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Maxwell's equations

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Equations describing classical electromagnetism

For thermodynamic relations, seeMaxwell relations.

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Maxwell's equations on a plaque on his statue in Edinburgh

Maxwell's equations, orMaxwell–Heaviside equations, are a set of coupledpartial differential equations that, together with theLorentz force law, form the foundation ofclassical electromagnetism, classicaloptics,electric andmagnetic 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 howelectric andmagnetic fields are generated bycharges,currents, and changes of the fields.^{[note 1]} The equations are named after the physicist and mathematicianJames 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 toOliver Heaviside.^[1]

Maxwell's equations may be combined to demonstrate how fluctuations in electromagnetic fields (waves) propagate at a constant speed in vacuum,c (299792458 m/s^[2]). Known aselectromagnetic radiation, these waves occur at various wavelengths to produce aspectrum of radiation fromradio waves togamma rays.

Inpartial differential equation form and acoherent system of units, Maxwell's microscopic equations can be written as (top to bottom: Gauss's law, Gauss's law for magnetism, Faraday's law, Ampère-Maxwell law) ${\begin{aligned}\nabla \cdot \mathbf {E} \,\,\,&={\frac {\rho }{\varepsilon _{0}}}\\\nabla \cdot \mathbf {B} \,\,\,&=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {B} &=\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\end{aligned}}$ With $\mathbf {E}$ the electric field, $\mathbf {B}$ the magnetic field, $\rho$ theelectric charge density and $\mathbf {J}$ thecurrent density. $\varepsilon _{0}$ is thevacuum permittivity and $\mu _{0}$ thevacuum permeability.

The equations have two major variants:

Themicroscopic 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 theatomic scale.
Themacroscopic 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 forequivalent alternative formulations. Versions of Maxwell's equations based on theelectric andmagnetic scalar potentials are preferred for explicitly solving the equations as aboundary value problem,analytical mechanics, or for use inquantum mechanics. Thecovariant formulation (onspacetime rather than space and time separately) makes the compatibility of Maxwell's equations withspecial relativity manifest.Maxwell's equations in curved spacetime, commonly used inhigh-energy andgravitational physics, are compatible withgeneral 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 theunification 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 aclassical limit of the more precise theory ofquantum electrodynamics.

Name	Integral equations	Differential equations
Gauss's law	$\oiint$ ${\vphantom {\oint }}_{\scriptstyle \partial \Omega }\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {1}{\varepsilon _{0}}}\iiint _{\Omega }\rho \,\mathrm {d} V$	$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$
Gauss's law for magnetism	$\oiint$ ${\vphantom {\oint }}_{\scriptstyle \partial \Omega }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$	$\nabla \cdot \mathbf {B} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$
Ampère–Maxwell law	${\begin{aligned}\oint _{\partial \Sigma }&\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}\left(\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\varepsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} \right)\\\end{aligned}}$	$\nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)$

Name	Integral equations	Differential equations
Gauss's law	$\oiint$ ${\vphantom {\oint }}_{\scriptstyle \partial \Omega }\mathbf {E} \cdot \mathrm {d} \mathbf {S} =4\pi \iiint _{\Omega }\rho \,\mathrm {d} V$	$\nabla \cdot \mathbf {E} =4\pi \rho$
Gauss's law for magnetism	$\oiint$ ${\vphantom {\oint }}_{\scriptstyle \partial \Omega }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$	$\nabla \cdot \mathbf {B} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {1}{c}}{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$	$\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$
Ampère–Maxwell law	${\begin{aligned}\oint _{\partial \Sigma }&\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}={\frac {1}{c}}\left(4\pi \iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +{\frac {\mathrel {\mathrm {d} }}{\mathrm {d} t}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} \right)\end{aligned}}$	$\nabla \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)$

Name	Integral equations (SI)	Differential equations (SI)	Differential equations (Gaussian system)
Gauss's law	$\oiint$ ${\scriptstyle \partial \Omega }$ $\mathbf {D} \cdot \mathrm {d} \mathbf {S} =\iiint _{\Omega }\rho _{\text{f}}\,\mathrm {d} V$	$\nabla \cdot \mathbf {D} =\rho _{\text{f}}$	$\nabla \cdot \mathbf {D} =4\pi \rho _{\text{f}}$
Ampère–Maxwell law	${\begin{aligned}\oint _{\partial \Sigma }&\mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=\\&\iint _{\Sigma }\mathbf {J} _{\text{f}}\cdot \mathrm {d} \mathbf {S} +{\frac {d}{dt}}\iint _{\Sigma }\mathbf {D} \cdot \mathrm {d} \mathbf {S} \\\end{aligned}}$	$\nabla \times \mathbf {H} =\mathbf {J} _{\text{f}}+{\frac {\partial \mathbf {D} }{\partial t}}$	$\nabla \times \mathbf {H} ={\frac {1}{c}}\left(4\pi \mathbf {J} _{\text{f}}+{\frac {\partial \mathbf {D} }{\partial t}}\right)$
Gauss's law for magnetism	$\oiint$ ${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$	$\nabla \cdot \mathbf {B} =0$	$\nabla \cdot \mathbf {B} =0$
Maxwell–Faraday equation (Faraday's law of induction)	$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {d}{dt}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$

Formulation	Homogeneous equations	Inhomogeneous equations
Fields Minkowski space	$\partial _{[\alpha }F_{\beta \gamma ]}=0$	$\partial _{\alpha }F^{\alpha \beta }=\mu _{0}J^{\beta }$
Potentials (any gauge) Minkowski space	$F_{\alpha \beta }=2\partial _{[\alpha }A_{\beta ]}$	$2\partial _{\alpha }\partial ^{[\alpha }A^{\beta ]}=\mu _{0}J^{\beta }$
Potentials (Lorenz gauge) Minkowski space	$F_{\alpha \beta }=2\partial _{[\alpha }A_{\beta ]}$ $\partial _{\alpha }A^{\alpha }=0$	$\partial _{\alpha }\partial ^{\alpha }A^{\beta }=\mu _{0}J^{\beta }$
Fields any spacetime	${\begin{aligned}&\partial _{[\alpha }F_{\beta \gamma ]}=\\&\qquad \nabla _{[\alpha }F_{\beta \gamma ]}=0\end{aligned}}$	${\begin{aligned}&{\frac {1}{\sqrt {-g}}}\partial _{\alpha }({\sqrt {-g}}F^{\alpha \beta })=\\&\qquad \nabla _{\alpha }F^{\alpha \beta }=\mu _{0}J^{\beta }\end{aligned}}$
Potentials (any gauge) any spacetime (with§topological restrictions)	$F_{\alpha \beta }=2\partial _{[\alpha }A_{\beta ]}$	${\begin{aligned}&{\frac {2}{\sqrt {-g}}}\partial _{\alpha }({\sqrt {-g}}g^{\alpha \mu }g^{\beta \nu }\partial _{[\mu }A_{\nu ]})=\\&\qquad 2\nabla _{\alpha }(\nabla ^{[\alpha }A^{\beta ]})=\mu _{0}J^{\beta }\end{aligned}}$
Potentials (Lorenz gauge) any spacetime (with topological restrictions)	$F_{\alpha \beta }=2\partial _{[\alpha }A_{\beta ]}$ $\nabla _{\alpha }A^{\alpha }=0$	$\nabla _{\alpha }\nabla ^{\alpha }A^{\beta }-R^{\beta }{}_{\alpha }A^{\alpha }=\mu _{0}J^{\beta }$

Formulation	Homogeneous equations	Inhomogeneous equations
Fields any spacetime	$\mathrm {d} F=0$	$\mathrm {d} {\star }F=\mu _{0}J$
Potentials (any gauge) any spacetime (with topological restrictions)	$F=\mathrm {d} A$	$\mathrm {d} {\star }\mathrm {d} A=\mu _{0}J$
Potentials (Lorenz gauge) any spacetime (with topological restrictions)	$F=\mathrm {d} A$ $\mathrm {d} {\star }A=0$	${\star }\Box A=\mu _{0}J$

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History of the equations

Conceptual descriptions

Gauss's law

Gauss's law for magnetism

Faraday's law

Ampère–Maxwell law

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

Key to the notation

Differential equations

Integral equations

Formulation in the SI

Formulation in the Gaussian system

Relationship between differential and integral formulations

Flux and divergence

Circulation and curl

Charge conservation

Vacuum equations, electromagnetic waves and speed of light

Macroscopic formulation

Bound charge and current

Auxiliary fields, polarization and magnetization

Constitutive relations

Alternative formulations

Solutions

Overdetermination of Maxwell's equations

Maxwell's equations as the classical limit of QED

Variations

Magnetic monopoles

See also

Explanatory notes

References

Further reading

Historical publications

External links

Modern treatments

Other