18 THE MAGNETIC CIRCUIT [ART. 9 



capacity; it would also largely eliminate sparking in commututing 

 machines. 



Prob. 17. A long iron rod, having a cross-section of 9.3 sq.cm., is 

 bent into a circular ring so that the ends almost touch each other. The 

 ring is wound with 500 turns of wire, the winding being concentrated 

 around the gap to minimize the leakage. When a current of 2.5 amperes 

 is sent through the winding a flux of 74.9 kilo-maxwells is established 

 in the circuit. Assuming the reluctance of the iron to be negligible, 

 calculate the clearance between the ends of the rod. 



Ans. Between 1.9 and 2.0 mm. 



Prob. 18. What is the length of the air-gap in the preceding problem 

 if the estimated reluctance of the iron part of the circuit is 2 milli-rels? 



Ans. 1.7mm. 



Prob. 19. A magnetic circuit consists of three parts, the reluctances 

 of which are (R, =0.004 rel, (R 2 = 0.005 rel, and (R 3 =0.013 rel. The 

 paths (R 2 and (R 3 are in parallel with each other and are in series with 

 (Ri. What is the total permeance of the circuit? 'Ans. 131.4 perms. 



Prob. 20. In the preceding problem let (R t be the reluctance of the 

 steel frame of an electric machine, (R 2 be that of two air-gaps, and the 

 armature, and (R 3 the leakage reluctance between two poles. The ratio 

 of the total flux in the frame to the useful flux through the armature 

 is called the leakage factor of the machine. What is its value in this 

 case? Ans. 1.38. 



Prob. 21. Referring to the two preceding problems let the air-gap be 

 reduced so as to reduce the leakage factor to 1.2. How many ampere- 

 turns will be required to produce a useful flux of 2.1 megalines in the 

 magnetic circuit under consideration? Ans. 12,950. 



Prob. 22. An iron ring having a cross-section* of 4 by 5 cms. is placed 

 inside of a hollow ring. This ring has a mean diameter of 32 cm., an 

 axial width of 11 cm., and a radical thickness of 8 cm.. How many 

 ampere-turns are required to produce a total flux of 47 kilolines (count- 

 ing that in the air as well as that in the iron), if the estimated relative 

 permeability of the iron is 1400? Hint: Let the average flux density in 

 the air be B a , and that in the iron be B^. We have two simultaneous 

 equations: 20fl; + (88-20)a=47, and J5 t -/J5o = 1400. Ans. 134. 



Prob. 23. What per cent of the total flux in the preceding problem 

 is in the air? Ans. 0.24 per cent. 



Prob. 24. Show that in a ring, such as is shown in Fig. 1, the flux 

 density, strictly speaking, is not uniform, but varies inversely as the 

 distance from the center. Solution: Take an elementary tube of flux 

 of a radius x. The magnetic intensity at any point within the tube is 

 H=M/2nx, and the flux density, according to eq. (16), J5 = fj.M/2nx. 



Prob. 25. What is the true permeance of a circular ring of rectangular 

 cross-section, the outside diameter of which is D lt the inside diameter D 2 , 

 and the axial width h? Solution: The permeance of an infinitesimal 

 tube of radius x is dp =fjMx/2nx. The permeances of all the tubes 



