60 
MR. S. BUTTERWORTH ON EDDY-CURRENT LOSSES 
co-ordinates (z, r, 0). 
are 
With the assumed conditions the electromagnetic equations 
• t} 1 cE • n 
— = - — , twit = 
r cd 
8E 
dr 
1 0 (Qr) 
r dr 
1 0P 
r 00 
= 4-7T&E 
L 
> 
( 1 ) 
in which P, Q represent the components of the magnetic force acting along and 
perpendicular to r, and E represents the electric force acting parallel to 2 . 
Eliminating P and Q, the equation to be satisfied by E is 
1 0/ 0E\ 1 0 2 E 
r dr\ dr) r 2 dd 2 
47T^.«E, . 
( 2 ) 
the normal solution of which is 
E = R (l cos nd + S„ sin nO, .(3) 
in which R„ and S„ are functions of r both satisfying the equation 
. (4) 
Writing A 2 = —4 Trkiw and putting x for A r, (4) may be written 
& 2 R fl + (a^—n 8 )R„ = 0,.(5) 
in which 
S- = x~ . 
ax 
This is the general differential equation for the Bessel functions, so that inside the 
cylinder the appropriate solution of (4) is 
R„ = A„J }1 (\r), .(6) 
the second solution being excluded, since the electric force is not infinite at the axis. 
Outside the cylinder k is zero, so that the solution of (4) is 
= B n r n + C n /r n . (6 a) 
except when n — 0, in which case 
R a = B 0 log £ r+ C 0 
(6b) 
In order to maintain the continuity of E and — at the boundary of the cylinder, 
A„, B n , C„ must satisfy the relations 
