182 THE MAGNETIC CIRCUIT [ART. 57 



which links with all the turns of the coil. The e.m.f. induced in 

 the coil by this flux, when the current changes, is - e t = n(d<P t /dt), 

 and the relation between the current and the flux is $t=(P c nii t 

 whore (P e is the permeance of the path of the complete linkages. 

 By repeating the reasoning given above in the case of a single loop 

 we find that 



W c =\ni<I> c , ....... (100) 



or 



...... (lOOa) 



where the subcript c signifies that the quantities refer to the com- 

 plete linkages only (Fig. 45) . Two forms only are retained, being 

 those that are of the most practical importance. 



The energy of the partial linkages is calculated in a similar 

 manner. Let J<P P be a small annular flux (Fig. 45) which links 

 with n p turns of the coil, where n p may be an integer or a fraction. 

 For these turns the linkage with AQp is a complete linkage, while 

 for the external (n n p ) turns it is no linkage at all and represents 

 no energy, because no e.m.f. is induced by J0 P in the turns external 

 to it. Thus, the energy due to the flux J$ p , according to eqs. 

 (100) and (lOOa), is equal to %n P iA$ P , or to %n p 2 i 2 J(P P . The 

 total energy of the partial linkages is the sum of such expressions, 

 over the whole flux <D P) or 



(101) 

 or 



W P =WZn P 2 4(P P ........ (lOlo) 



The total energy of the coil is 



0p], ..... (102) 



or 



TP=Ji*[n2<P e + SnpWp], .... (1020) 



where the first term on the right-hand side refers to the complete 

 linkages and the second to the partial linkages of the flux and the 

 current. In these expressions the current is in amperes, the fluxes 

 in webers, the permeances in henrys, and the energy in joules 

 (watt-seconds) . If other units are used the corresponding numeri- 

 cal conversion factors must be introduced. 



