CHAP. XIII] TORQUE AND TRACTIVE EFFORT. 241 



Two more expressions for the density of the energy can be written, 

 using the reluctivity v instead of the permeability /. 



In a uniform field the preceding expressions represent the 

 actual amounts of energy stored per cubic centimeter. In a 

 non-uniform field W is the density of energy at a point, or the 

 limit of the expression J W/ J V. This is analogous to what we 

 have in the case of a non-uniform distribution of matter, where 

 the density of matter at a point is the limit of the ratio of the 

 mass to the volume. Thus, the total energy stored in a non- 

 uniform field is 



(166) 



where the integration is to be extended over the volume of the 

 whole magnetic circuit. Similarly, from eqs. (164) and (165) 

 we get 



(167) 

 (168) 



These expressions are consistent with eqs. (102) and (102a) 

 as is shown in prob. 6 below. 



When fi is variable, the preceding formulae do not hold true, 

 and the density of energy is represented by eq. (19), Art. 16. 



Prob. 1. Deduce an expression for the magnetic energy stored in 

 the insulation of a concentric cable (Fig. 46), between the radii a and 6, 

 the length of the cable being I cm. and the current i. Hint: For an 

 infiiiitc.-imal shell of a radius x and (hicknesss dx \ve ha\ c : // i/2itx, 

 and ilV I'-./-/ dx. Ans. W -0.23/i' log (6/a)10- 8 joules. 



Prob. 2. Check the answer to the preceding problem by means 

 of eqs. (104) and (109). 



Prob. 3. In a concentric cable (Fig. 46) a-7 mm. and 6 is 20 nun. 

 What is the density of the energy at the inner and outer condu 

 \\heu i- 120 amp.' 



Ans. 4.68 and 0."7 micro joules per cu.rm. 



Prob. 4. Deduce expression (110) from eq. (167). 



Prob. 5. Taking the data from the various problems given in this 

 book a* typical. ~h.\v that ordinarily in generators and motors a large 

 propoi ic total energy of the field is stored in the air-gap. 



Prob. 6. Show that eqs. (166) to (168) are consistent with eqs. 

 (102) to (103a). Solution : Take an infinitesimal tube of partial linkages 



