215 217] ELECTROMAGNETISM. 421 



217. Force due to any Linear Current. If the potential 

 at a point P is fl and at a neighboring point Q is fl + SH, where 

 the distance PQ = Sh, and if H is the magnetic force at P, we 

 have 



(2) Sl = 



This change in the potential is the same as the change that 

 would be made in the potential at P by moving the whole circuit 

 parallel to itself the same distance Sh in the opposite direction. 

 The change Sfl is proportional to the change Sco made in the 

 solid angle subtended at P due to the motion of the circuit, which 

 is easily seen to be exactly the solid angle subtended at P by 

 the narrow ribbon of cylindrical surface whose edges are the 

 initial and final positions of the circuit, and whose generating 

 lines are equal and parallel to Sh. But any arc ds has described 

 in the motion an area dS of a parallelogram equal to 



(3) dS = dsSh sin (ds, Sh), 



and if n be the normal to this element of area, we have for the 

 element dSco of the solid angle subtended by it at P, 



,. dScos(nr) , -, . ^. 



(4) doco = -- --- - = dsoh sin (ds, oh) cos (nr), 



where r is the distance of the element from P. Consequently 

 integrating around the ribbon 



(6) sn 



If we consider that each element of the current of length ds 

 contributes to the field the potential dl and the force dH, we 

 have, by (2), 



/ \ JTTZI. fj-cr *i.\ j*r\ T rdsShsin(ds, Sh)cos(nr) 



(7) - dHSh cos (dH, Sh) = dSl = / ^J - - - ^- - . 



The numerator is the volume of the parallelepiped whose sides 

 are r, ds, Sh. It therefore vanishes if the direction of Sh coincides 

 with that of r. 



There is accordingly no component of the force in the direction 

 of r, or the force is perpendicular to r. In like manner if Sh has 



