ELECTROMOTIVE FORCE AND CURRENT. 13 



curve No. II. shows the change of current measured in amperes 

 per cycle. 



The second curve is obtained from the first by drawing 

 at a number of points tangents to curve No. I., as shown at P, 

 a horizontal line P N corresponding in length to J cycle, and 

 a vertical line through N to meet the tangent at M. The 

 length N M then shows the change in current which would 

 actually occur in cycle if the rate of increase or decrease 

 remained the same as it actually is at P. 



Four times the length of M N will consequently give the 

 height of the "rate of change " curve at the point corresponding 

 to the same angle as P. The scale for the dotted curve (Fig. 

 6) is taken four times as great as for the original curve, 

 so that when the length N M is plotted vertically it corresponds 

 to four times the change of current in J-cycle, i.e., to the rate 

 of change of current for a whole period. 



When N M is drawn downwards, it is plotted below the 

 base-line in curve II., and vice versa. 



It will be noticed that the curves I. and II. in Fig. 6 are 

 similar in shape, but that the second curve is displaced by 

 one- quarter of a period to the left of the curve I. From 

 this it is apparent that the change of current is most rapid 

 at the point where the current curve crosses the axis and 

 changes its direction. In other words, the rate of change is 

 greatest when the current is zero. The rate of change is 

 zero, on the other hand, at the moment when the current has 

 attained its maximum value in either direction. 



The maximum value of the rate of change of current 

 curve is seen to correspond to a change of 78'5 amperes per 

 cycle, i.e., the rate of change of current per cycle is 6-3 times 

 the maximum value of the current. More accurately, the 

 rate of change per cycle is 2 TT times the maximum current. 

 Since the current passes through n cycles per second the 

 maximum rate of change of current per second is 2 TT n x the 

 maximum value of the current.* 



* The reason for this exact relation between current and rate of 

 change of current is that the rate of change of the sine of an angle is 

 equal to its cosine when the angle is measured in units of circular 

 measure or radians. Since there are 2ir radians in 360, this corresponds 

 to a rate of change of 2w x the cosine, measured in rate of change per 

 cycle. The maximum value of the cosine is equal to the maximum 

 value of the sine, and the fact that cos = sin (0 4- 90) is shown by the 

 relative displacement of the curves by 90. , If points on curve I. are 

 represented by C sin 0, the corresponding points on curve II. will be 

 given by the expression 2?r (7 cos $. 



