A glance at this equation should show that the power may be large and the current small if the voltage is made large. Hence large losses can be avoided by keeping the current small and by using high voltages when large amounts of power are desired. A few cases are worked out to make this entirely clear.

Suppose that the power delivered to a line of 3 ohms in each of the following cases is 50 kilowatts:*

1. The difference of potential of the wires at the power-house end of the line is 500 volts.

Power = 50 kilowatts = 50,000 watts.

a loss of 60 per cent.

Voltage = 500.

Current = 100 amperes.

I2R (loss in line) = 1002 × 3 = 30,000 watts,

2. The difference of potential at the power house is 10,000 volts.

Power = 50 kilowatts = 50,000 watts.

Voltage = 10,000.

Current = 5 amperes.

I2R (loss) = 52 × 3 = 75 watts, a small loss.

For distribution of power for light-circuits in cities and towns, about 2200 volts are commonly used; but where the distances are much greater, higher voltages are employed. For example, the electrical power from the hydroelectric plant on the Mississippi River at Keokuk is delivered to the transmission lines at about 110,000 volts.

* To avoid complications these examples have been treated as though they were cases of the transmission of energy by direct currents. If alternating currents were used, and if average values were given for the quantities concerned, the power equation would be (see last paragraph of section 419)

Average power = average voltage X average current X power factor.

The power factor is different for different circuits, varying from about 0.8 to 0.95; in some cases it may be as low as 0.6. When the circuit is used almost entirely for lighting purposes, the power factor may be regarded, in all rough computations, as equal to 1.0.

433. Summary. Joule's law states that the heat developed in a conductor by an electric current is, by equation (33),

Heat energy in joules = I2Rt,

or, by equation (35),

Heat energy in calories = 0.24 12Rt (approximately).

The power, P, expended in heat is, by equation (36),

or, by equation (38),

P = I2R watts,

P = 0.24 I2R calories per second.

When the terminals of any piece of apparatus are maintained at a constant voltage, the rate of heating varies inversely as the resistance. When two circuits are connected in parallel, the greater heat will be developed in the one with the smaller resistance.

The energy losses along transmission lines are due to the heat developed in the conductors. Equation (36) gives the power loss.

On account of the fact that the heating of conductors depends on the square of the current, and thus increases very rapidly as the current increases, it is customary to transmit power at high voltage. When the voltage is high, the current may be relatively small; for the power is always proportional to the product of current and voltage.

PROBLEMS

1. A current of 20 amperes flows over a transmission line which has a resistance of 0.5 ohm. What is the rate of loss of heat in watts?

2. The heating-coils of an electric iron have a total resistance of 18 ohms. When the iron is connected to supply mains which have a difference of potential of 108 volts, how many calories will be developed per minute? (3. A heating-coil to which 500 watts is supplied is placed in 1000 gm. of water. How long will it take to raise the temperature of the water from 20° C. to 100° C. if all loss of heat to the surroundings is negligible?

4. Calculate the number of calories produced when a 25-watt lamp is run for an hour.

5. An electric toaster has a resistance of 20 ohms. (a) How much power in watts does it use when connected to a 100-volt circuit? (b) How many calories will it develop in 30 min. ?

6. Heat energy is supplied electrically at the rate of 75 cal./min. (a) Find the number of watts supplied. (b) If the E. M. F. is 2 volts, find the current.

17. A coil of wire having a resistance of 20 ohms is immersed in a vessel

containing 300 gm. of water at 20° C. The wire carries a current of 2 amperes for 10 min. If all the heat generated is used to warm the water, what is the final temperature of the water?

8. A transmission line has a resistance of 2 ohms. (a) How much power is lost when a current of 10 amperes flows in the line? (b) a current of 100 amperes?

19. How much will it cost to raise the temperature of 10 gal. of water (1 gal. has a mass of 3800 gm.) from 10° C. to 60° C. by electrical energy which is supplied at 10¢ per kilowatt-hour?

10. 50 kilowatts of electrical power is delivered to a long transmission line which has a resistance of 5 ohms. (a) What will be the power wasted in the line if the power is delivered at 10,000 volts? (b) at 1000 volts?

CHAPTER XXX

RESISTANCE; GENERAL FORM OF OHM'S LAW

Summary of the principal facts about the resistance of conductors, 434. Rheostats; resistance boxes, 435. Measurement of resistance; the Wheatstone-bridge method, 436. Resistances in series, 437. Resistances in parallel, 438. Proof of the law of parallel resistances, 439. Law of divided circuits; shunts, 440. Ohm's law; loss of potential due to resistance, 441. Ohm's law applied to a complete circuit; electromotive force, 442. Ohm's law applied to groupings of cells, 443. General form of Ohm's law, 444. Limitations of Ohm's law, 445. Counter electromotive force in circuits, 446. Summary, 447.

: 434. Summary of the principal facts about the resistance of conductors. 1. The resistance of a conductor has the same numerical value for large currents as for small currents, provided the temperature is not changed. It is this fact that makes Ohm's law so useful. If the resistance of a given piece of electricat apparatus has once been found, its value is known for other occasions.

(2) The resistance of a conductor varies inversely with the cross-sectional area.

(3) The resistance of a conductor is directly proportional to its length.

(4) The resistance of a conductor depends on the material (see section 413 for the relative resistances of a number of metals). Impurities in a metal usually tend to increase its resistance. Since alloys of metals have much greater resistance than pure metals, wires made of alloys are commonly used where a considerable resistance is needed; for example, in the coils of electric heaters. Metals have much lower resistance than liquids.

(5) The resistance of metallic conductors usually increases as the temperature rises. The resistance of liquids, such as those used in cells, decreases as the temperature rises. For example, the resistance of a storage battery is higher at low temperatures. The resistance of most of the pure metals increases at about the same rate-about 4 per cent for an increase of 10° C. Alloys usually have a small rate of change. In the case of a few alloys the resistance remains practically constant over a wide range of temperature. This is one of the reasons that certain alloys are used as conductors in many types of electrical instruments. If a, the "temperature coefficient," and Ro, the resistance at 0°C., are known, the resistance R, at a temperature t can be computed from R1 = R(1+at).

435. Rheostats; resistance boxes. In the laboratory it is often necessary to insert in a circuit resistance which can be increased or decreased in order to obtain a current of the desired magnitude.

For this purpose there are many types of commercial rheostats. They usually contain coils of wire with switches or sliding contacts so that different portions can be used (see Figs. 259 and 260).

Resistance boxes differ from rheostats in that the length and size of the wire in the coils are so chosen that the resistances have definite values, while in a rheostat the values of the resistances are usually not given. The coils in resistance boxes are usually made