Maxwell's equations are a set of coupled partial differential equations that describe how electric and magnetic fields are generated by electric charges and currents. Together with the Lorentz force law, they form the foundation of classical electromagnetism, classical optics, electric and magnetic 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 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. Maxwell's 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. The publication of the equations marked the unification of a theory for previously separately described phenomena: magnetism, electricity, light, and associated radiation. The modern form of the equations in their most common formulation is credited to Oliver Heaviside. 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.
Summary The two most commonly used variants of Maxwell's equations are the microscopic and macroscopic differential form in SI units.
Microscopic version in SI units In partial differential equation form and in SI units, Maxwell's microscopic equations are written as (top to bottom: Gauss's law, Gauss's law for magnetism, Faraday's law, Ampère-Maxwell law)
∇ ⋅ E = ρ ε 0 ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = μ 0 ( J + ε 0 ∂ E ∂ t ) {\displaystyle {\begin{array}{lcl}\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{array}}}
In these equations, E {\displaystyle \mathbf {E} } is the electric field, B {\displaystyle \mathbf {B} } the magnetic field, ρ {\displaystyle \rho } the total electric charge density, J {\displaystyle \mathbf {J} } the total current density, ε 0 {\displaystyle \varepsilon _{0}} the vacuum permittivity, μ 0 {\displaystyle \mu _{0}} the vacuum permeability, ∇ ⋅ {\displaystyle \nabla \cdot } the divergence operator, and ∇ × {\displaystyle \nabla \times } the curl operator.
Macroscopic version in SI units In partial differential equation form and in SI units, Maxwell's macroscopic equations are written as:
∇ ⋅ D = ρ f ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × H = J f + ∂ D ∂ t {\displaystyle {\begin{array}{lcl}\nabla \cdot \mathbf {D} &=&\rho _{f}\\\nabla \cdot \mathbf {B} &=&0\\\nabla \times \mathbf {E} &=&-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {H} &=&\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}\end{array}}}
In these equations, D {\displaystyle \mathbf {D} } is the electric displacement field, E {\displaystyle \mathbf {E} } the electric field, B {\displaystyle \mathbf {B} } the magnetic field, H {\displaystyle \mathbf {H} } the magnetic field intensity, ρ f {\displaystyle \rho _{f}} the free electric charge density, J f {\displaystyle \mathbf {J} _{f}} the free current density, ∇ ⋅ {\displaystyle \nabla \cdot } the divergence operator, and ∇ × {\displaystyle \nabla \times } the curl operator.
History of the equations
Conceptual descriptions
Gauss's law
Gauss's law describes the relationship between an electric field and electric charges: an electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarization of material. The coefficient of the proportion is the permittivity of free space.
Gauss's law for magnetism
Gauss's law for magnetism states that electric charges have no magnetic analogues, called magnetic monopoles; no north or south magnetic poles exist in isolation. Instead, the magnetic field of a material is attributed to a dipole, and the net outflow of the magnetic field through a closed surface is zero. Magnetic dipoles may be represented as loops of current or inseparable pairs of equal and opposite "magnetic charges". Precisely, the total magnetic flux through a Gaussian surface is zero, and the magnetic field is a solenoidal vector field.
Faraday's law
The Maxwell–Faraday version of Faraday's law of induction describes how a time-varying magnetic field corresponds to the negative curl of 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 change of the magnetic flux through the enclosed surface. The electromagnetic induction is the operating principle behind many electric generators: for example, a rotating bar magnet creates a changing magnetic field and generates an electric field in a nearby wire.
Ampère–Maxwell law
The original law of Ampère states that magnetic fields relate to electric current. Maxwell's addition states that magnetic fields also relate to changing electric fields, which Maxwell called displacement current. The integral form states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve. Maxwell's modification of Ampère's circuital law is important because the laws of Ampère and Gauss must otherwise be adjusted for static fields. As a consequence, it predicts that a rotating magnetic field occurs with a changing electric field. A further consequence is the existence of self-sustaining electromagnetic waves which travel through empty space. The speed calculated for electromagnetic waves, which could be predicted from experiments on charges and currents, matches the speed of light; indeed, light is one form of electromagnetic radiation (as are X-rays, radio waves, and others). Maxwell understood the connection between electromagnetic waves and light in 1861, thereby unifying the theories of electromagnetism and optics.
Microscopic formulation in terms of electric and magnetic fields (in vacuum version) In the electric and magnetic field formulation there are four equations that determine the fields for given charge and current distribution. A separate law of nature, the Lorentz force law, describes how the electric and magnetic fields act on charged particles and currents. By convention, a version of this law in the original equations by Maxwell is no longer included. The vector calculus formalism below, the work of Oliver Heaviside, has become standard. It is rotationally invariant, and therefore mathematically more transparent than Maxwell's original 20 equations in x, y and z components. The relativistic formulations are more symmetric and Lorentz invariant. For the same equations expressed using tensor calculus or differential forms (see § Alternative formulations). The differential and integral formulations are mathematically equivalent; both are 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.
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 total electric charge density (total charge per unit volume), ρ, and the total electric current density (total current per unit area), J. The universal constants appearing in the equations (the first two ones explicitly only in the SI formulation) are:
the permittivity of free space, ε0, and the permeability of free space, μ0, and the speed of light, c = ( ε 0 μ 0 ) − 1 / 2 {\displaystyle c=({\varepsilon _{0}\mu _{0}})^{-1/2}}
Differential equations In the differential equations,
the nabla symbol, ∇, denotes the three-dimensional gradient operator, del, the ∇⋅ symbol (pronounced "del dot") denotes the divergence operator, the ∇× symbol (pronounced "del cross") denotes the curl operator.
Integral equations In the integral equations,
Ω is any volume with closed boundary surface ∂Ω, and Σ is any surface with closed boundary curve ∂Σ, The equations are a little easier to interpret with time-independent surfaces and volumes. Time-independent surfaces and volumes are "fixed" and do not change over a given time interval. For example, since the surface is time-independent, we can bring the differentiation under the integral sign in Faraday's law:
d d t ∬ Σ B ⋅ d S = ∬ Σ ∂ B ∂ t ⋅ d S , {\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\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 using the differential version and using Gauss' and Stokes' theorems as appropriate.
∫ ∂ Ω {\displaystyle {\vphantom {\int }}_{\scriptstyle \partial \Omega }} is a surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed
∭ Ω {\displaystyle \iiint _{\Omega }} is a volume integral over the volume Ω,
∮ ∂ Σ {\displaystyle \oint _{\partial \Sigma }} is a line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed.
∬ Σ {\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): Q = ∭ Ω ρ d V , {\displaystyle Q=\iiint _{\Omega }\rho \ \mathrm {d} V,} where dV is the volume element. The net magnetic flux ΦB is the surface integral of the magnetic field B passing through a fixed surface, Σ: Φ B = ∬ Σ B ⋅ d S , {\displaystyle \Phi _{B}=\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} ,}
The net electric flux ΦE is the surface integral of the electric field E passing through Σ: Φ E = ∬ Σ E ⋅ d S , {\displaystyle \Phi _{E}=\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} ,}
The net electric current I is the surface integral of the electric current density J passing through Σ: I = ∬ Σ J ⋅ d S , {\displaystyle I=\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} ,} where dS denotes the differential 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 vector potential).
Formulation in the SI
Formulation in the Gaussian system
The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of ε0 and μ0 into the units (and thus redefining these). With a corresponding change in the values of the quantities for the Lorentz force law this yields the same physics, i.e. trajectories of charged particles, or work done by an electric motor. These definitions are often preferred in theoretical and high energy physics where it is natural to take the electric and magnetic field with the same units, to simplify the appearance of the electromagnetic tensor: the Lorentz covariant object unifying electric and magnetic field would then contain components with uniform unit and dimension. Such modified definitions are conventionally used with the Gaussian (CGS) units. Using these definitions, colloquially "in Gaussian units", the Maxwell equations become:
The equations simplify slightly when a system of quantities is chosen in the speed of light, c, is used for nondimensionalization, so that, for example, seconds and lightseconds are interchangeable, and c = 1. Further changes are possible by absorbing factors of 4π. This process, called rationalization, affects whether Coulomb's law or Gauss's law includes such a factor (see Heaviside–Lorentz units, used mainly in particle physics).
Relationship between differential and integral formulations The equivalence of the differential and integral formulations are a consequence of the Gauss divergence theorem and the Kelvin–Stokes theorem.
Flux and divergence
According to the (purely mathematical) Gauss divergence theorem, the electric flux through the boundary surface ∂Ω can be rewritten as
The integral version of Gauss's equation can thus be rewritten as
∭ Ω ( ∇ ⋅ E − ρ ε 0 ) d V = 0 {\displaystyle \iiint _{\Omega }\left(\nabla \cdot \mathbf {E} -{\frac {\rho }{\varepsilon _{0}}}\right)\,\mathrm {d} V=0}
Since Ω is arbitrary (e.g. an arbitrary small ball with arbitrary center), this is satisfied if and only if the integrand is zero everywhere. This is the differential equations formulation of Gauss equation up to a trivial rearrangement. Similarly rewriting the magnetic flux in Gauss's law for magnetism in integral form gives
which is satisfied for all Ω if and only if ∇ ⋅ B = 0 {\displaystyle \nabla \cdot \mathbf {B} =0} everywhere.
Circulation and curl
By the Kelvin–Stokes theorem we can rewrite the line integrals of the fields around the closed boundary curve ∂Σ to an integral of the "circulation of the fields" (i.e. their curls) over a surface it bounds, i.e.
∮ ∂ Σ B ⋅ d ℓ = ∬ Σ ( ∇ × B ) ⋅ d S , {\displaystyle \oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\iint _{\Sigma }(\nabla \times \mathbf {B} )\cdot \mathrm {d} \mathbf {S} ,}
Hence the Ampère–Maxwell law, the modified version of Ampère's circuital law, in integral form can be rewritten as
∬ Σ ( ∇ × B − μ 0 ( J + ε 0 ∂ E ∂ t ) ) ⋅ d S = 0. {\displaystyle \iint _{\Sigma }\left(\nabla \times \mathbf {B} -\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)\cdot \mathrm {d} \mathbf {S} =0.}
Since Σ can be chosen arbitrarily, e.g. as an arbitrary small, arbitrary oriented, and arbitrary centered disk, we conclude that the integrand is zero if and only if the Ampère–Maxwell law in differential equations form is satisfied. The equivalence of Faraday's law in differential and integral form follows likewise. The line integrals and curls are analogous to quantities in classical fluid dynamics: the circulation of a fluid is the line integral of the fluid's flow velocity field around a closed loop, and the vorticity of the fluid is the curl of the velocity field.
Charge conservation The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the Ampère–Maxwell law has zero divergence by the div–curl identity. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives:
0 = ∇ ⋅ ( ∇ × B ) = ∇ ⋅ ( μ 0 ( J + ε 0 ∂ E ∂ t ) ) = μ 0 ( ∇ ⋅ J + ε 0 ∂ ∂ t ∇ ⋅ E ) = μ 0 ( ∇ ⋅ J + ∂ ρ ∂ t ) {\displaystyle 0=\nabla \cdot (\nabla \times \mathbf {B} )=\nabla \cdot \left(\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +\varepsilon _{0}{\frac {\partial }{\partial t}}\nabla \cdot \mathbf {E} \right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +{\frac {\partial \rho }{\partial t}}\right)}
i.e.,
∂ ρ ∂ t + ∇ ⋅ J = 0. {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.}
By the Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary:
In particular, in an isolated system the total charge is conserved.
Vacuum equations, electromagnetic waves and speed of light
In a region with no charges (ρ = 0) and no currents (J = 0), such as in vacuum, Maxwell's equations reduce to:
∇ ⋅ E = 0 , ∇ × E + ∂ B ∂ t = 0 , ∇ ⋅ B = 0 , ∇ × B − μ 0 ε 0 ∂ E ∂ t = 0. {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} &=0,&\nabla \times \mathbf {E} +{\frac {\partial \mathbf {B} }{\partial t}}=0,\\\nabla \cdot \mathbf {B} &=0,&\nabla \times \mathbf {B} -\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}=0.\end{aligned}}}
Taking the curl (∇×) of the curl equations, and using the curl of the curl identity we obtain
μ 0 ε 0 ∂ 2 E ∂ t 2 − ∇ 2 E = 0 , μ 0 ε 0 ∂ 2 B ∂ t 2 − ∇ 2 B = 0. {\displaystyle {\begin{aligned}\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0,\\\mu _{0}\varepsilon _{0}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0.\end{aligned}}}
The quantity μ 0 ε 0 {\displaystyle \mu _{0}\varepsilon _{0}} has the dimension (T/L)2. Defining c = ( μ 0 ε 0 ) − 1 / 2 {\displaystyle c=(\mu _{0}\varepsilon _{0})^{-1/2}} , the equations above have the form of the standard wave equations
1 c 2 ∂ 2 E ∂ t 2 − ∇ 2 E = 0 , 1 c 2 ∂ 2 B ∂ t 2 − ∇ 2 B = 0. {\displaystyle {\begin{aligned}{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}-\nabla ^{2}\mathbf {E} =0,\\{\frac {1}{c^{2}}}{\frac {\partial ^{2}\mathbf {B} }{\partial t^{2}}}-\nabla ^{2}\mathbf {B} =0.\end{aligned}}}
Already during Maxwell's lifetime, it was found that the known values for ε 0 {\displaystyle \varepsilon _{0}} and μ 0 {\displaystyle \mu _{0}} give c ≈ 2.998 × 10 8 m/s {\displaystyle c\approx 2.998\times 10^{8}~{\text{m/s}}} , then already known to be the speed of light in free space. This led him to propose that light and radio waves were propagating electromagnetic waves, since amply confirmed. In the old SI system of units, the values of μ 0 = 4 π × 10 − 7 {\displaystyle \mu _{0}=4\pi \times 10^{-7}} and c = 299 792 458 m/s {\displaystyle c=299\,792\,458~{\text{m/s}}} are defined constants, (which means that by definition ε 0 = 8.854 187 8... × 10 − 12 F/m {\displaystyle \varepsilon _{0}=8.854\,187\,8...\times 10^{-12}~{\text{F/m}}} ) that define the ampere and the metre. In the new SI system, only c keeps its defined value, and the electron charge gets a defined value. In materials with relative permittivity, εr, and relative permeability, μr, the phase velocity of light becomes
v p = 1 μ 0 μ r ε 0 ε r , {\displaystyle v_{\text{p}}={\frac {1}{\sqrt {\mu _{0}\mu _{\text{r}}\varepsilon _{0}\varepsilon _{\text{r}}}}},}
which is usually less than c. In addition, E and B are perpendicular to each other and to the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. In turn, that electric field creates a changing magnetic field through Maxwell's mo