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Maxwell's Equations

Name	Differential form	Integral form
Gauss's law:	$\nabla \cdot \mathbf {D} =\rho$	$\oint _{S}\mathbf {D} \cdot \mathrm {d} \mathbf {A} =q=\int _{V}\rho \,\mathrm {d} V$
Gauss' law for magnetism (absence of magnetic monopoles):	$\nabla \cdot \mathbf {B} =0$	$\oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0$
Faraday's law of induction:	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$	$\oint _{C}\mathbf {E} \cdot \mathrm {d} \mathbf {l} =-\int _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$
Ampère's Circuital Law (with Maxwell's extension):	$\nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}$	$\oint _{C}\mathbf {H} \cdot \mathrm {d} \mathbf {l} =\int _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {A} +\int _{S}{\frac {\partial \mathbf {D} }{\partial t}}\cdot \mathrm {d} \mathbf {A}$

The following table provides the meaning of each symbol and the SI unit of measure:

Symbol	Meaning	SI Unit of Measure
$\mathbf {E}$	electric field	volt per meter or, equivalently, newton per coulomb
$\mu$	magnetic permeability of the medium	henries per meter, or newtons per ampere squared
$i_{N}$	net electrical current enclosed by an Amperian line	amperes
$q$	net electric charge enclosed by the Gaussian surface	coulombs
$\mathbf {H}$	magnetic field also called the auxiliary field	ampere per meter
$\mathbf {D}$	electric displacement field also called the electric flux density	coulomb per square meter
$\mathbf {B}$	magnetic flux density also called the magnetic induction also called the magnetic field	tesla, or equivalently, weber per square meter
$\ \rho \$	free electric charge density, not including dipole charges bound in a material	coulomb per cubic meter
$\mathbf {J}$	free current density, not including polarization or magnetization currents bound in a material	ampere per square meter
$\mathrm {d} \mathbf {A}$	differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S	square meters
$\mathrm {d} V\$	differential element of volume V enclosed by surface S	cubic meters
$\mathrm {d} \mathbf {l}$	differential vector element of path length tangential to contour C enclosing surface S	meters
$\nabla \cdot$	the divergence operator	per meter
$\nabla \times$	the curl operator	per meter

Although SI units are given here for the various symbols, Maxwell's equations are unchanged in many systems of units (and require only minor modifications in all others). The most commonly used systems of units are SI, used for engineering, electronics and most practical physics experiments, and Planck units (also known as "natural units"), used in theoretical physics, quantum physics and cosmology. An older system of units, the cgs system, is also used.

In order to complete the theory of electromagnetism we need to add another equation to Heaviside's group of four 'Maxwell's Equations'. The force exerted upon a charged particle by the electric field and magnetic field is given by the Lorentz force equation:

$\mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ),$

where $q\$ is the charge on the particle and $\mathbf {v} \$ is the particle velocity. This is slightly different when expressed in the cgs system of units below.

This extra equation appeared in cartesian format as equation (D) of the original eight 'Maxwell's Equations'.

Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms (where they undergo quantum mechanical effects as well). It is only in this averaged sense that one can define quantities such as the permittivity and permeability of a material. Below the microscopic, Maxwell's equations, ignoring quantum effects, are simply those of a vacuum — but one must include all atomic charges and so on, which is generally an intractable problem.