18 Dr. J. Larmor on Electromagnetic Induction 



broadside on to the direction of the force, with its poles placed 

 so as to oppose the motion. 



VI. We shall next consider the damping of the torsional 

 oscillations of a spherical conductor in a magnetic field. If 



— denote the period of the oscillations, we have to write e' Kt (o 



fC 



instead of eo in the equations. 



First, taking the case of a thin shell, we have 



K3>=-a<^~(P + P ); 

 hence 



" -AR.^±l=-taa>«e"'(A + Ao), 



or —cacose tKt . 



A= . 2n + l„ AoJ 

 iaa>se lKZ - 4 K 



47T 



and taking the real part only, we have, corresponding to 

 A cos s0 and co cos /ct, the real part of 



— cos $<j> —p (cos Kt — i sin ict) sin s<j> . 

 l+p 2 (cos2/^-tsin2/rt) °> 



where 



(2» + l)B 



p= — • 



^ 47ra&)5 



This 

 _ — (cos s(p +p cos #£ sin s<f>)(l +p 2 cos 2*tf) — p z sin rttfsin 2tf£sin 50 

 = ~~ l+p 4t + 2p 2 'cos2tct ' 



(1 +jt> 2 cos 2/ct) cos s0 + p(l +ff 2 ) cos Kt sin 50 . 



l+p 4 + 2^ 2 cos2^ ~ A °* * V 14 ) 



It is the part of this expression containing sin s0 that fur- 

 nishes the retarding couple. Thus, in a field whose potential is 



Xl = A Q ( - ) Y„ COS 50, 



we find 



Kl+P 2 )cos^ (w + *)! 



Couple = A . i +p 4 + 2p 2 cos 2/er *• ^z^J] ; 



and in the case where the force is uniform and equal to F we 

 find 



p(l + p 2 ) COS Kt 



Couple =«F . 1+^ + 2^068 2^ 

 where now p = 3R 47rae», and the angular velocity is 



