Maxwell's equations

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]

For thermodynamic relations, see Maxwell relations.

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 partial differential equation form and SI units, Maxwell's microscopic equations can be written as ${\begin{aligned}\nabla \cdot \mathbf {E} \,\,\,&={\frac {\rho }{\varepsilon _{0}}}\\\nabla \cdot \mathbf {B} \,\,\,&=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {B} &=\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\end{aligned}}$ With $\mathbf {E}$ the electric field, $\mathbf {B}$ the magnetic field, $\rho$ the electric charge density and $\mathbf {J}$ the current density. $\varepsilon _{0}$ is the vacuum permittivity and $\mu _{0}$ the vacuum permeability.

The equations have two major variants:

The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest. Maxwell's equations in curved spacetime, commonly used in high-energy and gravitational physics, are compatible with general relativity.^{[note 2]} In fact, Albert Einstein developed special and general relativity to accommodate the invariant speed of light, a consequence of Maxwell's equations, with the principle that only relative movement has physical consequences.

The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. Since the mid-20th century, it has been understood that Maxwell's equations do not give an exact description of electromagnetic phenomena, but are instead a classical limit of the more precise theory of quantum electrodynamics.

the total electric (total charge per unit volume), $ρ$ , and

charge density

the total electric (total current per unit area), $J$ .

current density

For homogeneous materials, $ε$ and $μ$ are constant throughout the material, while for inhomogeneous materials they depend on within the material (and perhaps time).^[16]^: 463

location

For isotropic materials, $ε$ and $μ$ are scalars, while for anisotropic materials (e.g. due to crystal structure) they are .^[15]^: 421^[16]^: 463

tensors

Materials are generally , so $ε$ and $μ$ depend on the frequency of any incident EM waves.^[15]^: 625^[16]^: 397

dispersive

In the tensor calculus formulation, the $F αβ$ is an antisymmetric covariant order 2 tensor; the four-potential, $A α$ , is a covariant vector; the current, $J α$ , is a vector; the square brackets, $[ ]$ , denote antisymmetrization of indices; $\partial α$ is the partial derivative with respect to the coordinate, $x α$ . In Minkowski space coordinates are chosen with respect to an inertial frame; $(x α) = (ct, x, y, z)$ , so that the metric tensor used to raise and lower indices is $η αβ = diag(1, -1, -1, -1)$ . The d'Alembert operator on Minkowski space is $◻ = \partial α \partial α$ as in the vector formulation. In general spacetimes, the coordinate system $x α$ is arbitrary, the covariant derivative $\nabla α$ , the Ricci tensor, $R αβ$ and raising and lowering of indices are defined by the Lorentzian metric, $g αβ$ and the d'Alembert operator is defined as $◻ = \nabla α \nabla α$ . The topological restriction is that the second real cohomology group 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.

electromagnetic tensor

In the formulation on arbitrary space times, $F = 1 / 2 F αβ d x α \land d x β$ is the electromagnetic tensor considered as a 2-form, $A = A α d x α$ is the potential 1-form, $J=-J_{\alpha }{\star }\mathrm {d} x^{\alpha }$ is the current 3-form, $d$ is the exterior derivative, and ${\star }$ is the Hodge star on 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 ${\star }$ 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 $\Box =(-{\star }\mathrm {d} {\star }\mathrm {d} -\mathrm {d} {\star }\mathrm {d} {\star })$ 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.

differential form

The Maxwell equations can also be formulated on a spacetime-like Minkowski space where space and time are treated on equal footing. The direct spacetime formulations make manifest that the Maxwell equations are relativistically invariant. Because of this symmetry, the electric and magnetic fields are treated on equal footing and are recognized as components of the Faraday tensor. This reduces the four Maxwell equations to two, which simplifies the equations, although we can no longer use the familiar vector formulation. In fact the Maxwell equations in the space + time formulation are not Galileo invariant and have Lorentz invariance as a hidden symmetry. This was a major source of inspiration for the development of relativity theory. Indeed, even the formulation that treats space and time separately is not a non-relativistic approximation and describes the same physics by simply renaming variables. For this reason the relativistic invariant equations are usually called the Maxwell equations as well.

Each table below describes one formalism.

Other formalisms include the geometric algebra formulation and a matrix representation of Maxwell's equations. Historically, a quaternionic formulation^[17]^[18] was used.

Solutions[edit]

Maxwell's equations are partial differential equations that relate the electric and magnetic fields to each other and to the electric charges and currents. Often, the charges and currents are themselves dependent on the electric and magnetic fields via the Lorentz force equation and the constitutive relations. These all form a set of coupled partial differential equations which are often very difficult to solve: the solutions encompass all the diverse phenomena of classical electromagnetism. Some general remarks follow.

As for any differential equation, boundary conditions^[19]^[20]^[21] and initial conditions^[22] are necessary for a unique solution. For example, even with no charges and no currents anywhere in spacetime, there are the obvious solutions for which E and B are zero or constant, but there are also non-trivial solutions corresponding to electromagnetic waves. In some cases, Maxwell's equations are solved over the whole of space, and boundary conditions are given as asymptotic limits at infinity.^[23] In other cases, Maxwell's equations are solved in a finite region of space, with appropriate conditions on the boundary of that region, for example an artificial absorbing boundary representing the rest of the universe,^[24]^[25] or periodic boundary conditions, or walls that isolate a small region from the outside world (as with a waveguide or cavity resonator).^[26]

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.

Numerical methods for differential equations can be used to compute approximate solutions of Maxwell's equations when exact solutions are impossible. These include the finite element method and finite-difference time-domain method.^[19]^[21]^[27]^[28]^[29] For more details, see Computational electromagnetics.

Overdetermination of Maxwell's equations[edit]

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 $\nabla \cdot \nabla \times \mathbf {B} \equiv 0,\nabla \cdot \nabla \times \mathbf {E} \equiv 0$ , 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[edit]

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.

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, :10.1007/978-94-015-8422-7_16, ISBN 978-90-481-4525-6