CHAP. IIIl HYSTERESIS AND EDDY CURRENTS 39 



To prove that the energy lost per cubic unit of iron per cycle 

 of magnetisation is represented by the area of the hysteresis loop, 

 we first write down the expression for the energy returned to the 

 electric circuit during an infinitesimal change of flux in the 

 part AC of the cycle. Let the flux in the ring at the instant under 

 consideration be webers, and the magnetomotive force ui amp- 

 ere-turns, where i is the instantaneous value of the current , and n 

 is the total number of turns on the exciting winding. The instan- 

 taneous induced e.m.f., due to a decrease of the flux by <i<I> during 

 an infinitesimal element of time <lt seconds, is e= nd0/dtvo\t. 

 The sign minus is necessary because e is positive (in the direction 

 of the current) when dd> is negative, that is to say. when the flux 

 decreases. The electric energy corresponding to this voltage is 



dW = eidt= nidffl watt-seconds (joules). 



Hence, the total energy returned to the electric circuit during the 

 part AC of the cycle is 



W= I nidffl, 



J A 



or. interchanging the limits of integration, 



W= C A nid<J>. 



Since all the parts of the ring undergo the same process, and 

 the curve in 1'ii:. 7 is plotted for a unit cube of the material, it is 

 of interest to find the loss of energy per cubic centimeter of mate- 

 rial. If S is the cross-section and / the mean length of the lines 

 of force in the iron, we have that the volume 



V = Sl cubic centimeters. 



Dividing the expression for the energy by this equation, we find 

 that the energy in watt-seconds per cubic centime'- M is 



, . . . (110 



H is in ampere-turns per centimeter, and B is in m 

 per square cent in 



Hut II -Hi is the area of an infinitesimal strip, such as is shown 



by hatchii ig. 7. Consequently, tin- right-hand side . 



