\section{The Maxwell Equations of the Electromagnetic Field}

The mathematical theory of electromagnetism was developed
and published in 1864 by James Clerk Maxwell.
He described the known electric and magnetic effects
in terms of four equations relating the electric and
magnetic fields and their
sources---charged particles and electric currents.
The development of this theory was a supreme achievement in
the history of science.
Maxwell's theory is still used today by physicists and
electrical engineers.
The theory was further developed in the 20th century
to account for the quantum theory of light.
But even in quantum electrodynamics Maxwell's equations
remain valid although their interpretation is somewhat
different from the classical theory.
In any case, Maxwell's theory continues today to be an
essential part of theoretical physics.

A knowledge of calculus and vectors is necessary for
a full understanding of the Maxwell equations.
However, the essential structure of the theory can be understood
without going into the mathematical details.
Each equation is expressed most powerfully as a partial differential
equation relating variations of the electric and magnetic fields,
with respect to variations of position or time, and the charge
and current densities in space.

\paragraph{Gauss's law.}
Gauss's law, written as a field equation,
is $\bnabla\cdot\vec{E}=\rho/\epsilon_{0}$.
(The symbol $\bnabla\cdot$ denotes the divergence operator.)
Here $\rho(\vec{x},t)$ is the charge per unit volume in the
neighborhood of $\vec{x}$ at time $t$;
$\epsilon_{0}$ is a constant of nature equal to
$8.85 \times 10^{-12}$\,C$^{2}$/Nm$^{2}$.
Gauss's law relates the electric field $\vec{E}(\vec{x},t)$
and the charge density.
The solution for a charged particle $q$ at rest is
$\vec{E}(\vec{x})=kq\vec{e}/r^{2}$
where $r$ is the distance from the charge to $\vec{x}$,
$\vec{e}$ is the direction vector,
and $k=1/(4\pi\epsilon_{0})$;
this is the familiar inverse square law of electrostatics.
Electric field lines diverge at a point charge.

\paragraph{Gauss's law for magnetism.}
The analogous equation for the magnetic field
is $\bnabla\cdot\vec{B}=0$.
There are no magnetic monopoles---particles that act as a point
source of $\vec{B}(\vec{x},t)$.
Unlike the electric field lines, which may terminate on charges,
the magnetic field lines always form closed curves because
magnetic charges do not exist.
There is no divergence of magnetic field lines.

\paragraph{Faraday's law.}
The field equation that describes Faraday's law of electromagnetic
induction is $\bnabla\times\vec{E}=-\partial{\vec{B}}/\partial{t}$.
The quantity $\bnabla\times\vec{E}$,
called the curl of $\vec{E}(\vec{x},t)$,
determines the way that the vector field $\vec{E}$ curls around
each direction in space.
Also, $\partial{\vec{B}}/\partial{t}$
is the rate of change of the magnetic field.
This field equation expresses the fact that a magnetic
field that varies in time implies an electric field that
curls around the change of the magnetic field.
It is equivalent to Faraday's statement
that the rate of change of magnetic field
flux through a surface $S$ is equal to an electromotive
force (EMF) around the boundary curve of $S$.

\paragraph{The Amp\`{e}re-Maxwell law.}
In a system of steady electric currents the magnetic field is
constant in time and curls around the current in directions
defined by the right-hand rule.
The field equation that expresses this field $\vec{B}$
(Amp\`{e}re's law) is $\bnabla\times\vec{B}=\mu_{0}\vec{J}$,
where $\vec{J}(\vec{x})$ is the current
per unit area at $\vec{x}$ and $\mu_{0}$ is a constant
equal to $4\pi \times 10^{-7}$\,Tm/A.\footnote{%
The units are T = tesla for magnetic field and
A = ampere for electric current.}
But Amp\`{e}re's law is incomplete, because it does
not apply to systems in which the currents and fields
vary in time.
Maxwell deduced from mathematical considerations a
generalization of Amp\`{e}re's law,
\[
\bnabla\times\vec{B}
=\mu_{0}\vec{J}
+\mu_{0}\epsilon_{0}\frac{\partial{\vec{E}}}{\partial{t}},
\]
in which the second term on the right side is called the
displacement current.
The displacement current is a necessary term in order
for the system of four partial differential equations
to be self-consistent.
The Amp\`{e}re-Maxwell law implies that $\vec{B}$
curls around either electric current ($\vec{J}$)
or changing electric field ($\partial\vec{E}/\partial{t}$).
The latter case is analogous to electromagnetic induction
but with $\vec{E}$ and $\vec{B}$ reversed;
a rate of change in one field induces circulation in the
other field.

Maxwell's introduction of the displacement current was a daring
theoretical prediction.
At that time there was no experimental
evidence for the existence of displacement current.
Laboratory effects predicted by the displacement current
are very small and their observation was not possible
with the apparatus available at that time.
However, the Maxwell equations, including the
displacement current, make a striking prediction---that
light consists of electromagnetic waves.
The fact that Maxwell's theory explains the
properties of light, and other forms of
electromagnetic radiation, provides the evidence
for the existence of the displacement current.