\section{Applications of Electromagnetic Induction} A number of important inventions are based on the phenomenon of electromagnetic induction. Two that have great technological significance will be described here. \subsection{The electric generator} \begin{figure}[p] \begin{center} \includegraphics[width=\textwidth]{./eps/ElGen.eps} \end{center} \caption{The design principle of an electric generator. The magnetic flux through the coil varies as the coil rotates and an EMF is induced in accord with Faraday's law. The voltage difference $V$ between the slip rings varies sinusoidally in time (inset graph). \label{fig:ElGen}} \end{figure} An electric generator (or dynamo) is a machine that generates an electromotive force (EMF) by electromagnetic induction. One or more coils of wire rotate in a magnetic field. The magnetic flux through a coil changes as the coil rotates, inducing an EMF around the coil by Faraday's law, ${\cal E}=-d\Phi/dt$. Figure \ref{fig:ElGen} shows schematically the design principle of an electric generator. \fbox{\bf Fig.\,7} In the model, the square coil rotates about the vertical axis that bisects the coil. The model field $\vec{B}$ is the constant field between north and south poles of ferromagnets. As the coil rotates, the flux through it varies sinusoidally as a function of time. Then there is an induced alternating EMF around the coil. The model generator in Fig.\,\ref{fig:ElGen} has the ends of the coil in contact with two slip rings. The voltage difference between the rings is equal to the EMF ${\cal E}$ around the coil. Thus the device generates an alternating voltage $V$ (shown in the inset graph) with constant frequency equal to the rotational frequency of the coil. If the rings are connected by a wire with resistance $R$, an alternating current (AC) $I$ will occur in the wire with $V=IR$ by Ohm's law. An external torque must be applied to the coil of the generator to maintain the constant rotation. If the slip rings are connected to an electric appliance, then energy is supplied by the generator because the induced EMF drives an alternating current in the appliance. But energy cannot be created---only converted from one form to another. If no external torque is applied, the generator coil will slow down and stop as its rotational kinetic energy is converted to the work done by the appliance. The slowing of the rotation is an example of magnetic braking: A conductor moving in a magnetic field experiences a force opposite to the motion. To maintain a constant rotation of the generator coil (and hence AC power to the appliance) the coil must be driven by a torque exerted by an external device. In a portable generator the external torque is produced by a gasoline-powered engine. In an electric power plant the external torque is produced by a turbine driven by a flow of hot steam; the steam is produced in a boiler, e.g., from burning coal in a coal-fired power plant or from the heat of nuclear reactions in a nuclear reactor. The electric generators producing electric power around the world are large and complex machines; but the fundamental physics in these devices is simple: Faraday's law and the design principle in Fig.\,\ref{fig:ElGen}. \subsection{The transformer} \begin{figure}[p] \begin{center} \ing{./eps/transformer.eps} \end{center} \caption{The design principle of a transformer. Two coils of conducting wire are wrapped around an iron core. The number of turns is different in the two coils, so the output EMF $V_{2}$ is different from the input EMF $V_{1}$. \label{fig:trans}} \end{figure} A transformer takes an input alternating EMF $V_{1}$, e.g., from an electric generator, and makes an output EMF $V_{2}$ that might be supplied to an electric appliance. (See Fig.\,\ref{fig:trans}.) The frequencies of $V_{1}$ and $V_{2}$ are equal but the amplitudes are different. A step-down transformer has $V_{2}V_{1}$; high-voltage applications require such a transformer. The physical phenomenon acting in a transformer is electromagnetic induction. Figure \ref{fig:trans} shows a simple transformer: Two coils of conducting wire are wrapped around a toroidal ferromagnetic core. \fbox{\bf Fig.\,8} The alternating current driven in the primary coil by the input EMF $V_{1}(t)$ creates a strong alternating magnetic field $\vec{B}(t)$, according to Amp\`{e}re's law. Because of the ferromagnetism, $\vec{B}$ extends around the toroid. Then the changing magnetic flux through the secondary coil induces an EMF by Faraday's law, ${\cal E}=-d\Phi/dt$; this EMF is the output voltage $V_{2}$. If $N_{1}$ is the number of turns of wire in the primary coil and $N_{2}$ is the number of turns in the secondary coil, then the ratio of the voltages is $V_{2}/V_{1}=N_{2}/N_{1}$. Transformers are a crucial part of the electric power grid. An electric power plant creates a 3-phase alternating EMF supplied to the grid. But the electric appliances that will use the power are located far from the power plant. Therefore a current must occur over very long transmission lines. The amount of lost power $P_{\rm lost}$, dissipated in resistance of the transmission wires, is given by the formula $P_{\rm lost}=I^{2}R$ where $I=$ current and $R=$ resistance of the line. The power supplied to users is $P_{\rm used}=IV$, where $V$ is the transmitted EMF. The ratio of power lost to power used is \[ \frac{P_{\rm lost}}{P_{\rm used}} =\frac{\left(P_{\rm used}/V\right)^{2}\,R}{P_{\rm used}} =\frac{P_{\rm used}R}{V^{2}}, \] i.e., inversely proportional to the square of the line voltage. Therefore the transmission of electric power is most efficient if the voltage $V$ is high. A typical power-plant generator might produce an alternating EMF with amplitude $168$\,kV, i.e., root mean square EMF $120$\,kV.\footnote{% The unit kV, kilovolt, is 1000 volts.} But that voltage would be changed in a step-up transformer at the power plant to a very high voltage, e.g., $345$\,kV. High-voltage transmission lines carry the electric power across large distances. Then step-down transformers are required at the end of a transmission line in order to reduce the voltage for applications. A transformer substation might reduce the voltage to $4.8$\,kV for overhead lines into residential areas. A final transformer mounted on the utility pole reduces the voltage to $110$\,V for the wires to a single house.