32 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1954 



For a rigorous general treatment, these postulates must be expressed 

 in differential form (the field equations), and solutions obtained in which 

 they are satisfied at all points within the field. They may be applied 

 directly, however, in certain cases of simple symmetry, and this same 

 treatment, to some measure of approximation, may be used more gen- 

 erally. To do this, the lines of induction measuring B can be considered 

 as grouped into tubes of induction. Each such tube is, from the proper- 

 ties of B, continuous, and the integral of B over a cross section of the tube 

 is a constant quantity ^, characterizing this tube. Over a length of tube 

 M bounded at its ends by two surfaces over each of which H is constant 

 exists a difference in magnetic potential, AJF, equal to the line integral 



/ 



HdL Then A3^ = <^A(R where A(R is the reluctance of this portion of the 



tube, determined by its dimensions and the permeability ii of the me- 

 dium. In particular, for a tube of uniform cross-section a over length /, 

 A(R = (/{ixa). More generally, if two ec^uipotential surfaces can be identi- 

 fied, bounding a region of constant permeability (such as an air gap), 

 the solution of Laplace's equation for the region bounded by these 

 surfaces permits the evaluation of the flux between them, and thus of 

 the reluctance A(R. 



It follows that if the pattern of the field can be recognized, so that 

 the boundaries of some major tube of force can be determined, the 

 reluctance of its several sections can be evaluated, and their sum ZA(R 

 or (R, is the ratio iF/(^. where ip is the flux of the tube and JF the magneto- 

 motive force of the coil linking it. 



The possibility of recognizing the approximate pattern of the field 

 results from the high values of the permeability of magnetic materials 

 to that for air, jia. The ratio /x/Ma is in excess of 1000 under normal 

 conditions of operation. Hence the reluctances of air paths are large 

 compared with those through magnetic material, the tubes of induction 

 tend to follow a path through the iron, and the major changes in mag- 

 netic potential occur where they pass through air gaps. In an electro- 

 magnet such as that of Fig. 3(a), the major tube of induction follows a 

 path through the core and armature, and the potential drops balancing 

 the applied magnetomotive force $J appear principally at the air gap 

 separating surface 1 from surface 2, and at the heel gap separating sur- 

 faces 3 and 4. There will l)e little potential difference between 4 and 5, 

 but the large potential difference lietween this region and surface 1 will 

 result in a leakage field between them. 



Thus the total flux v? linking the coil can be considered as di\'ided into 

 tubes of induction ^.4 and ip^c following the paths indicated in Fig. 3(a). 



