(For further details as to this line the reader is referred to ‘Nature, Vol. xxxiii. p. 13, whence this brief notice has been derived.) 10. Distribution of Potential in the Circuit of a Dynamo and Motor. In what follows we shall speak of a dynamo of constant E.M.F. E, and a motor of reverse E.M.F. e. The reader can easily apply the same methods to the similar case of voltaic piles and electrolytic cells (se: Chapter XV. § 9). As in Chapter XIII. so here, the ordinates represent potentials; the abscissæ represent resistances; and the whole diagram exhibits the fall of potential, that occurs in obedience to Ohm's law, throughout the circuit. We have for convenience represented the E.M.F. E of the dynamo as a AV occurring abruptly at one point in the armature. This is not the case; for the E.M.F. E is the sum of a number of E.M.F.s occurring in the different turns of wire. But in what follows, this simple assumption will not lead us to any erroneous results; and without it we could have no such simple diagrams as those here given. (i.) Dynamo; open circuit.-In fig. i. the points A and B represent the terminals (or brushes) of the dynamo; A x and y B y I B FIG. i. together measure the internal resistance R, of the dynamo; the ordinate xy measures the E.M.F. E. Since there is no current, it is clear that Ax and y B are horizontal, or that the points A and B have a difference of ordinate measured by xy. In other words, the E.M.F. E may be measured by the AV of the terminals A and B when the circuit is open. (ii.) Dynamo; closed circuit; no other E.M.F. in the circuit. Now let the circuit be closed through an external wire whose resistance is measured by the abscissa ba' cf fig. ii. A' are really one point, the circuit being complete. and y A' make an angle with the axis Or such portional to tan þ, or to x v Here A and The lines A x that C is pro as explained in Chapter XIII. It is clear that in this case the points A and B have a difference This means that of ordinates that is less than the ordinate xy. the AV between the terminals will not measure the total E.M.F. E of the dynamo, but will be less than this latter; there being some fall of potential down the internal resistance R1, according Direction of current B to Ohm's law. FIG. ii. We may conveniently designate the AV between A and B by the symbol E. It will not be hard to find what Let R, be the internal resistance a b, and R, the external resistance ba'. Then, since there is in the whole circuit a total fall of potential measured by E, there will in the portion R, be a fall measured . E, by Ohm's law. This expression can be written by R1 Hence, E falls short of E by the amount R, C; or B We can therefore find out the value of the E.M.F. of a dynamo (supposing it not to be shunt-wound) when it is running and when the circuit is closed, by finding the AV between its terminals and by adding to this the product of its resistance into the current. If measures be made in ampères, ohms, and volts, there will be no 'constants' involved in these expressions. (iii.) Dynamo and motor; a reverse E.M.F.-In fig. iii. we have the case of a circuit that comprises a dynamo and motor. A and B are the terminals of the dynamo, C and D those of the The resistance R, of the former is measured by the motor. 2 The re abscissa a b, and the resistance R2 of the latter by ca. maining abscissæ be and da' together represent the resistance of the connecting wires. Ax, y z, and w A', are of course parallel. The ordinate xy represents the E.M.F. F. of the dynamo, while zw represents the reverse E.M.F. e of the motor. current is given by the expression. E-e gram, it is proportional to A a a' The ; or, in the dia Using the same notation as in (ii.) above, we may say that the AV E between the terminals A B is related to E by the equation B A E=E+R, C. C D In a similar way we have for the AV E between the terminals of the motor the relation since here the fall of potential due to Ohm's law is added to the fall measured by e. The reader should notice the fact, at first sight somewhat paradoxical, that in the dynamo the E.M.F. (represented for convenience by the abrupt rise xy) appears to be opposed to the current; while in the motor the E.M.F. (similarly represented by zw) appears to be acting with the current. These apparent difficulties disappear upon a little consideration. up the electricity from r to y; and it is the advantage thus gained that enables the current to flow in the direction indicated by the arrow. Hence the driving E.M.F. E is represented suitably in the diagram by the rise ry. In the case of the motor we may observe that if the current is to do work, it must be by being, as it were, let down an 'electrical hill. Hence we must occur in the opposite direction to that in which y Occurs; the current must fall down z w. Moreover, such a fall as z w rightly represents a reverse E.M.F., or one which opposes the current, for the following reason. The current is proportional to tan (see Chapter XIII.); and hence the current is diminished, if the inclination of the lines Ar, yz, and wA', to the axis Or be diminished. Now the fall zw does diminish the angle 4, as is easily seen if we compare fig. iii. with a figure in which zw is removed, xy and a a' remaining unaltered. Hence e is rightly represented by the fall z w. § 11. Work done per Second upon a Dynamo as Related to the Velocity of Rotation. The mechanical work per second or activity expended in driving a dynamo is, if we neglect friction, measured in watts by the product E C. The manner in which the magnitude of this product depends upon the velocity of rotation of the armature varies according to the nature of the dynamo. (i.) Case of magneto, or other constant-field, machine. - Here we have, by the formula of Chapter XXIII. § 2, that E is proportional to v; and hence that, when there is no other E.M.F. in the circuit, C is also proportional to v. Therefore the work is proportional to v2. (ii.) Case of series, or other varying-field, machines.-- In these the E.M.F. is first affected directly by 7, and then is affected also indirectly, inasmuch as increase of current increases the field-strength. Hence the work is proportional to some higher power of v than the second power. Since however the increase in field-strength, corresponding to a given increase in the current, depends upon the degree to which the cores of the field-magnets are already saturated, it is not possible to express in any simple and yet exact manner the relation between the activity E C and the velocity of rotation v. CHAPTER XXV. VARIOUS APPLICATIONS OF ELECTRICITY; TELEGRAPHS, § 1. Introductory.-In the present Chapter we shall describe, but necessarily in a very brief manner, various applications of 'electricity'; these applications being for convenience divided into groups, as indicated in the heading of the Chapter. From the nature of the subject there will not be, as there was in most of the previous Chapters, any order or progress from group to group. The whole forms a somewhat miscellaneous collection, illustrative of various principles explained earlier in the Course. TELEGRAPHS AND ELECTRIC SIGNALLING. § 2. General Principle of Telegraphy.-Let us consider two stations A and B, more or less remote from one another, connected by an insulated wire, the circuit being completed either through a return wire, or by means of large metallic plates buried in the soil, through the earth itself. It is clear that from either station can be sent currents that will pass through the other station. And if each station be provided, not only with a battery and with a suitable instrument for making or breaking current, but also with a receiving instrument on which the current acts to deflect a needle, sound a bell, or in other ways attract attention, it is clear that we have the means of exchanging signals and messages between the two stations. Note on the return wire.'—We may remark that as a rule a return wire is not employed, the circuit being completed by means of large metallic plates buried in the earth, as is seen in the diagram to § 4. It is not necessary to suppose that the current does actually return through the earth. For the plates will be kept at what will be approximately zero potential (or the same |