320 MB. S. H. BURBUBY ON THE INDUCTION OF ELECTRIC 



37. It is usual to treat the sheet with <r/h constant, and therefore without making 

 it self-inductive. We have generally at any point on the sheet 



, . 



dz dz 



And, therefore, on the sheet 



rf ? _ JL 2K dG _ o^dG 



dt" 2v dz ' dt ~~ 2v dz ' 

 and 



d dfl d_(<K* <H\ 



dt dv dt \dx dy) 



d_dG d_d_ 

 ~ dx dt dy dt 



a_ d /dG dY\ <r d dfl 

 2ir dz \dx dy) 2ir dz dv 



We can now treat the problem directly by the method of (16). 

 For we have by induction in the absence of resistance 



d dflo d dfl _ 

 ~&k dv ~*~ ~dt dv ~ ' 



if I1 be the magnetic potential of the field. 



Therefore, generally, to determine the value of (I on the sheet, 



o d^ dfl _ a_ d dfl 

 dt dv dt dv 2-jr dz dv 



For example, if the external field is constant as regards time, and the sheet rotates 

 with uniform angular velocity o> round the axis of z, 



and, when the motion has become steady, 



_ 



dt dv ^Jd dv ' 



where 6 is the angle between a plane through the axis and the point considered, and 

 a fixed plane through the axis. 



