a crank and by turning the latter had caused the slider to assume a reciprocating motion across the lines of the field. In that case, when the crank is vertical, that is, parallel to the lines of the field, the speed of the slider is a maximum, and therefore its electro-motive force is also a maximum. As the crank approaches either of its dead points, where it is horizontal, the speed of the slider and its electro-motive force diminish and become zero at the moment the motion is reversed. From what was said in the preceding chapter, it will also be seen that the direction in which the electro-motive force acts depends on the direction of motion, and the current produced must therefore be alternating. If we plot the angles of the crank on the horizontal, starting from any given position, say, for instance, from its position at the end of the stroke, and the electro-motive forces. on the vertical, we obtain a graphic representation of the relation between these two quantities. In a uniform field, where the electro-motive force depends only on the speed of the slider, but not on its position in the field, the electromotive force is evidently proportional to the sine of the angle of the crank, and is given by the expression ω E = F da sin a, where a is the circumferential speed of the crank, and a its angular position, the other symbols being the same as before. It will be seen that E = 0, for a = 0 and α = π, whilst for T α == or α = π E attains its greatest numerical value, 2' being positive or negative according to the sign of the angle. The same relations obtain in the ideal alternating current dynamo, Fig. 13. If the crank is in the position shown, the wire is in the middle of the S pole piece and cuts the lines of force at maximum speed; if the crank is MAXIMUM ELECTRO-MOTIVE FORCE. 55 vertical, the wire moves parallel to the lines, and its rate of cutting lines is zero. This position corresponds to the end of the stroke with an oscillating slider. When the crank is again horizontal, but pointing to the left, the wire is in the middle of the N pole piece, and again its speed across the lines, or its rate of cutting lines, and the electro-motive force are maxima, but the current will be in an opposite direction to what it was at first. If the crank be turned in the direction indicated by the arrow, the current will leave the machine at the contact spring B, during the time the crank is on the right-hand side of 'the vertical diameter, and it will flow from B2 through the external circuit, and enter the machine at B1 during the time the crank is on the left-hand side of the vertical diameter. Let n be the number of revolutions per n 1 minute, then 2 rw, the circumferential speed of 60 the wire, and the maximum of electro-motive force, irrespective of sign, is evidently Now 2 r d is the total space swept by the wire, and F 2 r d is the total number of lines passing through that space; let z be that number, and we find for the maximum of electro-motive force the expression, During one half revolution the electro-motive force increases from zero to this maximum, and then decreases again to zero. As far as practical applications of the dynamo are concerned, it is not the maximum electro-motive force which we require to know, but the mean electro motive force, which acting during the same time as the variable electro-motive force, would cause the same quantity of electricity to flow through the circuit. Let, at any given moment, the wire occupy a position defined by the angle a from the vertical, and let it advance through an angle da during the time ♪ t, then the quantity of electricity flowing through the whole circuit of resistance R is evidently α During one half revolution a increases from zero to л, and the integral of the above expression taken between these limits gives 9 = 2 Fdr The time occupied in this transfer of q units of electricity is t = W and if, during that time, a constant electromotive force E1 were acting, the quantity transferred would be Ε' πρ If this quantity is equal to 9, we con sider E1 the average electro-motive force, and its value is given by the equation 1 Since F dw is the maximum electro-motive force generated at the moment when the wire is cutting the lines of the field at right angles, we have also 2 1 П MEAN ELECTRO-MOTIVE FORCE. 57 Inserting the value for E from equation 1, we find the average electro-motive force z being, as before, the total number of lines contained in n the space swept by the wire, whilst is the number of revolutions per second. 60 2 In the ideal alternating current dynamo represented in Fig. 13, the wire in which the currents are generated is arranged to one side of the spindle only. We could easily improve the design by carrying the wire symmetrically to the other side of the spindle, but insulated from it, and attach its end to a second metal sleeve insulated from M. The contact spring or brush B2 would then have to be set so as to touch this second sleeve, and since the electro-motive forces created in the two wires are at any moment in the same direction as regards the circuitalthough opposite as regards a fixed point in space-this improved dynamo with two wires will give double the electro-motive force of the original arrangement. We could still further increase the electro-motive force by coiling the wire several times round the axis, forming a rectangular coil, each convolution being insulated from its neighbours, and if the number of turns counted on both sides of the spindle is Nt, the average electro-motive force will be For most practical purposes, and especially for the transmission of energy, alternating currents are, however, not so convenient as continuous currents, and to produce them it will be necessary to add to our dynamo a device by which the currents are all directed to flow in the same sense as far as the external circuit is concerned. Such a device is the commutator, and its action can be explained by reference to Fig. 14. In the position shown, the electro-motive force created in the wire a b will be directed towards the observer, and that created in the wire c d will be directed from the observer. The ends of these wires are joined at the back by a cross connection a c, and at the front by two wires d f and b 9, the two halves of a metal cylinder, which for the purpose Fig. 14. to N Bi IDEAL CONTINUOUS CURRENT DYNAMO. of insulation are secured on a wooden hub. The electromotive forces created in d c and a b tend to draw a current from the line in the direction of the arrow to the brush B1, thence through f d, c a, b g, to the brush B2, and out again into the external circuit. This process will go on until the crank reaches the lower vertical position, the strength of the current meanwhile decreasing to zero. When the crank is vertical, each brush touches simultaneously both halves of the metal cylinder or commutator, as it is technically termed, and a moment later the connections become reversed, the brush B now touching the half cylinder to which the wire ƒ is at |