CHAP. XIII] TORQUE AND TRACTIVE EFFORT 255 



Prob. 16. Derive formula (171) for the lifting magnet by means 

 of the principle of virtual displacements. Solution: The reluctance 

 of the air-gap is <R=/(M), so that ofcR/ds = l/(M), hence, according 

 to eq. (178) F- -i^V(M) - -AB'/2n. The minus sign indicates that 

 tin- stress is one of tension. 



Prob. 17. Derive expression (172) from formula (179). 



Prob. 18. Derive the formula for the repulsion between the wingings 

 in a transformer from eq. (182). 



Prob. 19. Derive from eq. (182) the force of repulsion between two 

 infinitely long, parallel, cylindrical conductors placed at a distance 

 of 6 meters apart, and forming an electric circuit (Fig. 47). 



Ans. 2.04i'(//6) X 10~ 8 kg. for / meters of the loop. 



Prob. 20. What deformation of the windings may be expected 

 during a severe short-circuit of a core-type or a cruciform type trans- 

 former (Figs. 12 and 14) with cylindrical eoils, (a) when the centers 

 of the coils are on the same horizontal line, and (fr) when one of the 

 windings is mounted somewhat higher than the other? 



Prob. 21. Show that in a constant-current rotary magnet 

 63) 7'a= Const. that is, the torque in the different positions of the 

 armature is inversely proportional to the air-gap at the entering pole- 

 tip. Hint: d& pwrdff/a, where w is the dimension parallel to the 



shaft. 



Prob. 22. State Kelvin's law when mechanical work is done ayain*t 

 the forces of the magnetic field. 



Prob. 23. A 60-cycle, 8-amp., series arc-lamp magnet has a stroke 

 of 32 mm.; the reactive voltage consumed in the initial position is 9 

 v.. and in the final poition 20 v. What is the average pull ? 



Ans. 372 grams. 



72. The Torque in Generators and Motors. The magnetic 

 circuits considered in the preceding articles of this chapter are 

 singly excited, that is, they have but one exciting electric circuit. 

 From the point of view of mechanical forces this also applies 

 to each air-gap in a transformer, because, neglecting the mag- 

 net i/ing current, the primary and the secondary coils may be 

 combined into equivai ageooflfl <\n.r>i). On the other 



hand, a L r < nerator or a motor under load has a doubly-excited 

 magnetic circuit, the useful field being linked with both the 

 field and the armature windings. The two iu.in.fs. not being 

 in direct opposition in span-, the flux is deflected from the shortest 



to 9 incl.; 8. P. Thompson, On tin* Pn-drtcnnination of I'luiiKcr Eloctro- 

 magn - L904, VoL 1, p Jane, 



Ueber I i.'ignetc, EUcktroiecl Mtuchintmbau, Vol. 28 d-'Ki . 



p. 833; K. \\ik !;. .momical DWIK" "f Din-.t- I 



magnets; Trarui. Amer. In*t. EUct. Eng9. t Vol. 30 (1911). 



