COFFIN. — THE SELF-INDUCTANCE OF COILS. 791 



where w is 2 ir X frequency, and 77 the maximum value of 77. Outside 

 the coil, 77 must be constant, but as it vanishes at infinity it is zero 

 everywhere. Hence, for 



r%r 2 77=0. 



In the material of the coil, Maxwell's equations are 



±Trq=VVH, ^E=-VVF, snxdq = <rF i (3) 



from which follow SV77 = and SVq = (4) 



where q is the current density, 



F is the electric force, 

 o- is the specific conductivity. 



Eliminating q and F, by taking the curl of the first equation (3) 



47rFV 9 = VWVH= VSV77- \ 2 H 



= - V 2 77 by (4) 



and from the second and third of (3) 



VWq = a-VVF=-o-^ 



1 dt 



J TT 



therefore 4tt <t -=- = V 2 77. (5) 



dt v J 



Now assume 77 in the material of the coil to be given by a function £7" of r 

 alone (on account of symmetry), multiplied by U Q e 1Mt , or 77= H Ue lult 

 where U =■ 1 when r = r x 



£7=0 " r = r 2 . 



The differential equation then which £7 must satisfy by equation (5) 

 becomes in polar co-ordinates 



d 2 U \dU J0rr n 



Tir + -- 7 --f£ 2 tf=0 (6) 



dr* r dr v ' 



where k = — i 4 tt <t w. 



The solutions of this equation are the Bessel's functions of order zero, 

 J and K and hence 



H=H {A 7 (h r) + BK (k r)} e M (7) 



where A and B are arbitrary constants. 



