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CHAPTER XXVIII

POTENTIAL DIFFERENCES; WORK AND POWER
IN CIRCUITS

The quantitative relation between quantity and current, 407. Potential: an analogy, 408. Differences of potential: voltage, 409. Voltmeters, 410. Voltages of cells, 411. Resistance; Ohm's law, 412. Resistances of different materials, 413. The voltage drop along wires, 414. Summary of facts regarding the difference of potential, 415. Quantitative definition of difference of potential, 416. Measurement of difference of potential, 417. Work done in a part of a circuit, 418. Power expended in a part of a circuit, 419. Other units for measuring work, 420. Practical examples, 421. Work done in a complete circuit: electromotive force, 422. Summary, 423.

The student should now be ready to learn the quantitative meaning of some of the more common concepts in use in practical applications of electricity. In this chapter special attention is given to those involved in the measurement of power and work, which play so important a part in electrical practice.

407. The quantitative relation between quantity and current. Popularly the word current, as applied to the flow of the water in a river, is used in two different senses: sometimes it refers to the speed of the water, and sometimes to the total quantity of water passing in a given time. For example, if more water flows during a given time in a wide, sluggish stream than in one that is narrow but swift, the current of the wide stream is said to be larger. With this meaning, the current of the river may be measured in gallons per minute or in cubic feet per minute. In electricity the term is used in this latter sense, referring not to the velocity of the charge but to the total quantity passing in a unit of time.

Hence current of electricity is the rate of flow of charge or, in the case of a uniform flow, the charge passing in 1 second.

If I is the average current, the total quantity of electricity, Q,

passing along a wire in t seconds, is given by the equation

In using this equation both Q and I must be measured in the same system of units. The current I is usually measured either in C.G.S. electromagnetic units or in amperes. When referring to static charges, we usually measure Q in C.G.S. electrostatic units, but rarely when dealing with the flow of electricity. Expressed in practical units equation (10) becomes

Coulombs = amperes X seconds.

(11)

408. Potential: an analogy. Most of the common applications - of electricity today are cases of the transfer or transformation of energy. When one turns on an electric light, he connects the lamp to a source of energy. This energy flows into the filament and is there converted into other forms of energy-heat and light. The electric-light meters found in so many houses are instruments that indicate the amount of energy which has been delivered; and it is energy that is paid for, since the unit commonly used, the kilowatt-hour, is a unit of energy. The electric stove and the electric motor are instruments for the transformation of electrical energy into other forms of energy. So common is this transformation that it is often necessary to estimate or to compute the amount of electrical energy that is needed. In no other branch of human knowledge are measurements more important than in electricity. But one cannot make estimates or computations of electrical energy without knowing something about potential. For this an analogy will be helpful.

Suppose one wished to find out something about the amount of energy available from a given water supply; for example, a reservoir located at some higher place. Could the available energy be determined if one knew only the quantity of water that could be used? No; more information than this is needed. In additior to knowing the quantity of water, one must know the elevation of the reservoir, or the pressure in the water mains due to the height of the reservoir. If the pressure is small, a large quantity of water

may be required to supply a certain amount of energy; if the pressure is high, a smaller quantity of water suffices. Two things must be known: the quantity and the pressure.

A similar condition exists in estimating electrical work. The available energy cannot be estimated merely from a knowledge of the quantity of electricity. Something else must be known, something analogous to pressure in the case of water power. This * needed concept is called potential. In electrical usage the term potential, as will be pointed out later, has a definite quantitative meaning; but at present the student may regard it as a term analogous to water pressure. A simple case will illustrate. Water always tends to flow from higher to lower elevations or from higher to lower pressures. In a similar way a positive charge of electricity tends to flow from higher to lower potentials. In order that one may obtain water from the water mains, the pressure in the mains is kept greater than atmospheric pressure. For a somewhat similar reason the two wires of a house-lighting circuit are kept at different potentials, the difference usually being a little over 100 volts. When the two wires are connected through a lamp, a current flows from the wire at higher potential to the one at lower potential and lights the lamp.

Potential is such an important and fundamental concept in electricity that the student should form a definite idea of its character and use. For this reason he should study very carefully the following sections.

409. Differences of potential: voltage. In considering cases of the flow of water from one elevation to another we are not especially concerned with the elevation of these two places above the mean sea level. What we are interested in is the difference of elevation of the two places. The same thing is true of the use of pressures. For example, in stating the pressure of steam or water it is customary to give only the difference in pressure between the steam or water and the air outside. The same sort of thing is true in electricity. In no practical problem do we need to know the absolute potential of any conductor: it is only the difference in potential between that conductor and some other one that must be known.

In practical work, difference in potential is usually measured in volts, and the difference of potential is often called the voltage, a term that refers to a difference in condition of two conductors. For example, when it is said that the voltage of the light circuit is 110 volts, what is meant is that the difference in potential of the two wires which enter the house is 110 volts.

Before undertaking to learn the precise definition of difference of potential and how it is used in computing work and power, it will be to the student's advantage to learn a number of facts about potential differences. These

will be given in the follow

Ming sections.

410. Voltmeters. Several different types of commercial instruments are now available for measuring difference of potential. The divergence of the leaves of the ordinary electroscope is a measure of the difference of potential existing between the leaves and the case of the instrument. Similarly, the electrostatic voltmeter, in which a delicately balanced charged

vane moves when attracted or repelled by another charged conductor, measures difference of potential (Fig. 254). The scales in the commercial forms are ordinarily graduated so that they read volts, the indicated number being the difference in potential between two terminals mounted on the instrument. The common type of voltmeter that we find on switchboards and in laboratories also has two terminals, and the deflection of the index indicates the difference of potential between these two terminals.

If for the time being we regard the difference of potential (voltage) as that which is indicated by some form of voltmeter, we can readily ascertain a number of facts regarding potential. If the student wants to find out something about water pressure, he can use a pressure gauge and find the facts by actual test. In a similar way he can use an instrument that indicates potential difference, to test out a number of cases.

411. Voltages of cells. In section 384 it was pointed out that the terminals of a dry cell are charged. The experiment described there showed also that the terminals are not at the same potential; for the instrument used, the electroscope, indicates difference in potential. But it is much easier to show that the terminals of the cell are not at the same potential by using an ordinary voltmeter.

1. The difference of potential between the two terminals of a dry cell is about 1.5 volts. Touching these terminals does not discharge them. If a small lamp is lighted by connecting it across them, the terminals will still be at different potentials, since these potentials are maintained by chemical action in the cell.

FIG. 255

+

2. In section 384 an experiment was described which showed that the carbon terminal of a dry cell was positively charged, and the zinc terminal negatively charged. Since positive electricity tends to go from a higher to a lower potential, it follows that the carbon terminal is at a higher potential than the zinc terminal.

3. When a number of dry cells are connected in series, larger differences of potential are obtained. To connect them in series the carbon terminal (usually the central one) of the first cell is connected to the zinc of the second, the carbon of the second to the zinc of the third, and so on. A voltmeter will show that if each cell has a difference of 1.5 volts, two cells will have twice that, three cells three times, and so on. The student should have very little difficulty in seeing why this is true. The carbon of the first cell in Fig. 255* is 1.5 volts higher than the zinc terminal, and the carbon of the second cell is 1.5 volts higher than the second zinc. But the zinc of the second is at the same potential as the carbon of the first; hence the carbon of the second is 3.0 volts

* The figure shows also the conventional method of drawing cells.b