PEIRCE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNET. 157 



through the circuit corresponding to a current of intensity I h <f> (I), 

 and if the resistance of the circuit is ft, the differential equation which 

 determines the growth of the current is 



E j- = ft I or „ TiT = dt. 



dt E — ft I 



Since 4> is known, the coefficient of dl is known after values have been 

 assigned to the constants E and ft. If with a given E, ft has the 

 value r, the curve obtained by plotting the coefficient of dl against / 

 will have a shape something like that of the line KCDP of Figure 51, 

 which has the line / = E/r for an asymptote. If with the same value 

 of the electromotive force ft has the value (r + h), the curve will have a 

 shape something like that of the line KB DA, which has the vertical 

 asymptote /= E / (r + /i) 

 which passes through Q. If 

 with the core in the state for 

 which the diagram is drawn, 

 the circuit be closed at the 

 time t = 0, and if the resis- 

 tance be (r -f- h), the time 

 required for the current to 

 attain any value 1' less than 

 E/(r -f- h) is proportional to 

 the shaded area under the 

 curve KB DA from the ordi- 

 nate axis up to the vertical 

 line x = I'. If, however, the 

 resistance of the circuit had been r, the time required for the current 

 to grow to the intensity /' would be represented on the same scale by 

 the area under the curve KCDP from x = 0, to x = F. If the circuit 

 were closed when its resistance was (r -\- h), and if the current were 

 allowed practically to reach its final value for this resistance, as repre- 

 sented by the line OE, and if then the resistance h were suddenly 

 shunted out, the current would grow to its new final value at a rate 

 determined by the fact that the time required to reach the current OH 

 must be equal, on the scale of the diagram, to the area EFPH. If the 

 circuit had been closed first when its resistance was r, the time required 

 for the current to grow from the intensity OE to the intensity Oil 

 would still be equal, on the scale used, to the area EFPH, and the 

 shape of the current curve, from E/(r-A- k) on, would be the same as 

 before. Of course the N of this theory need not be the same as the 

 N of the statical hysteresis diagram for the given magnet ; it might 



CURRENT. 



FlGUKE 51. 



