CHAP. X] ENERGY AND INDUCTANCE 181 



inside the wire may be disregarded. The electromagnetic energy 

 possessed by the loop is equal to the electrical energy spent in 

 establishing the current i in the loop against the induced e.m.f. 

 Let it and <D t be the instantaneous values of the current in 

 amperes and the flux in webers at a moment t during the period of 

 building up the flux, and let e t be the instantaneous applied 

 voltage. Let the flux increase by dQ t during an infinitesimal 

 interval dt\ then the electrical energy (in joules) supplied from 

 the source of power to the magnetic field is 



dW =i t e t dt =i t (d<P t /dt)dt =i t d0 t , 



where e t = d<D t /dt is the instantaneous e.m.f. induced in the 

 loop by the changing flux. The total energy supplied from the 

 electrical source during the period of building up the field to its 

 final value is 



(98) 



In a medium of constant permeability the integration can be 

 easily performed, because the flux is proportional to the current, 

 or, according to eq. (2) in Art. 5, <D t =(Pit, where (P is the permeance 

 of the magnetic circuit, in henrys. Eliminating by means of this 

 relation either i t or <P t from eq. (98) we can obtain any one of the 

 following three expressions for the electromagnetic energy stored 

 in the loop: 



(99) 



In the first form, eq. (99) expresses the fact that the magnetic 

 energy stored in a loop is equal to one-half the product of the cur- 

 rent by the flux; in the second form, it shows that the stored 

 energy is proportional to the square of the current and to the per- 

 meance of the magnetic circuit. Both forms are of importance in 

 practical applications. 



Take now the more general case of a coil of n turns (Fig. 45) ; 

 the flux which links with a part of the turns is now of a magnitude 

 comparable with that of the llux which links with all the turns of 

 the coil. We shall consider the complete linkages and the partial 

 linkages separately. Consider first the energy due to the flux 



