1)6 



Dr. W. F. G. Swann 



071 



Appendix. 



(I) The magnetic field due to a sphere of radius a rotating with 

 angular velocity w and charged to' a surface density s. 



The current clue to the rotation of 

 a zone of width adu. subtending an 

 angle a with the axis of rotation at 

 the centre of the sphere, is 



ccas . a sin a . du. 



The magnetic potential due to such 

 a current at a point Q whose polar 

 coordinates are r . (j, is (see Maxwell's 



'Electricity and Magnetism,' vol. ii. 

 p. 333) 



^ 1 



r» + l 



Sft = SwA. sin" a . da f -L- . a — P„'(«) . P„ W , 



n = l it -\- L / 



where P„ stands for the ordinary spherical harmonic of 

 degree n. 



The potential due to the rotating sphere is 



£1 = 2™*"$ "V;\ +l . W) f'P/^ sin 3 a. da. 



Writing the integral as I n , we readily find by the application 

 of the ordinary processes of spherical harmonics. I„ = 



except when n = \ } in which case T„= -. 



4 a A 



.*. f>= TT7rso) — cos 6. 

 .) ?•- 



The horizontal component of the fi 



eld is 





ITS 5 w 



sin0. 



At the surface of the sphere and on the equator, we have 



1 

 H ~ ., 7rsiiQ). or it' \ represents the potential of the sphere, 



we ha\ e 



11 



