430 The Magnetic Field produced by Electric Currents [OH. xm 



be in the plane determined by the two lines r and ds, so that the whole force 

 must have a component out of the plane of r and ds. It is almost incon- 

 ceivable that such a force could be the result of pure action at a distance, so 

 that we are led to attribute the forces acting on a circuit conveying a current 

 to action through the medium. 



Action between two circuits. 



499. Before leaving this question, however, mention must be made of 

 various attempts to resolve the forces between two circuits into forces between 

 pairs of elements. 



If the currents, say of strength i, i', are replaced by their equivalent shells, 

 the mutual potential energy of these shells is, by 423, 446, 



where e is the angle between the two elements ds, ds and r is their distance 

 apart. The forces tending to move the circuits in any specified way may be 

 obtained by differentiation. 



It is obvious that these forces can be accounted for if we suppose the 

 elements dsds' to act on one another with forces of which the mutual poten- 

 tial energy is 



^^ / cos e 7 7 , 

 dsds . 



This, however, is not the most general way of decomposing the resultant 

 force. Obviously we shall get the same form for W if we assume for the 

 mutual potential energy of the two elements 



">j j > /cose d 2 d> \ 

 -n'dsds -- + 'Hr/) 

 V r dsds/ 



where <f> is any single valued function of position of the elements ds, ds. 

 Clearly < must have the physical dimensions of a length. Following Helm- 

 holtz, let us take <j> /cr, where K is a constant, as yet undetermined. We 

 have 



3 2 , x A 9 d 3w,, d , 9 , d\ 



5~o~/ ( r ) = (I*- + m 5- + n 5- U V- / + m zr> + n *-, ) r - 

 dsds x \ dx dy dzj\ dx dy dz J 



VT dr X x' 



Now - 



sothat 



, 



cxox r r 3 



8 = (x-x')(y'-y) 

 dxdy' r 3 



