﻿Plane, Cylindrical, and Spherical Current-Sheets. 201 



Hence at the surface of the sheet 



(1 <1Q 1 d dfl 2 

 dt dz dt dz 



(remembering that y 2 fl l = and v 2 I2 2 = 0). 



4. Now let 12 be the magnetic potential due to the external 

 or inducing system, a' and ft" the magnetic potentials on the 

 two sides due to the sheet itself, so that 



Qtsfio+iy, n 2 =n + n". ... (4) 



Then fl and its differential coefficients with respect to z 

 are continuous in crossing the sheet, while from the symmetry 

 of the field due to the sheet 



n'(.v, i/ ,z) = -tl"(.v, !/ ,-z), (5) 



so fl 



that d n n' n+l d n £l" 

 dz n ~~ ( } dz" ' 





Therefore the surface condition (3) gives 





d da' d da i dm 1 



dt dz + dt dz 2ttCc dz*' ' ' 



. (6) 



d da" d da i dm" 



dt dz ' dt dz " ' 2ttCc dz 2 ' ' 



• (7) 



as the surface conditions satisfied by the potential at the two 

 sides of the sheet. 



Surface Conditions at a Cylindrical Sheet. 



5. Using cylindrical coordinates, let r=a be the equation 

 of the middle surface of a cylindrical current-sheet, and let 

 its thickness c be so small that the corresponding elements of 

 its bounding surfaces and middle surface, viz. 



(a — ^c) d0dz, (a-f- Jc) d6 dz, and adddz 



are to be regarded as equal. 



If a ly a 2 are the magnetic potentials inside and outside the 



