Maxwell's equations

[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]).

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.

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 integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.

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 finite element analysis.

Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by using the differential version and using Gauss' and Stokes' theorems as appropriate.

With a corresponding change in the values of the quantities for the Lorentz force law this yields the same physics, i.e. trajectories of charged particles, or work done by an electric motor.

These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the electromagnetic tensor: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension.

Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives:

This perpetual cycle allows these waves, now known as electromagnetic radiation, to move through space at velocity c. 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.

In the macroscopic equations, the influence of bound charge Qb and bound current Ib is incorporated into the displacement field D and the magnetizing field H, while the equations depend only on the free charges Qf and free currents If.

This reflects a splitting of the total electric charge Q and current I (and their densities ρ and J) into free and bound parts:

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.

When an electric field is applied to a dielectric material its molecules respond by forming microscopic electric dipoles – their atomic nuclei move a tiny distance in the direction of the field, while their electrons move a tiny distance in the opposite direction.

The bound charge is most conveniently described in terms of the polarization P of the material, its dipole moment per unit volume.

The direct spacetime formulations make manifest that the Maxwell equations are relativistically invariant, where space and time are treated on equal footing.

Because of this symmetry, the electric and magnetic fields are treated on equal footing and are recognized as components of the Faraday tensor.

Maxwell equations in formulation that do not treat space and time manifestly on the same footing have Lorentz invariance as a hidden symmetry.

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.

Other formalisms include the geometric algebra formulation and a matrix representation of Maxwell's equations.

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.

In some cases, Maxwell's equations are solved over the whole of space, and boundary conditions are given as asymptotic limits at infinity.

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.

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 the Lorentz force law (along with the rest of classical electromagnetism) are extraordinarily successful at explaining and predicting a variety of phenomena.

This is known as semiclassical theory or self-field QED and was initially discovered by de Broglie and Schrödinger and later fully developed by E.T.