Maxwell's equations

  • maxwell's equations (mid-left) as featured on a monument in front of warsaw university's center of new technologies

    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. 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.[note 1] 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. known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. 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 first used the equations to propose that light is an electromagnetic phenomenon.

    the equations have two major variants. the microscopic maxwell 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" maxwell 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 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, 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 previously described phenomena: magnetism, electricity, light and associated radiation. 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.

  • conceptual descriptions
  • formulation in terms of electric and magnetic fields (microscopic or in vacuum version)
  • relationship between differential and integral formulations
  • charge conservation
  • vacuum equations, electromagnetic waves and speed of light
  • macroscopic formulation
  • alternative formulations
  • relativistic formulations
  • solutions
  • overdetermination of maxwell's equations
  • maxwell's equations as the classical limit of qed
  • variations
  • see also
  • notes
  • references
  • historical publications
  • further reading
  • external links

Maxwell's equations (mid-left) as featured on a monument in front of Warsaw University's Center of New Technologies

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. 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.[note 1] 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. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to γ-rays. 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 first used the equations to propose that light is an electromagnetic phenomenon.

The equations have two major variants. The microscopic Maxwell 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" Maxwell 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 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, 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 previously described phenomena: magnetism, electricity, light and associated radiation. 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.