ELECTROMAGNETS. MAGNETISM OF IRON. 



353 



field, and W is the work done by the field upon a pole of strength 

 m as the pole travels along the path. 



Proof. The product mA is the average value along the path 

 of the component parallel to the path of the force & (= m$C) 

 with which the field acts on the pole. Therefore ImA is the 

 work done on the pole by the field as the pole travels from one 

 end of the path to the other. That is : 



W=lmA 



Dividing both members of this equation by m, and remember- 

 ing that IA of according to equation (i), we have equation (2). 



The following discussion of magnetomotive force is based upon the method of calcu- 

 lus. The usual definition of magnetomotive force along a path is that it is the -work 

 per unit pole done by the magnetic field 

 upon a magnet pole which is carried along 

 the path. From this definition of magneto- 

 motive force it may be shown that the mag- 

 netomotive force along a path is equal to the 

 line integral of the magnetic field along the 

 path, as follows : Consider an element A/ of 

 a path pp', Fig. I. Let ctf represent the 

 intensity of the magnetic field at this element 

 and E the angle between &C and A/. Then 

 the component of df parallel to A/ is <9if cos e. 

 Let a magnet test pole of strength m be 

 moved along A/. The force with which the 

 magnetic field acts upon this pole is mctf, and 

 the component of this force parallel to A/ 

 is mfffcos e, so that mctfcos e X A/ is the work A W done by the field on the 

 pole as the pole moves along A/. That is : 



A W= m&Ccos e - A/ 



and the total work done by the field on the pole while the pole is moved along the 

 path from / to ff is : 



Fig. 1. 



W= 



W 



A/ 



A/ 



The sum 2c?fcos e A/ is called the line integral of the magnetic field along the 

 path pp f . The quotient S^fcos e A/ -r- / is the average value along the path of 

 the component of <9f parallel to the path, / being the length of the path. 



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