# Maxwell's equations

Maxwell equationsMaxwell equationMaxwell’s equationsMaxwell's equationequationsMaxwell's field equationsMaxwell's theoryelectromagnetic theoryMaxwellelectromagnetism
Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.wikipedia
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### James Clerk Maxwell

MaxwellJ. C. MaxwellJames Maxwell
The equations are named after the physicist and mathematician James Clerk Maxwell, who published an early form of the equations that included the Lorentz force law between 1861 and 1862.
Maxwell's equations for electromagnetism have been called the "second great unification in physics" after the first one realised by Isaac Newton.

### Covariant formulation of classical electromagnetism

Formulation of Maxwell's equations in special relativityspecial relativity formcan be rewritten
The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest.
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.

### Special relativity

special theory of relativityrelativisticspecial
The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest.
The incompatibility of Newtonian mechanics with Maxwell's equations of electromagnetism and, experimentally, the Michelson-Morley null result (and subsequent similar experiments) demonstrated that the historically hypothesized luminiferous aether did not exist.

### Quantum mechanics

quantum physicsquantum mechanicalquantum theory
Versions of Maxwell's equations based on the electric and magnetic potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics.
Ludwig Boltzmann independently arrived at this result by considerations of Maxwell's equations.

### Lorentz force

magnetic forceLorentz force lawLorentz
Versions of Maxwell's equations based on the electric and magnetic potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics. Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
By eliminating ρ and J, using Maxwell's equations, and manipulating using the theorems of vector calculus, this form of the equation can be used to derive the Maxwell stress tensor σ, in turn this can be combined with the Poynting vector S to obtain the electromagnetic stress–energy tensor T used in general relativity.

### Maxwell's equations in curved spacetime

curved space Maxwell equationscurved spacetime Maxwell equationscurved spacetime Maxwell field equations
Maxwell's equations in curved spacetime, commonly used in high energy and gravitational physics, are compatible with general relativity.
These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime.

### Gauss's law for magnetism

Gauss' law for magnetismGauss's lawfor magnetism
Gauss's law for magnetism states that there are no "magnetic charges" (also called magnetic monopoles), analogous to electric charges.
In physics, Gauss's law for magnetism is one of the four Maxwell's equations that underlie classical electrodynamics.

### Classical field theory

field equationsclassical field theoriesfield theory
Since the mid-20th century, it has been understood that Maxwell's equations are not exact, but a classical limit of the fundamental theory of quantum electrodynamics.
The first field theories, Newtonian gravitation and Maxwell's equations of electromagnetic fields were developed in classical physics before the advent of relativity theory in 1905, and had to be revised to be consistent with that theory.

### Gauss's law

Gauss' lawGauss lawGauss
Gauss's law describes the relationship between a static electric field and the electric charges that cause it: a static electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through any closed surface is proportional to the charge enclosed by the surface.
It is one of Maxwell's four equations, which form the basis of classical electrodynamics.

Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays.
His mathematical theory, now called Maxwell's equations, predicted that a coupled electric and magnetic field could travel through space as an "electromagnetic wave".

### Electromagnetic spectrum

spectrumspectraspectral
Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays.
During the 1860s James Maxwell developed four partial differential equations for the electromagnetic field.

### Displacement current

displacementdisplacement currentsMaxwell displacement current
Ampère's law with Maxwell's addition states that magnetic fields can be generated in two ways: by electric current (this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's addition", which he called displacement current).
appearing in Maxwell's equations that is defined in terms of the rate of change of

### Magnetic potential

magnetic vector potentialvector potentialmagnetic scalar potential
Versions of Maxwell's equations based on the electric and magnetic potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics.
If electric and magnetic fields are defined as above from potentials, they automatically satisfy two of Maxwell's equations: Gauss's law for magnetism and Faraday's Law.

### Speed of light

clight speedspeed of light in vacuum
An important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in a vacuum.
The classical behaviour of the electromagnetic field is described by Maxwell's equations, which predict that the speed c with which electromagnetic waves (such as light) propagate through the vacuum is related to the distributed capacitance and inductance of the vacuum, otherwise respectively known as the electric constant ε 0 and the magnetic constant μ 0, by the equation

### Electromagnetic induction

inductionmagnetic inductioninduced
This aspect of electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field, which in turn generates an electric field in a nearby wire.
Faraday's law was later generalized to become the Maxwell–Faraday equation, one of the four Maxwell equations in his theory of electromagnetism.

### Oliver Heaviside

HeavisideHeaviside, OliverHeaviside|Heaviside's operators
The vector calculus formalism below, the work of Oliver Heaviside, has become standard.
Oliver Heaviside FRS (18 May 1850 – 3 February 1925) was an English self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations (equivalent to Laplace transforms), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.

### Electric field

electricelectrostatic fieldelectrical field
Gauss's law describes the relationship between a static electric field and the electric charges that cause it: a static electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through any closed surface is proportional to the charge enclosed by the surface. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of the fields. The Maxwell–Faraday version of Faraday's law of induction describes how a time varying magnetic field creates ("induces") an electric field.
However, since the magnetic field is described as a function of electric field, the equations of both fields are coupled and together form Maxwell's equations that describe both fields as a function of charges and currents.

### Ampère's circuital law

Ampère's lawAmpere's lawAmpere's circuital law
Ampère's law with Maxwell's addition states that magnetic fields can be generated in two ways: by electric current (this was the original "Ampère's law") and by changing electric fields (this was "Maxwell's addition", which he called displacement current).
James Clerk Maxwell (not Ampère) derived it using hydrodynamics in his 1861 paper "" and it is now one of the Maxwell equations, which form the basis of classical electromagnetism.

### Electromagnetic wave equation

Therefore, these equations allow self-sustaining "electromagnetic waves" to travel through empty space (see electromagnetic wave equation).
The electromagnetic wave equation derives from Maxwell's equations.

The Maxwell–Faraday version of Faraday's law of induction describes how a time varying magnetic field creates ("induces") an electric field.
The Maxwell–Faraday equation (listed as one of Maxwell's equations) describes the fact that a spatially varying (and also possibly time-varying, depending on how a magnetic field varies in time) electric field always accompanies a time-varying magnetic field, while Faraday's law states that there is EMF (electromotive force, defined as electromagnetic work done on a unit charge when it has traveled one round of a conductive loop) on the conductive loop when the magnetic flux through the surface enclosed by the loop varies in time.

electromagnetic waveelectromagnetic waveselectromagnetic
Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. Therefore, these equations allow self-sustaining "electromagnetic waves" to travel through empty space (see electromagnetic wave equation).
According to Maxwell's equations, a spatially varying electric field is always associated with a magnetic field that changes over time.

### Classical electromagnetism

classical electrodynamicselectrodynamicsclassical
Maxwell's equations are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
From Maxwell's equations, it is clear that ∇ × E is not always zero, and hence the scalar potential alone is insufficient to define the electric field exactly.

### Electromagnetism

electromagneticelectrodynamicselectromagnetic force
Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.
In Faraday's law, magnetic fields are associated with electromagnetic induction and magnetism, and Maxwell's equations describe how electric and magnetic fields are generated and altered by each other and by charges and currents.

### Vacuum permittivity

permittivity of free spaceelectric constantvacuum electric permittivity
'Counting' the number of field lines passing through a closed surface yields the total charge (including bound charge due to polarization of material) enclosed by that surface, divided by dielectricity of free space (the vacuum permittivity).
Likewise, ε 0 appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources.

### Spacetime

space-timespace-time continuumspace and time
The covariant formulation (on spacetime rather than space and time separately) makes the compatibility of Maxwell's equations with special relativity manifest.
A consequence of Maxwell's theory of electromagnetism is that light carries energy and momentum, and that their ratio is a constant:.