ELECTROMAGNETS. MAGNETISM OF IRON. 77 



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

 of the component parallel to the path of the force /(= mch) 

 with which the field acts on the pole, the meaning of A being 

 explained in Art. 46. 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 = ImA (2) 



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

 ing that I A = cf according to equation (i), Art. 46, we have 

 equation (i) of this article. 



The above proposition is usually taken as the definition of 

 magnetomotive force along a path, and from this definition it 

 may be shown that the magnetomo- 

 tive force along a path is equal to 

 what is called the line integral* of the 

 magnetic field along the path, as fol- 

 lows: Consider an element A/ of a 

 path pp', Fig. 54. Let efC represent 

 the intensity of the magnetic field at 

 this element and e the angle between 

 eft and A/. Then the component of SK 

 parallel to A/ is 3f* cos e. 



Let a magnet test pole of strength m be moved along AZ. 

 The force with which the magnetic field acts upon this pole is 

 mdt, and the component of this force parallel to A/ is mdi cos e, 

 so that mdi cos e X A/ is the work AW done by the field on the 

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



ATF" = m3f cos e-A/ 

 from which, by integration, we have: 



cos dl 



* A very simple and complete discussion of line and surface integrals such as 

 are used in electromagnetic theory is given in Chapter IX of Franklin, MacNutt 

 and Charles's Calculus, South Bethlehem, Pa, 1913. 



