MORE USES OF COMPLEX NUMBERS
generators can produce voltages of any value over a wide range, from a few volts to many thousands of volts. When Hoover Dam, on the Colorado River in Nevada near Las Vegas went on line in 1936, for example, it delivered electrical energy to Los Angeles at a potential of 275,000 volts. The unit of voltage is named after the Italian physicist Alessandro Volta (1745–1827), who constructed the first battery in 1800. The simplest electrical circuits contain only resistors, an electrical component commonly made of carbon in the form of a small cylinder with a wire terminal at each end by which it can be connected to other components. It is mathematically defined to be any device that obeys the relationship v 5 iR, i.e., Ohm’s law, where v is the instantaneous voltage difference between the two terminals, i is the instantaneous current in the resistor, and R is the value of the resistor. R is measured in ohms if v and i are measured in volts and amperes, respectively. There are two other two-terminal components that are commonly found in electrical circuits, the capacitor and the inductor. They obey the relationships shown in figure 5.4, where C and L denote the values of those components, in farads and henrys, respectively. The figure also shows the symbols used by electrical engineers when drawing circuit diagrams. The details of how these various components are physically constructed are not important in this example—we’ll be interested only in their mathematical definitions. These units are named in honor of the American physicist Joseph Henry (1797– 1878) and the English physicist and chemist Michael Faraday (1791–1867). The voltage-current relationships for the various components give us a physical interpretation of what is meant by saying a capacitor has a value of 20 microfarads (20 1026 F), or that an inductor has a value of 500 millihenrys (500 1023 H). The first case means any electrical device that has the property of conducting twenty amperes of current at a particular instant of time if, at that instant, the voltage drop across the device’s terminals is changing at the rate of one million volts per second, and such high rates of change are not at all uncommon in certain electronic circuits for very brief intervals of time.
Figure 5.4. The three standard electrical components used in circuits.
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