483-485] Work done in threading a Circuit 417 



WORK DONE IN THREADING A CIRCUIT. 



485. In fig. 121 let the thick line represent a circuit in which a current 

 is flowing,and let the thin line through 



the point P represent the outline of ,x'' "^v^ 



any equivalent magnetic shell, P /' 

 being any point in the shell. Let us 

 imagine that we thread the circuit by { 

 any closed path beginning and ending I 

 at P, this path being represented by 

 the dotted line in the figure. At every 



point of this path except P, we have a ^.. ---'' 



full knowledge of the magnetic forces. FIG. 121. 



It will be convenient to regard the shell as having a definite, although 

 infinitesimal, thickness at P. Let 7+, 71 denote the points in 

 which the path intersects the positive and negative faces of the 

 shell. Then we may say that the forces are known at all points of 

 the path, except over the small range 7+71. 



The original current can, however, be represented by any 

 number of equivalent magnetic shells, for any shell is capable of 

 representing the current, provided only it has as boundary the 

 circuit in which the current is flowing. FIG. 122. 



Let any other equivalent shell cut the path in the points Q+Q-. From 

 our knowledge of the forces exerted by this shell, we can determine the 

 forces exerted by the current at all points of the path except those within 



the range Q+Q In particular we can determine the forces over the range 



7+71, and it is at once obvious that on passing to the limit and making the 

 range 7+71 infinitesimal, the forces at the points P+, 71, and at all points on the 

 infinitesimal range 7+71 must be equal. Obviously the forces are also finite. 



The work done on a unit pole in taking it round the complete circuit 

 from 71 back to 71, is accordingly the same as that done in taking it from 71 

 round the path to 7+. This can be calculated by supposing the forces to be 

 exerted by the first equivalent shell, for the path is entirely outside this 

 shell. If the potential due to the shell is flp+ at 7+ and is Hp_ at 71, the 

 work done is Hp + Hp_. 



Now H, the potential of the shell at any point, is, as we know ( 419), 

 equal to iw, where &> is the solid angle subtended by the shell and i is the 

 current, measured in .electromagnetic units. The change in the solid angle 

 as we pass from 71 to 7+ is, as a matter of geometry, equal to 4-Tr. Thus 



Hp + - Hp_ = 4?' (410). 



The work done in taking a unit pole round the path described is accord- 

 ingly 4<7ri. 



j. 27 



