Maxwell's equations

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. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the 'speed of light'. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also first used the equations to propose that light is an electromagnetic phenomenon.3D Euclidean space + time ∇ × E + ∂ B ∂ t = 0 {displaystyle {egin{aligned} abla imes mathbf {E} +{frac {partial mathbf {B} }{partial t}}=0end{aligned}}} ∇ × B − 1 c 2 ∂ E ∂ t = μ 0 J {displaystyle {egin{aligned} abla imes mathbf {B} -{frac {1}{c^{2}}}{frac {partial mathbf {E} }{partial t}}&=mu _{0}mathbf {J} end{aligned}}} 3D Euclidean space + time E = − ∇ φ − ∂ A ∂ t {displaystyle {egin{aligned}mathbf {E} &=-mathbf { abla } varphi -{frac {partial mathbf {A} }{partial t}}end{aligned}}} ( − ∇ 2 + 1 c 2 ∂ 2 ∂ t 2 ) A + ∇ ( ∇ ⋅ A + 1 c 2 ∂ φ ∂ t ) = μ 0 J {displaystyle {egin{aligned}left(- abla ^{2}+{frac {1}{c^{2}}}{frac {partial ^{2}}{partial t^{2}}} ight)mathbf {A} +mathbf { abla } left(mathbf { abla } cdot mathbf {A} +{frac {1}{c^{2}}}{frac {partial varphi }{partial t}} ight)&=mu _{0}mathbf {J} end{aligned}}} 3D Euclidean space + time E = − ∇ φ − ∂ A ∂ t {displaystyle {egin{aligned}mathbf {E} &=-mathbf { abla } varphi -{frac {partial mathbf {A} }{partial t}}\end{aligned}}} ( − ∇ 2 + 1 c 2 ∂ 2 ∂ t 2 ) A = μ 0 J {displaystyle {egin{aligned}left(- abla ^{2}+{frac {1}{c^{2}}}{frac {partial ^{2}}{partial t^{2}}} ight)mathbf {A} &=mu _{0}mathbf {J} end{aligned}}} space + timespace (with topological restrictions) + time E i = − ∂ A i ∂ t − ∂ i ϕ = − ∂ A i ∂ t − ∇ i ϕ {displaystyle {egin{aligned}E_{i}&=-{frac {partial A_{i}}{partial t}}-partial _{i}phi \&=-{frac {partial A_{i}}{partial t}}- abla _{i}phi \end{aligned}}} space (with topological restrictions) + timeAny space + timeAny space (with topological restrictions) + timeAny space (with topological restrictions) + timeMinkowski spaceMinkowski spaceMinkowski spaceAny spacetimeAny spacetime (with topological restrictions) Any spacetime (with topological restrictions)Any spacetimeAny spacetime (with topological restrictions)Any spacetime (with topological restrictions) 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. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at a constant speed (c) in the vacuum, the 'speed of light'. Known as electromagnetic radiation, these waves may occur at various wavelengths to produce a spectrum from radio waves to γ-rays. The equations are named after the physicist and mathematician James Clerk Maxwell, who between 1861 and 1862 published an early form of the equations that included the Lorentz force law. He also 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. 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. 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. Gauss's law describes the relationship between a static electric field and the electric charges that cause it: The 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. Picturing the electric field by its field lines, this means the field lines begin at positive electric charges and end at negative electric charges. '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). Gauss's law for magnetism states that there are no 'magnetic charges' (also called magnetic monopoles), analogous to electric charges. 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'. In terms of field lines, this equation states that magnetic field lines neither begin nor end but make loops or extend to infinity and back. In other words, any magnetic field line that enters a given volume must somewhere exit that volume. 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. The Maxwell–Faraday version of Faraday's law of induction describes how a time varying magnetic field creates ('induces') 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 decrease of the magnetic flux through the enclosed surface. The dynamically induced electric field has closed field lines similar to a magnetic field, unless superposed by a static (charge induced) electric field. 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. 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). In integral form, the magnetic field induced around any closed loop is proportional to the electric current plus displacement current (proportional to the rate of change of electric flux) through the enclosed surface.