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. 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.
Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields propagate at a constant speed in vacuum, c. 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 a coherent system of units, Maxwell's microscopic equations can be written as
With the electric field, the magnetic field, the electric charge density and the current density. is the vacuum permittivity and the vacuum permeability.
The equations have two major variants:

The microscopic equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale.
The macroscopic equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic-scale charges and quantum phenomena like spins. However, their use requires experimentally determined parameters for a phenomenological description of the electromagnetic response of materials.

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 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. 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.

History of the equations

Conceptual descriptions

Gauss's law

describes the relationship between an electric field and electric charges: an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of free space.

Gauss's law for magnetism

states that electric charges have no magnetic analogues, called magnetic monopoles; no north or south magnetic poles exist in isolation. Instead, the magnetic field of a material is attributed to a dipole, and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field.

Faraday's law

The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds to the negative curl of an electric field. In integral form, it states that the work per unit charge required to move a charge around a closed loop equals the rate of change of the magnetic flux through the enclosed surface.
The electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field and generates an electric field in a nearby wire.

Ampère–Maxwell law

The original law of Ampère states that magnetic fields relate to electric current. Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.
Maxwell's modification of Ampère's circuital law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields. As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space.
The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, matches the speed of light; indeed, light is one form of electromagnetic radiation. Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.

Formulation in terms of electric and magnetic fields (microscopic or in vacuum version)

In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate law of nature, the Lorentz force law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. The vector calculus formalism below, the work of Oliver Heaviside, has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations in x, y and z components. The [|relativistic formulations] are more symmetric and Lorentz invariant. For the same equations expressed using tensor calculus or differential forms.
The differential and integral formulations are mathematically equivalent; both are useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely local and are a more natural starting point for calculating the fields in more complicated situations, for example using finite element analysis.

Key to the notation

Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated.
The equations introduce the electric field,, a vector field, and the magnetic field,, a pseudovector field, each generally having a time and location dependence.
The sources are

the total electric charge density,, and
the total electric current density,.

The universal constants appearing in the equations are:

the permittivity of free space,, and
the permeability of free space,, and
the speed of light,
Differential equations

In the differential equations,

the nabla symbol,, denotes the three-dimensional gradient operator, del,
the symbol denotes the divergence operator,
the symbol denotes the curl operator.
Integral equations

In the integral equations,

is any volume with closed boundary surface, and
is any surface with closed boundary curve,

The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law:
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.

is a surface integral over the boundary surface, with the loop indicating the surface is closed
is a volume integral over the volume,
is a line integral around the boundary curve, with the loop indicating the curve is closed.
is a surface integral over the surface,
The total electric charge enclosed in is the volume integral over of the charge density : where is the volume element.
The net magnetic flux is the surface integral of the magnetic field passing through a fixed surface, :
The net electric flux is the surface integral of the electric field passing through :
The net electric current is the surface integral of the electric current density passing through : where denotes the differential vector element of surface area, normal to surface..
Formulation in the SI

Name	Integral equations	Differential equations
Gauss's law
Gauss's law for magnetism
Maxwell–Faraday equation
Ampère–Maxwell law