CHAP. XJ ENERGY AND INDUCTANCE 183 



Some new light is thrown upon these relations by using the 

 m.m.f . M instead of the ampere-turns ni. Namely, eqs. (102) and 

 (102a) become: 



T7=i[M c c + SMpJ0 P ], .... (103) 

 or 



. . . . (103a) 



These expressions are analogous to those for the energy stored in 

 an electrostatic circuit, viz., $EQ, and \E 2 C (see The Electric Cir- 

 cuit). The m.m.f. M c is analogous to the e.m.f. E\ the magnetic 

 flux C is analogous to the electrostatic flux Q, and the permeance 

 (P c is analogous to the permittance C. 



We can assume as a fundamental law of nature the fact that 

 with a given steady current the magnetic field is distributed in such 

 a way that the total electromagnetic energy of the system is a 

 maximum. All known fields obey this law, and, in addition, it can 

 be proved by the higher mathematics. Eq. (102a) shows that this 

 law is fulfilled when the sum n c 2 (P c + n P 2 <P P is a maximum. When 

 the partial linkages are comparatively small, the energy stored is a 

 maximum when the permeance (P c of the paths of the total linkages 

 is a maximum. This fact is made use of in the graphical method 

 of mapping out a magnetic field, in Art. 41 above. 



Prob. 5. The no-load saturation curve of an 8-pole electric gen- 

 erator is a straight line such that when the useful flux is 10 megalines 

 per pole the excitation is 7200 amp .-turns per pole; the leakage factor 

 is 1.2. Show that at this excitation there is enough energy stored 

 in the field to supply a small incandescent lamp with power for a few 

 minutes. 



Prob. 6. Explain the function and the diagram of connections of 

 a field-discharge switch. 



Prob. 7. Prove that the magnetic energy stored in an apparatus 

 containing iron is proportional to the area between the saturation curve 

 and the axis of ordinates. The saturation curve- is understood to give 

 the total flux plotted against the exciting ampere-turns as absciss. 

 Hint: See Art. Hi. 



Prob. 8. Deduce expression (102a) directly, by writing down an 

 expression for the total instantaneous e.m.f. induced in a coil (Fig. 45). 



Prob. 9. Explain the reason for which, in the formula) deduced above, 

 it is permissible to consider n to be a fractional number. 



68. Inductance as the Coefficient of Stored Energy, or the 

 Electrical Inertia of a Circuit. K<j. (l()2a) shows that in a mag- 

 netic circuit of constant permeability the stored energy is proper- 



