SELF-INDUCTION OF TWO PARALLEL WIRES. 121 



From this follows, to within a constant depending on the position 

 of the return current, 



(7) H = - 2/x I - = - 



For a series of parallel currents the value of H at each point is 

 the sum of the values which result from the different currents. 



Let us consider two parallel currents of constant densities a- and 

 a-' in cylindrical conductors whose sections are S and S'. The 

 potential energy of these currents reduces in the present case to 



W = - \Hwdxdydz. 



If we restrict the field to the space comprised between two 

 parallel planes at unit distance, and perpendicular to the current, 

 the energy relative to this portion of the field becomes 



--((^wdxd = -f 



~2jJ WXy 2j 



The integral should be extended over the whole plane, but it only 

 differs from zero in points in which is the current w that is. to say, 

 for the sections S and S' of the conductor. If we denote by H and 

 H' the values of the electromotive force which arise from the currents 

 o-S and o-'S', and we denote by an index to the sign / the spaces 

 which correspond to the different integrals, we may write 



2 W= |HoVS' + IHWS + Ho-^S+ HWS'. 



Js' Js Js Js' 



(3) 



If the second conductor S' is only the return wire of the first 

 current, we have I + 1' = 0, or o-S + o-'S' = 0. In this case the 

 common constant of the different integrations gives a zero term in 

 the value of the energy. At the point P', where is the element */S', 

 the portion SH of the electromotive force due to the elementary 

 current ov/S at the distance r, is, from equation (7), 



8H= - 



