\section{Electromagnetic Waves}

Electromagnetism includes phenomena involving charges,
currents, magnets, and the fields $\vec{E}$ and $\vec{B}$.
Light is also an electromagnetic phenomenon.
Light and other forms of radiation are described
by field theory as electromagnetic waves.
Therefore optics---the science of light---is a part
of electromagnetism.

The Maxwell equations describe the behavior of
the electric field $\vec{E}(\vec{x},t)$ and magnetic
field $\vec{B}(\vec{x},t)$.
The time-dependent fields influence each other,
even in a vacuum where no charge or current is present,
through Faraday's law
($\bnabla\times\vec{E}=-\partial\vec{B}/\partial{t}$)
and the displacement current
($\bnabla\times\vec{B}=\mu_{0}\epsilon_{0}\partial\vec{E}/\partial{t}$).
The four field equations are all satisfied in vacuum
if both $\vec{E}(\vec{x},t)$ and $\vec{B}(\vec{x},t)$
are waves traveling in space.
However, the electric and magnetic waves are co-dependent,
i.e., intricately related in several ways.

Figure \ref{fig:EMwave} shows a snapshot in time of the
fields for a section of an electromagnetic wave in vacuum.
\fbox{\bf Fig.\,9}
The field vectors
are shown at points along a line parallel to the
direction of propagation of the wave (z axis).
The wavelength is constant (at least for the section
of wave shown).
The field vectors $\vec{E}$ and $\vec{B}$ vary
in magnitude as a function of position along the
line of propagation.
The magnitudes are related by $B=E/c$
where $c$ is the wave speed.\footnote{%
The speed of light in vacuum is $c=2.998\times 10^{8}$\,m/s.}
The directions of $\vec{E}$ and $\vec{B}$ are orthogonal,
in the x and y directions respectively.
The field components oscillate sinusoidally with $z$.
An electromagnetic wave is a transverse wave because the
fields oscillate in directions orthogonal to the direction
of propagation.

\begin{figure}[p]
\begin{center}
\includegraphics[width=0.9\textwidth]{./eps/EMwave.eps}
\end{center}
\caption{Electric and magnetic field vectors in a
section of a linearly polarized electromagnetic wave.
\label{fig:EMwave}}
\end{figure}

Figure \ref{fig:EMwave} shows the field vectors along
only a single line parallel to the direction of
propagation $\bwidehat{z}$.
The full wave fills a three-dimensional volume.
In a linearly polarized plane wave the field vectors are
constant on planar wave fronts orthogonal to $\bwidehat{z}$,
e.g., with $\vec{E}$ and $\vec{B}$ pointing in the
$\bwidehat{x}$ and $\bwidehat{y}$ directions, respectively,
uniformly over the wave front.
All properties of polarized light are then described
by the wave theory.
As another 3D example, a short radio antenna radiates a
spherical wave; far from the antenna $\vec{E}$ and $\vec{B}$
are orthogonal to one another on spherical wave fronts.

A wave is a structure that is extended in space.
The parts of the wave also vary in time.
Figure \ref{fig:EMwave} is a snapshot showing the wave
as a function of position at an instant of time.
As time passes, the fields at any fixed position
$\vec{x}$ will change.
For a linearly polarized plane wave,
$\vec{E}$ at $\vec{x}$ oscillates in time between
orientations parallel and antiparallel to
a fixed polarization direction ($\bwidehat{x}$) which is
orthogonal to the line of propagation ($\bwidehat{z}$);
$\vec{B}$ varies similarly in a direction ($\bwidehat{y}$)
orthogonal to both $\vec{E}$ and the line of propagation.
The combined variations in space and time imply that the
wave structure moves as a whole in the direction of propagation.
So a snapshot at a later time would look just the same as
Fig.\,\ref{fig:EMwave} except translated by some distance
in the direction of propagation.
In other words, the shape is constant but the positions
of the nodes (points where $\vec{E}=0$ and $\vec{B}=0$)
move in space.

\paragraph{The electromagnetic spectrum.}
Electromagnetic waves have an infinite range of
wavelength $\lambda$.
Visible light has $\lambda$ from $400$\,nm (violet)
to $700$\,nm (red).
Ultraviolet and infrared light have shorter and longer
wavelengths, respectively.
Microwaves have $\lambda$ from $1$\,mm to $1$\,m,
with important technological uses,
e.g., communications and radar.
X-rays have very short wavelengths, $\lambda<1$\,nm.
All these forms of electromagnetic radiation have
the same wave speed in vacuum,
which can be evaluated from the field theory,
$c=1/\sqrt{\mu_{0}\epsilon_{0}}=2.998\times 10^{8}$\,m/s.