538, 539] Plane Current-sheets 471 



by equation (483), so that equation (486) becomes 



and similarly, at the negative face of the sheet, we have the equation 



d 9 , < 



(488). 



Finite Current-sheets. 



539. Suppose that in an infinitesimal interval any pole of strength m 

 moves from P to Q. This movement may be represented by the creation 

 of a pole of strength m at P and of one of strength + m at Q. Thus 

 the most general motion of the inducing field may be replaced by the crea- 

 tion of a series of poles. The simplest problem arises when the inducing 

 field is produced by the sudden creation of a single pole, and the solution 

 of the most general problem can be obtained from the solution of this simple 

 problem by addition. 



From equations (487) and (488) it is clear that -7-^(0 + 1') remains 



Ctt 02 



finite on both surfaces of the sheet during the sudden creation of a new 



o 

 pole, so that (ft + H') remains unaltered in value over the whole surface 



of the sheet. Let the increment in (fl + ft') at any point in space be 



denoted by A, then A is a potential of which the poles are known in the 

 space outside the sheet, and of which the value is known to be zero over 

 the surface of the sheet. The methods of Chapter vill. are accordingly 

 available for the determination of A : the required value of A is the 

 electrostatic potential when the current-sheet is put to earth in the 



OQ> 



presence of the point charges which would give a potential -^ if the sheet 



oz 



were absent. 



Physically, the fact that =- (O + 1') remains unaltered over the whole 



oz 



surface of the sheet means that the field of force just outside the sheet 

 remains unaltered, and hence that currents are instantaneously induced in 

 the sheet such that the lines of force at the surfaces of the sheet remain 

 unaltered. 



The induced currents can be found for any shape of current-sheet for 

 which the corresponding electrostatic problem can be solved*, but in general 

 the results are too complicated to be of physical interest. 



* See a paper by the author, "Finite Current- sheets," Proc. Lond. Math. Soc. Vol. xxxi. 

 p. 151. 



