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 partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio technologies, including power generation, electric motors, wireless communication, cameras, televisions, computers etc. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of each other. One important consequence of the equations is that they demonstrate how fluctuating electric and magnetic fields propagate at 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, and first proposed 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 details. However, their use requires experimentally determining parameters for a phenomenological description of the electromagnetic response of materials.

The term "Maxwell's equations" is often 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 spacetime formulations (i.e., on spacetime rather than space and time separately), are commonly used in high energy and gravitational physics because they make the compatibility of the equations with special and general relativity manifest. [note 1] In fact, Einstein developed special and general relativity to accommodate the absolute speed of light that drops out of the Maxwell 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 field theory approximation of some aspects of the fundamental theory of quantum electrodynamics, although some quantum features, such as quantum entanglement, are completely absent and in no way approximated. (For example, quantum cryptography has no approximate version in Maxwell theory.) In many situations, though, deviations from Maxwell's equations are immeasurably small. Exceptions include nonclassical light, photon–photon scattering, quantum optics, and many other phenomena related to photons or virtual photons.

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

In the electric and magnetic field formulation there are four equations. The two inhomogeneous equations describe how the fields vary in space due to sources. Gauss's law describes how electric fields emanate from electric charges. Gauss's law for magnetism describes magnetic fields as closed field lines not due to magnetic monopoles. The two homogeneous equations describe how the fields "circulate" around their respective sources. Ampère's law with Maxwell's addition describes how the magnetic field "circulates" around electric currents and time varying electric fields, while Faraday's law describes how the electric field "circulates" around time varying magnetic fields.

A separate law of nature, the Lorentz force law, describes how the electric and magnetic field act on charged particles and currents. A version of this law was included in the original equations by Maxwell but, by convention, is no longer included.

The precise formulation of Maxwell's equations depends on the precise definition of the quantities involved. Conventions differ with the unit systems, because various definitions and dimensions are changed by absorbing dimensionful factors like the speed of light c. This makes constants come out differently. The most common form is based on conventions used when quantities measured using SI units, but other commonly used conventions are used with other units including Gaussian units based on the cgs system, [1] Lorentz–Heaviside units (used mainly in particle physics), and Planck units (used in theoretical physics).

The vector calculus formulation below has become standard. It is mathematically much more convenient than Maxwell's original 20 equations and is due to Oliver Heaviside. [2] [3] The differential and integral equations formulations are mathematically equivalent and are both useful. The integral formulation relates fields within a region of space to fields on the boundary and can often be used to simplify and directly calculate fields from symmetric distributions of charges and currents. On the other hand, the differential equations are purely local and are a more natural starting point for calculating the fields in more complicated (less symmetric) situations, for example using finite element analysis. [4] For formulations using tensor calculus or differential forms, see alternative formulations. For relativistically invariant formulations, see relativistic formulations.

Formulation in SI units convention

Name Meaning Integral equations Differential equations
Gauss's law The electric flux leaving a volume is proportional to the charge inside. ${\displaystyle {\scriptstyle \partial \Omega }}$ ${\displaystyle \mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {1}{\varepsilon _{0}}}\iiint _{\Omega }\rho \,\mathrm {d} V}$ ${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}}$
Gauss's law for magnetism There are no magnetic monopoles; the total magnetic flux through a closed surface is zero. ${\displaystyle {\scriptstyle \partial \Omega }}$ ${\displaystyle \mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}$ ${\displaystyle \nabla \cdot \mathbf {B} =0}$
Maxwell–Faraday equation ( Faraday's law of induction) The voltage induced in a closed loop is proportional to the rate of change of the magnetic flux that the loop encloses. ${\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} }$ ${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$
Ampère's circuital law (with Maxwell's addition) The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) that the loop encloses. ${\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\mu _{0}\varepsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} }$ ${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)}$

Formulation in Gaussian units convention

Gaussian units are a popular system of units that are part of the centimetre–gram–second system of units (cgs). When using Gaussian units it is conventional to use a slightly different definition of electric field Ecgs = c−1 ESI. This implies that the modified electric and magnetic field have the same units (in the SI convention this is not the case making dimensional analysis of the equations different: e.g. for an electromagnetic wave in vacuum ${\displaystyle |\mathbf {E} |_{\mathrm {SI} }=|c|_{\mathrm {SI} }|\mathbf {B} |_{\mathrm {SI} }}$). The Gaussian system uses a unit of charge defined in such a way that the permittivity of the vacuum ε0 = 1/4πc, hence μ0 = /c. These units are sometimes preferred over SI units in the context of special relativity, [5]:vii in which the components of the electromagnetic tensor, the Lorentz covariant object describing the electromagnetic field, have the same unit without constant factors. Using these different conventions, the Maxwell equations become: [6]

Name Integral equations Differential equations Meaning
Gauss's law ${\displaystyle {\scriptstyle \partial \Omega }}$ ${\displaystyle \mathbf {E} \cdot \mathrm {d} \mathbf {S} =4\pi \iiint _{\Omega }\rho \,\mathrm {d} V}$ ${\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }$ The electric flux leaving a volume is proportional to the charge inside.
Gauss's law for magnetism ${\displaystyle {\scriptstyle \partial \Omega }}$ ${\displaystyle \mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}$ ${\displaystyle \nabla \cdot \mathbf {B} =0}$ There are no magnetic monopoles; the total magnetic flux through a closed surface is zero.
Maxwell–Faraday equation ( Faraday's law of induction) ${\displaystyle \oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {1}{c}}{\frac {d}{dt}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} }$ ${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$ The voltage induced in a closed loop is proportional to the rate of change of the magnetic flux that the loop encloses.
Ampère's circuital law (with Maxwell's addition) ${\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}={\frac {1}{c}}\left(4\pi \iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +{\frac {d}{dt}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} \right)}$ ${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)}$ The magnetic field integrated around a closed loop is proportional to the electric current plus displacement current (rate of change of electric field) that the loop encloses.

Key to the notation

Symbols in bold represent vector quantities, and symbols in italics represent scalar quantities, unless otherwise indicated.

The equations introduce the electric field, E, a vector field, and the magnetic field, B, a pseudovector field, each generally having a time and location dependence. The sources are

The universal constants appearing in the equations are

Differential equations

In the differential equations,

Integral equations

In the integral equations,

• Ω is any fixed volume with closed boundary surface ∂Ω, and
• Σ is any fixed surface with closed boundary curve ∂Σ,

Here a fixed volume or surface means that it does not change over time. The equations are correct, complete and a little easier to interpret with time-independent surfaces. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law:

${\displaystyle {\frac {d}{dt}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =\iint _{\Sigma }{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {S} \,,}$

Maxwell's equations can be formulated with possibly time-dependent surfaces and volumes by substituting the left hand side with the right hand side in the integral equation version of the Maxwell equations.

• ${\displaystyle {\scriptstyle \partial \Omega }}$ is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed
• ${\displaystyle \iiint _{\Omega }}$ is a volume integral over the volume Ω,
• ${\displaystyle \oint _{\partial \Sigma }}$ is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed.
• ${\displaystyle \iint _{\Sigma }}$ is a surface integral over the surface Σ,
• The total electric charge Q enclosed in Ω is the volume integral over Ω of the charge density ρ (see the "macroscopic formulation" section below):
${\displaystyle Q=\iiint _{\Omega }\rho \ \mathrm {d} V,}$
where dV is the volume element.
${\displaystyle I=\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} ,}$
where dS denotes the vector element of surface area S, normal to surface Σ. (Vector area is sometimes denoted by A rather than S, but this conflicts with the notation for magnetic potential).
Other Languages
Alemannisch: Maxwell-Gleichungen
беларуская (тарашкевіца)‎: Раўнаньні Максўэла
Bahasa Indonesia: Persamaan Maxwell
íslenska: Jöfnur Maxwells
lietuvių: Maksvelo lygtys
Bahasa Melayu: Persamaan Maxwell
Nederlands: Wetten van Maxwell
norsk nynorsk: Maxwells likningar
Simple English: Maxwell's equations
slovenčina: Maxwellove rovnice
slovenščina: Maxwellove enačbe