

$\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}} {\partial t}$

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

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

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

Detail

Charge density and the electric field

$\nabla \cdot \mathbf{D} = \rho$ ,

where $ρ$ is the free electric charge density (in units of C/m³), not including dipole charges bound in a material, and $\mathbf{D}$ 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:

$\oint_A \mathbf{D} \cdot d\mathbf{A} = Q_\mbox{enclosed}$

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

In a linear material, $\mathbf{D}$ is directly related to the electric field $\mathbf{E}$ via a material-dependent constant called the permittivity, $ε$ :

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

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 $ε 0$ , and appears in:

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

where, again, $\mathbf{E}$ is the electric field (in units of V/m), $ρ t$ is the total charge density (including bound charges), and $ε 0$ (approximately 8.854 pF/m) is the permittivity of free space. $ε$ can also be written as $\varepsilon_0 \cdot \varepsilon_r$ , where $ε r$ is the material's relative permittivity or its dielectric constant.

Compare Poisson's equation.

The structure of the magnetic field

$\nabla \cdot \mathbf{B} = 0$

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

Equivalent integral form:

$\oint_A \mathbf{B} \cdot d\mathbf{A} = 0$

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

Note: 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 assumption that there are no magnetic monopoles.

A changing magnetic field and the electric field

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

Equivalent integral Form:

$\oint_{s} \mathbf{E} \cdot d\mathbf{s} = - \frac {d\Phi_{\mathbf{B}}} {dt}$ where $\Phi_{\mathbf{B}} = \int_{A} \mathbf{B} \cdot d\mathbf{A}$

where

Φ_B 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 $\mathcal{E}$ , 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.

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 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.

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

This equation relates the electric and magnetic fields, but it also has a 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.

Note: 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

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

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: J = ∫ρ_qvdV 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 ρ_q.

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:

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

Equivalent integral form:

$\oint_s \mathbf{B} \cdot d\mathbf{s} = \mu_0 I_\mbox{encircled} + \mu_0\varepsilon_0 \int_A \frac{\partial \mathbf{E}}{\partial t} \cdot d \mathbf{A}$

s is the edge of the open surface A (any surface with the curve s as its edge will do), and I_encircled is the current encircled by the curve s (the current through any surface is defined by the equation: I_{through A} = ∫_AJ·dA).

Note: 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 on a more symmetrical form, as follows:

$\nabla \cdot \mathbf{E} = 4\pi\rho$

$\nabla \cdot \mathbf{B} = 0$

$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}$

$\nabla \times \mathbf{B} = \frac{1}{c} \frac{ \partial \mathbf{E}} {\partial t} + \frac{4\pi}{c} \mathbf{J}$

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

$\nabla \cdot \mathbf{E} = 0$

$\nabla \cdot \mathbf{B} = 0$

$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}$

$\nabla \times \mathbf{B} = \frac{1}{c} \frac{\partial \mathbf{E}}{\partial t}$

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

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

where $q \$ is the charge on the particle and $\mathbf{v} \$ is the particle velocity. Note that this is slightly different from the SI-unit expression above. For example, here the magnetic field $\mathbf{B} \$ has the same units as the electric field $\mathbf{E} \$ .

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

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:

$J^\beta = \partial_\alpha F^{\alpha\beta} \,\!$ ,

and

$0 = \partial_\gamma F_{\alpha\beta} + \partial_\beta F_{\gamma\alpha} + \partial_\alpha F_{\beta\gamma} \,\!$

where J is the 4-current, F is the field strength tensor (Faraday tensor) (written as a 4 × 4 matrix), and $\partial_\alpha = (\partial/\partial ct, \nabla)$ is the 4-gradient (so that $\partial_\alpha \partial^\alpha$ is the d'Alembertian operator). (The α in the first equation is implicitly summed over, according to Einstein notation.) The first tensor equation expresses the two inhomogeneous Maxwell's equations: Gauss' law and Ampere's law with Maxwell's correction. The second 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 ρ and the current density J. In terms of the 4-potential (as a contravariant vector) $\tilde{A}^{\alpha} = \left(\phi, \mathbf{A} c \right)$ , where φ is the electric potential and A is the magnetic vector potential in the Lorenz gauge $\left ( \partial_\alpha \tilde{A}^\alpha = 0 \right )$ , F can be expressed as:

$F^{\alpha\beta} = \partial^\alpha \tilde{A}^\beta - \partial^\beta \tilde{A}^\alpha \,\!$

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

$F^{\alpha\beta} = \left( \begin{matrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{matrix} \right) .$

The fact that both electric and magnetic fields are combined into a single tensor expresses the 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 can appear as a magnetic field in another frame, and vice versa.

(See Electromagnetic four-potential for the relationship between the d'Alembertian of the four-potential and the four-current, expressed in terms of the older vector operator notation).

Note that different authors sometimes employ different sign conventions for the above tensors and 4-vectors (which does not affect the physical interpretation). Note also that F^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 changes in some of F's components; more complex metric dualities are encountered in general relativity.

Maxwell's equations in terms of differential forms

In a vacuum, where ε and μ are constant everywhere, Maxwell's equations simplify considerably 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 then reduce to the Bianchi identity

$d\bold{F}=0$

where d is the exterior derivative, and the source equation

$d*\bold{F}=*\bold{J}$

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

$d*\bold{J}=0$

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 the Royal 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 ISBN 0-486-60637-6).
John David Jackson, Classical Electrodynamics (Wiley, New York, 1998).
Edward M. Purcell, Electricity and Magnetism (McGraw-Hill, New York, 1985).
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". Advanced Classical Electromagnetism, PHY387K. University of Texas at Austin, Fall 1996.

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