\section{Electromagnetic Induction}

The previous sections were concerned with static systems of
electric charge or current, i.e., in which the fields
$\vec{E}(\vec{x})$ and $\vec{B}(\vec{x})$ do not change
in time.
Some relationships between electricity and magnetism
were discussed:
a steady electric current produces a magnetic field
(Amp\`{e}re's law);
a magnetic field exerts a force on any electric
charge moving across the field lines.
However, for static fields electric and magnetic phenomena
appear to be rather distinct.

Time-dependent fields will be described next:
The electric field $\vec{E}(\vec{x},t)$ and magnetic field
$\vec{B}(\vec{x},t)$ are functions of time $t$ as well as position $\vec{x}$.
In dynamic systems the two fields affect each other significantly.
Therefore electric and magnetic phenomena are connected,
and must be described by a unified theory.
Electricity and magnetism are then combined into electromagnetism.

The phenomenon of electromagnetic induction 
was discovered in 1831 by Michael Faraday
in England and independently by Joseph Henry
in the United States.
The effect is that when a magnetic field changes in time,
an electric field is induced in directions that curl
around the change of the magnetic field.
This phenomenon has important technological applications.

Electromagnetic induction may be observed directly
in simple physical demonstrations.
Figure \ref{fig:DemoEMI} shows schematically a coil of
conducting wire C connected to a galvanometer G.
\fbox{\bf Fig.\,6}
The galvanometer acts as a current indicator:
When no current flows around C the galvanometer needle
points toward the central (zero current) position;
when current flows the needle is deflected.
No current source (such as a battery) is connected to the wire coil.
In Fig.\,\ref{fig:DemoEMI}, M is a magnet that can be moved toward
or away from the coil C.
When the magnet is at rest no current flows in C and
the galvanometer needle points in the central direction.

\begin{figure}[p]
\begin{center}
\ing{./eps/DemoEMI.eps}
\end{center}
\caption{Schematic apparatus for demonstrations
of electromagnetic induction.
\label{fig:DemoEMI}}
\end{figure}

If the magnet in Fig.\,\ref{fig:DemoEMI} moves toward
the coil, the galvanometer needle will be deflected in
one direction, indicating that a current is flowing in C.
The current exists while M is moving.
When the motion of M ceases, the needle will return to the
central position indicating that the current has stopped.
If the demonstration is repeated with the magnet moving
away from the coil, the galvanometer needle will be
deflected in the opposite direction while M moves.
These demonstrations show directly that a change in the
magnetic field through the coil induces a current
around the coil.

The magnetic field in the demonstration might be varied
in a different way.
Suppose the bar magnet M in Fig.\,\ref{fig:DemoEMI}
is replaced by an electromagnet that does not move.
Electric current $I(t)$ in the electromagnet solenoid
produces a magnetic field $\vec{B}(\vec{x},t)$ according
to Amp\`{e}re's law.
If the solenoid current is constant then $\vec{B}$ is constant
in time and there is no current in the coil C.
But if the solenoid current changes,
the magnetic field changes.
A deflection of the galvanometer needle will be observed
while the magnetic field is varying in time.
A current around the sensing coil C is again induced
when the magnetic field through the coil is changing.

These demonstrations show that when a magnetic field
$\vec{B}(\vec{x},t)$ changes in time,
a current is induced in a conductor that
is present in the field.
However, the induced current is actually a secondary effect.
The current in C is created by an induced electric field
$\vec{E}(\vec{x},t)$, and $\vec{E}$ is the primary effect.
Electromagnetic induction is fundamentally a phenomenon of
the fields, $\vec{B}(\vec{x},t)$ and $\vec{E}(\vec{x},t)$.
A magnetic field that changes in time induces
an electric field in directions curling around
the change of the magnetic field.
If there happens to be a conducting coil present,
as for example C in Fig.\,\ref{fig:DemoEMI},
then the induced electric field drives an electric current
around C.
But the primary effect is induction of the electric field.
The induced electric field $\vec{E}(\vec{x},t)$
exists while the magnetic field $\vec{B}(\vec{x},t)$
is varying in time.

The apparatus shown in Fig.\,\ref{fig:DemoEMI} is only
schematic.
The induced current in C would be very small for an
ordinary bar magnet M moving at reasonable velocities.
A practical demonstration would require a sensing coil
with many turns of wire and a sensitive galvanometer.
The effect might be increased by putting an iron core
inside the coil to enhance the magnetic field.

Faraday and Henry performed laboratory experiments similar to
the demonstrations illustrated in Fig.\,\ref{fig:DemoEMI}
in their discoveries of electromagnetic induction.
Faraday described the results of his detailed studies in
terms of the lines of force---his concept of a magnetic
field filling space around a magnet.
In modern language,
a statement summarizing his observations is

\begin{quotation}
\noindent
{\bf Faraday's Law:} When the flux of magnetic
field through a loop changes in time,
an electromotive force (EMF) is induced
around the loop.
\end{quotation}

\noindent
This statement is Faraday's law of electromagnetic induction.
In equation form, $d\Phi/dt=-{\cal E}$
where $\Phi$ is the magnetic flux through the loop,
$d\Phi/dt$ is the rate of change of the flux, 
and ${\cal E}$ is the electromotive force around the loop.
The ``flux'' is a quantitative measure of the
field lines passing through the loop, defined by the
product of the component of the magnetic field vector
perpendicular to a surface bounded by the loop
and the area of the surface.

The ``loop'' in Faraday's law is a closed curve,
e.g., a circle.
The loop may be a conducting loop,
in which case the induced EMF drives a current;
or it may just be an imaginary curve in space.
In either case the EMF is an induced electric
field curling around the change of the magnetic field.

Another, related demonstration may be carried out with the simple
apparatus of Fig.\,\ref{fig:DemoEMI}.
Instead of moving the magnet M and holding the coil C fixed,
suppose the coil is moved while the magnet is held fixed.
Again a current will be observed around C.
The phenomenon in this case is called ``motional EMF;''
an electromotive force is induced in a conductor that
moves relative to a nonuniform magnetic field.
In the language of Faraday, when a conducting wire moves
through magnetic lines of force an induced current flows
in the wire.
Evidently any change of the magnetic flux through a
conducting loop will induce a current in the loop.

Yet another way to change the magnetic flux through the coil C in
Fig.\,\ref{fig:DemoEMI} is to change the orientation of the coil.
In Fig.\,\ref{fig:DemoEMI} the plane of the coil is shown
perpendicular to the bar magnet.
In this orientation the magnetic field lines
pass straight through C;
the flux is the product of the magnetic field strength
${B}$ and the area $A$ of the loop, $\Phi=BA$.
Now suppose the coil rotates about a vertical axis,
with M fixed.
Then the flux of magnetic field through the coil changes
as the plane of the loop is at a varying angle to the field vector.
When the plane of the loop is parallel to the field lines
the flux is zero because no field lines pass through the coil.
While the coil is rotating,
a deflection of the galvanometer needle
will be observed, consistent with Faraday's law,
because of the changing flux.
This is another example of motional EMF:
The conducting wire moves through the magnetic field lines
and there is an induced current in the wire.