484 Dynamical Theory of Currents [CH. xvi 



The energy of the two currents is known to be 



E = (Li* + 2 Jfu v + JW S ) ..................... (512). 



Let us suppose, for the sake of generality, that this consists of kinetic 

 energy T and potential energy W. Then, assuming for the moment that the 

 mechanism of these currents is dynamical, in the sense that Lagrange's 

 equations may be applied, we shall have a dynamical system of energy 

 T+W, and one of the coordinates may be taken to be r, the distance apart 

 of the circuits. 



The Lagrangian equation corresponding to the coordinate r is found to 

 be (cf. equation (508)), 



ddT\ d(T-W) 



and since we know that, in the equilibrium configuration, 

 d fdT\ ..,dM 



T* (-^ } = > R = - n -*- > 



dt\drj dr 



we obtain on substitution in equation (513), 



W)_ ..,dM 



~ l 



From equation (512) we see that the right-hand member is the value of 



dE d(T+W) 3W 



- , or or -- ? . Hence our equation shews that -= = 0, from which we 



dr dr dr 



deduce that W= 0. In other words, assuming that a system of steady 

 currents forms a dynamical system, the energy of this system must be 

 wholly kinetic. 



This result compels us also to accept that the energy of a system of 

 magnets at rest must also be wholly kinetic. We shall discuss this result 

 later. For the present we confine our attention to the case of electric 

 phenomena only. We have found that if the mechanism of these pheno- 

 mena is dynamical (the hypothesis upon which we are going to work), then 

 the energy of electric currents must be kinetic. 



Induction of Currents. 



556. Let us consider a number of currents flowing in closed circuits. 

 Let the strengths of the currents be i l} i 2 , ... and let the number of tubes of 

 induction which cross these circuits at any instant be N lt N 2 ,..., so that if 

 the magnetic field arises entirely from the currents, we have (cf. 503) 



.(514). 

 etc., 



