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. 
${\scriptstyle \partial \Omega }$ $\mathbf {E} \cdot \mathrm {d} \mathbf {S} ={\frac {1}{\varepsilon _{0}}}\iiint _{\Omega }\rho \,\mathrm {d} V$ 
$\nabla \cdot \mathbf {E} ={\frac {\rho }{\varepsilon _{0}}}$ 
Gauss's law for magnetism 
There are no
magnetic monopoles; the total magnetic flux through a closed surface is zero. 
${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$ 
$\nabla \cdot \mathbf {B} =0$ 
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. 
$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}={\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$ 
$\nabla \times \mathbf {E} ={\frac {\partial \mathbf {B} }{\partial t}}$ 
Ampère'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. 
$\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +\mu _{0}\varepsilon _{0}{\frac {\mathrm {d} }{\mathrm {d} t}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S}$ 
$\nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)$ 
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 E_{cgs} = c^{−1} E_{SI}. 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 $\mathbf {E} _{\mathrm {SI} }=c_{\mathrm {SI} }\mathbf {B} _{\mathrm {SI} }$). The Gaussian system uses a unit of charge
defined in such a way that the permittivity of the vacuum ε_{0} = 1/4πc, hence μ_{0} = 4π/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 
${\scriptstyle \partial \Omega }$ $\mathbf {E} \cdot \mathrm {d} \mathbf {S} =4\pi \iiint _{\Omega }\rho \,\mathrm {d} V$ 
$\nabla \cdot \mathbf {E} =4\pi \rho$ 
The electric flux leaving a volume is proportional to the charge inside. 
Gauss's law for magnetism 
${\scriptstyle \partial \Omega }$ $\mathbf {B} \cdot \mathrm {d} \mathbf {S} =0$ 
$\nabla \cdot \mathbf {B} =0$ 
There are no
magnetic monopoles; the total magnetic flux through a closed surface is zero. 
Maxwell–Faraday equation (
Faraday's law of induction) 
$\oint _{\partial \Sigma }\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}={\frac {1}{c}}{\frac {d}{dt}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S}$ 
$\nabla \times \mathbf {E} ={\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$ 
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) 
$\oint _{\partial \Sigma }\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}={\frac {1}{c}}\left(4\pi \iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} +{\frac {d}{dt}}\iint _{\Sigma }\mathbf {E} \cdot \mathrm {d} \mathbf {S} \right)$ 
$\nabla \times \mathbf {B} ={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right)$ 
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 timeindependent surfaces. For example, since the surface is timeindependent, we can bring the
differentiation under the integral sign in Faraday's law:

 ${\frac {d}{dt}}\iint _{\Sigma }\mathbf {B} \cdot \mathrm {d} \mathbf {S} =\iint _{\Sigma }{\frac {\partial \mathbf {B} }{\partial t}}\cdot \mathrm {d} \mathbf {S} \,,$
Maxwell's equations can be formulated with possibly timedependent surfaces and volumes by substituting the left hand side with the right hand side in the integral equation version of the Maxwell equations.
 ${\scriptstyle \partial \Omega }$ is a
surface integral over the boundary surface ∂Ω, with the loop indicating the surface is closed
 $\iiint _{\Omega }$ is a
volume integral over the volume Ω,
 $\oint _{\partial \Sigma }$ is a
line integral around the boundary curve ∂Σ, with the loop indicating the curve is closed.
 $\iint _{\Sigma }$ 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):


 $Q=\iiint _{\Omega }\rho \ \mathrm {d} V,$
 where dV is the
volume element.


 $I=\iint _{\Sigma }\mathbf {J} \cdot \mathrm {d} \mathbf {S} ,$
 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).