Maxwell’s equations are a set of coupled partial differential equations that, together

7 min readMay 29, 2021

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. 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.

An important consequence of Maxwell’s equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed © in a vacuum. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum of light from radio waves to gamma rays.

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 (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.

Contents
1 Conceptual descriptions
1.1 Gauss’s law
1.2 Gauss’s law for magnetism
1.3 Faraday’s law
1.4 Ampère’s law with Maxwell’s addition
2 Formulation in terms of electric and magnetic fields (microscopic or in vacuum version)
2.1 Key to the notation
2.1.1 Differential equations
2.1.2 Integral equations
2.2 Formulation in SI units convention
2.3 Formulation in Gaussian units convention
3 Relationship between differential and integral formulations
3.1 Flux and divergence
3.2 Circulation and curl
4 Charge conservation
5 Vacuum equations, electromagnetic waves and speed of light
6 Macroscopic formulation
6.1 Bound charge and current
6.2 Auxiliary fields, polarization and magnetization
6.3 Constitutive relations
7 Alternative formulations
8 Relativistic formulations
9 Solutions
10 Overdetermination of Maxwell’s equations
11 Maxwell’s equations as the classical limit of QED
12 Variations
12.1 Magnetic monopoles
13 See also
14 Notes
15 References
16 Historical publications
17 Further reading
18 External links
18.1 Modern treatments
18.2 Other
Conceptual descriptions
Gauss’s law
Main article: Gauss’s law
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. This means that the net flux 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).

Gauss’s law for magnetism: magnetic field lines never begin nor end but form loops or extend to infinity as shown here with the magnetic field due to a ring of current.
Gauss’s law for magnetism
Main article: Gauss’s law for magnetism
Gauss’s law for magnetism states that there are no “magnetic charges” (also called magnetic monopoles), analogous to electric charges.[1] Instead, the magnetic field due to materials is generated by a configuration called a dipole, and the net outflow of the magnetic field through any closed surface is zero. Magnetic dipoles are best represented as loops of current but resemble positive and negative ‘magnetic charges’, inseparably bound together, having no net ‘magnetic charge’. Equivalent technical statements are that the sum total magnetic flux through any Gaussian surface is zero, or that the magnetic field is a solenoidal vector field.[note 3]

https://vexare4629.medium.com/martin-morocky-born-31-march-1983-known-by-his-902c97847c76
https://arrabble.medium.com/mshasho-clothing-is-a-namibian-clothing-brand-founded-db8543163031
https://arrabble.medium.com/martin-morocky-born-31-march-1983-known-by-his-stage-name-as-the-dogg-646979461508
https://vexare4629.medium.com/namibian-music-awards-441b2392403b
https://vexare4629.medium.com/namibia-n%C9%99%CB%88m%C9%AAbi%C9%99-about-this-soundlisten-n%C3%A6%CB%88-16-17-officially-the-republic-of-namibia-e21df090d8c8
https://arrabble.medium.com/an-electromagnetic-field-also-em-field-is-a-classical-i-e-non-quantum-6fc1a608d934
https://jacap31175.medium.com/in-nuclear-physics-and-particle-physics-the-weak-interaction-d752c8a72d9
https://dedecab107.medium.com/classical-electromagnetism-or-classical-electrodynamics-c9d3b3bb0533

Maxwell’s equations are a set of coupled partial differential equations that, together

Written by Dedecab