# 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. 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 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 determining 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 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 limit of the fundamental theory of quantum electrodynamics.

## Formulation in terms of electric and magnetic fields (microscopic or 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, conversely, 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 included no longer. The vector calculus formalism below, due to Oliver Heaviside,[1][2] has become standard. It is manifestly rotation invariant, and therefore mathematically much more transparent than Maxwell's original 20 equations in x,y,z components. The relativistic formulations are even more symmetric and manifestly Lorenz invariant. For the same equations expressed using tensor calculus or differential forms, see alternative formulations.

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.[3]

### Formulation in SI units convention

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

### Formulation in Gaussian units convention

The definitions of charge, electric field, and magnetic field can be altered to simplify theoretical calculation, by absorbing dimensioned factors of ε0 and c into the units of calculation, by convention. With a corresponding change in convention for the Lorenz 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.[4]:vii Such modified definitions are conventionally used with the Gaussian (CGS) units. Using these definitions and conventions, colloquially "in Gaussian units",[5] 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 through a closed surface equals (4π times) the charge inside an enclosed volume .
Gauss's law for magnetism ${\displaystyle {\scriptstyle \partial \Omega }}$ ${\displaystyle \mathbf {B} \cdot \mathrm {d} \mathbf {S} =0}$ ${\displaystyle \nabla \cdot \mathbf {B} =0}$ The magnetic flux through a closed surface is zero (i.e. there are no magnetic monopoles)
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 work per unit charge required to move a charge around a closed loop is proportional to

the rate of decrease of the magnetic flux through an enclosed surface.

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 induced around a closed loop is proportional (with the same constant) to (4π times) the electric current plus the rate of change of electric flux through an enclosed surface.

Note that the equations are particularly readable when length and time are measured in compatible units like seconds and lightseconds i.e. in units such that c = 1 unit of length/unit of time. Ever since 1983, metres and seconds are compatible except for historical legacy since by definition c = 299 792 458 m/s (≈ 1.0 feet/nanosecond).

Further cosmetic changes, called rationalisations, are possible by absorbing factors of 4π depending on whether we want Coulomb's law or Gauss law to come out nicely, see Lorentz-Heaviside units (used mainly in particle physics). In theoretical physics it is often useful to choose units such that Plancks constant, the elementary charge, and even Newton's constant are 1. See Planck units.

### 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 using the differential version and using Gauss and Stokes formula appropriately.

• ${\displaystyle {\scriptstyle \partial \Omega }}$ 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).
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