\section{The field concept}

In field theory,
electric and magnetic forces are described as the effects of
electric and magnetic fields.
The theory of electromagnetic phenomena is entirely
based on the field concept. 

Any electrically charged particle $q$ creates an
associated electric field $\vec{E}$.
The field extends throughout the space around $q$,
and varies with the position $\vec{x}$ in space.
If the particle is at rest (in some frame of reference)
then its field is independent of time.
In a system with many charges, the full electric field is
the sum of the electric fields of the individual charges.
Thus the electric field in an electrostatic system,
denoted mathematically by $\vec{E}(\vec{x})$, is
a function of position $\vec{x}$
and depends on the locations and
charge strengths of all the charged particles.

The electric field extends throughout a volume of space.
It exerts a force on any charge in the space.
To consider the field concept, suppose
two charged particles, $q_{1}$ and $q_{2}$, are located at
positions $\vec{x}_{1}$ and $\vec{x}_{2}$, respectively.
(See Fig.\,\ref{fig:Edip}.)
The electric field at an arbitrary point $\vec{x}$ is
\[
\vec{E}(\vec{x})=\frac{kq_{1}}{r_{1}^{2}}\vec{e}_{1}
+\frac{kq_{2}}{r_{2}^{2}}\vec{e}_{2}
\]
where ${r}_{1}$ is the distance from $q_{1}$ to $\vec{x}$,
$\vec{e}_{1}$ is the unit vector in the direction from
$q_{1}$ to $\vec{x}$, and $q_{2}$ and $\vec{e}_{2}$
are the analogous quantities for $q_{2}$.
A small test charge $q$ placed at $\vec{x}$ will experience
a force $\vec{F}=q\vec{E}(\vec{x})$.
Since $\vec{E}(\vec{x})$ is the sum of the fields due to $q_{1}$
and $q_{2}$, the force $\vec{F}$ on the test charge is the sum
of the two forces exerted on $q$.
The charges $q_{1}$ and $q_{2}$ also experience forces
due to the presence of each other.
For example, the force on $q_{2}$ due to $q_{1}$ is
$q_{2}\vec{E}_{1}(\vec{x}_{2})$ where $\vec{E}_{1}$
is the field due to $q_{1}$ alone.
The field due to a charged particle is inversely
proportional to the square of the distance from
the particle, so the force between charges obeys
the inverse-square law observed by Coulomb.

\begin{figure}[p]
\begin{center}
\ing{./eps/DipE.eps}
\caption{The electric field lines of a system of
two charges with equal but opposite charge.
\label{fig:Edip}}
\end{center}
\end{figure}

In field theory,
the force on a charged particle $q$ is attributed to the field
$\vec{E}(\vec{x})$ created by the other charges.
Thus the field concept is significantly different
from ``action at a distance.''
The force on the particle,
equal to $q\vec{E}(\vec{x})$, is exerted by the field at the
position of $q$, rather than by direct actions of the distant
charges.
In other words, an electrostatic system consists of two
physical entities:
a set of charged particles and
an electric field $\vec{E}(\vec{x})$.
The field is just as real as the particles.

Figure \ref{fig:Edip} illustrates the electric field for a
system of two charged particles with equal but opposite charges.
\fbox{\bf Fig.\,3}
The curves in Fig.\,\ref{fig:Edip},
called the electric field lines, represent the field.
The electric field is a vector at each point throughout
the space around the charges.
A positive test charge $q$ in the field would experience a force in
the direction of the field vector at its position.
We visualize the field by drawing the field lines,
which are curves that are everywhere tangent
to the field vector directions.

Electric charges at rest create an electric field $\vec{E}(\vec{x})$.
Ferromagnets and electric currents create another field---the
magnetic field $\vec{B}(\vec{x})$.
Figure \ref{fig:BfromI} illustrates the magnetic fields
for two elementary current sources:
(a) a small current loop and (b) a long current-carrying wire.
\fbox{\bf Fig.\,4}
In a ferromagnet the atoms behave as small
current loops with a uniform orientation.
The combined atomic fields make up the field
of the magnet.

\begin{figure}[p]
\parbox{0.49\textwidth}
{\includegraphics[width=0.40\textwidth]
{./eps/BfromIring.eps}}
\hfill
\parbox{0.49\textwidth}
{\includegraphics[width=0.30\textwidth]
{./eps/BfromIwire.eps}}
\caption{Magnetic field of (a) a small current
loop and (b) a long straight wire segment.
\label{fig:BfromI}}
\end{figure}

\begin{figure}[p]
\begin{center}
\ing{./eps/Lforce.eps}
\caption{The magnetic force on a moving charged
particle (or current segment).
The direction of the force is perpendicular to both
the particle velocity $\vec{v}$ (or current)
and the magnetic field:
$\vec{F}=q\vec{v}\times\vec{B}$.
A charged particle moving on a plane perpendicular
to a uniform magnetic field moves along a circle.
\label{fig:LForce}}
\end{center}
\end{figure}

Both $\vec{E}$ and $\vec{B}$ are force fields,
which extend throughout a volume of space.
But they exert distinct and different forces.
The electric field $\vec{E}$ exerts a force on a charge
$q$ in the direction of the field vector.
The magnetic field $\vec{B}$ exerts a force on moving
charges or current-carrying wires.
The direction of the magnetic force is perpendicular to
both the current and the field, as illustrated in
Fig.\,\ref{fig:LForce}.
\fbox{\bf Fig.\,5}
The magnetic field also exerts forces
on the poles of a ferromagnet.
The direction of the force is parallel (for a north pole)
or antiparallel (for a south pole) to the vector $\vec{B}$.
A compass needle aligns with the magnetic field
because the equal but opposite forces on the two
poles of the needle compose a torque that twists
the needle toward alignment with the field.
The magnetic forces on the current elements around a
small current loop also compose a torque;
in equilibrium the plane of the loop is oriented
perpendicular to the $\vec{B}$ vector.
The combined torques on the atoms in
a ferromagnet make up the net torque on the magnet.

The field concept was first stated by Michael Faraday.
From years of experimental studies on electricity and magnetism,
Faraday had formed the idea that a physical entity extends
throughout the space outside charges or magnets, and exerts
forces on other charges or magnets in the space.
He referred to this extended entity as the ``lines of force.''
The term {\em Electromagnetic Field} was coined by James Clerk
Maxwell, the renowned theoretical physicist who developed
a mathematical theory of electromagnetism based
on the field concept.