so that an increase of the external resistance from b to b causes the power received by the external circuit to be diminished by only th. And an examination of the second form of the formula shows that we may go on increasing until r equals b, that is, we may go on

29

Fig. 182. Curve connecting the Power received by an External Circuit and the Resistance of that Circuit.

increasing the external resistance from 6 to 2b without the power given to the external circuit being less than the maximum by more than 1th.

Finally, then, we may conclude that, if the external circuit be a simple resistance, having any value between

b

and 26 ohms, the power that it will receive will not

be less than 89 per cent. of the maximum power, viz.

The curve in Fig. 182 shows the connection between x, the external resistance, and W, the power received by the external circuit, equal lengths being taken to represent an ohm and a watt, and values of 7 volts and 2 ohms being arbitrarily taken for the fixed values of E and b. The curve is very flat in the neighbourhood of x equal to b, which illustrates what has been proved above, viz. that while a generator of fixed E.M.F. and resistance b ohms gives the maximum power to an external circuit, composed of a simple resistance x ohms, when x equals b, the power received by the external circuit will not be materially diminished if x has any value from about 6 to 11⁄2 b.

If we were dealing with the power given by the generator to a portion only of the circuit, and if this portion consisted of a simple resistance of x ohms, then in all the preceding equations we should have to substitute bl for b where l ohms was the fixed resistance of the leads connecting the generator with the portion of the circuit under consideration. And the curve in Fig. 182 would then connect the power given to this portion of the external circuit when x was the resistance of this portion alone, and when the resistance of the battery and leads together was 21 ohms.

If the external circuit, instead of consisting of a simple resistance, contains an apparatus possessing a back E.M.F. of E' volts, then we saw in § 125, page 394, that, apart from heating due to resistance, a generator having a fixed E.M.F. of E volts will cause the rate of transformation of electric energy in this apparatus to be a maximum

when this back E. M.F. is adjusted so that it equals

E

2

Now this rate of transformation of energy we have seen equals

E E'

× E' watts,

b + l + m

where b, and m are the resistances in ohms of the generator, the leads and the apparatus in question, and by applying to this expression exactly the reasoning that was used in connection with the expression A (E — A b) at the beginning of the section, it can be shown that when

Hence, the proportional diminution of the rate of transformation of electric energy due to a back E.M.F. in an apparatus equals the square of the proportional change in the back E.M.F. of the apparatus from the value that makes the rate of transformation a maximum.

For example, if E 20 volts

=

then, apart from the production of heat due to resistance, the rate of transformation of electric energy will be a maximum when the back E.M.F. is 10 volts. If, however, the back E.M.F. is 15 or 5 volts, that is, 50 per cent. greater, or 50 per cent. less, than 10 volts, the rate of transformation of electric energy

2

respectively, which are each only (50),

100

× 10 watts,

the maximum rate of transformation.

Example 101.-If a battery of 50 cells in series, each having an E.M.F. of 2.2 volts and a resistance of 0.05 ohm, be giving the maximum power to an external circuit, what is the current that flows, and by how much per cent. will the power given to the outside circuit be

reduced if the circuit be altered so that the current flowing is diminished by 20 per cent. ?

Answer.-22 amperes;

By 4 per cent.

Example 102.-If the external circuit in the last question consist of a simple resistance, what is the value of this resistance when it receives maximum power, and by how much per cent. will the power given to the external circuit be reduced if its resistance is (a) 50 per cent. smaller, (b) 40 per cent. larger, than that which corresponds with maximum power?

Answer.-2.5 ohms;

By 6.25 per cent., By 2-778 per cent.

Example 103.-A generator having a fixed E.M.F. of 220 volts drives a motor. What should be the back E. M. F. of the motor so that it may develop the greatest mechanical power, and by how much will the power it develops be reduced if the back E.M.F. be increased by rds above this value? Answer. -110 volts;

By ths.

127. Efficiency.-When, by means of any machine, or contrivance, one form of energy is converted into another form, some heat is produced; hence, if heat energy is not the form in which the energy is required after the conversion, some portion of the energy which has been used up has been converted into a useless form as far as the object in question is concerned, and may, therefore, be regarded as wasted. Consequently, in all cases the amount of useful energy produced is less than the amount of energy used up. For example, when a machine is used to do work there is a waste of energy in the heating of the bearings of the machine; if falling water is employed to turn a water wheel there is in addition waste of energy in the eddies set up in the water, in the splash of the water against the blades of the wheel as well as in the friction of the water stream against the sides of the channel which guides it to the wheel. When oil, wax,

gas, &c., are consumed as illuminants only a very small fraction of the available energy is converted into the special form of energy, called light, which affects the retina of the eye, and the greater part is wasted in heat, whose action on the eye does not differ from its action on other parts of the body. Again, in a battery a certain amount of chemical energy is wasted in the heat produced by "local action" (see § 132, page 428), which goes on even when the battery is producing no useful current; further, on the battery being used to send a current through some external circuit a portion of the chemical energy that is converted into the electric energy is always wasted in heating the battery in consequence of its resistance.

The value of any machine or contrivance for effecting a conversion of one form of energy into another depends first on the rate at which energy in a useful form is evolved by it-that is, the useful power the machine develops, and which is sometimes called its " useful activity"-secondly, the value of the contrivance depends on the ratio of the amount of useful energy produced to the amount used up in the process, and this ratio is called the "efficiency."

Efficiency, then, is expressed by a simple number, less than unity, such as, 0.63, 75 per cent., 84 per cent. Sometimes, however, it is found convenient to employ different units of energy, or of power, in the numerator and denominator of the fraction which represents the efficiency. For example, while the true efficiency of a glow lamp does not generally exceed 0·01—that is, not more than one-hundredth of the electric energy supplied to it is converted into light-the efficiency of a glow lamp is sometimes spoken of as candle per watt, meaning that an electric power of 4 watts must be supplied to the lamp to produce an illumination of 1 candle.

When any current generator developing an E.M.F. of E volts and having a resistance of b ohms is sending a current of A amperes round any circuit, the ratio