470 Induction of Currents in Continuous Media [CH. xv 



Let P denote the potential at any point of a distribution of poles of 

 strength <J>, so that 



(482), 



where dx'dy' is any element of the sheet. The magnetic potential at any 

 point outside the current-sheet of the field produced by the currents is then 



H = -~ .............................. (483). 



02 



If o- is the resistance of a unit square of the sheet at any point, and 

 u, v the components of current, we have, by Ohm's Law, 



X = (TU,- Y=(7V. 



The components u, v are readily found to be given by 



d d<& 



u = ^ , v = = , 

 dy dx 



so that we have the equations 



v 83> v 83> 



z -*9j: *=- ff fc ..................... < 484) 



true at every point of the sheet. 

 Hence, by equation (466), 



dc dY dX /8 2 <I> 



-dt=^-w = ~ (7 (^ + w) ............... ( } - 



The total magnetic field consists of the part of potential O due to the 

 currents and a part of potential (say) O', due to the magnetic system by which 

 the currents are induced. Thus the total magnetic potential is H + 1', and 

 at a point just outside the current-sheet (taking fju 1), 



dc_d d 



dt~d:td~z ( ^ + Ll)> 

 and equation (485) becomes 



The function P (equation (482)) is the potential of a distribution of poles 

 of surface density 4> on the sheet. Hence at a point just outside the sheet 



and on its positive face, ^ = 2?r^>. 



dz 



Hence, since P satisfies Laplace's equation outside the sheet, we have 

 (just outside the positive face of the sheet), 



8 2 <E = 1 /8 3 P 8 3 P 



df ~ 2?r \d?fa + dy*d 



1 8 3 P 



2?r 8^ 3 



J^ 



27T 



