IN CYLINDRICAL CONDUCTORS, ETC. 
93 
strand in question is situated. This field is normal to the axis of stranding and tangential 
to the cylinder on which the strand is wound. 
( b ) A field H 2 due to the remaining turns in the coil. This field will also be assumed 
to be normal to the axis of stranding, and will have two components Ifi cos 0 tangential 
to the cylinder ana H, sin 0 normal to the cylinder, 0 being the angular position of the 
element under consideration. 
Thus an element of one strand is situated in a nett field whose components tangential 
and normal to the winding cylinder are TI, -f- H 2 cos 0, H 2 sin 0. These may further 
be resolved into components along and at right angles to the direction of the element, 
viz. :— 
Hi = (H,+ H 2 cos 0) sin a, 
H b = (Hj + ELj cos 6) cos a, H 2 sin 0. 
Now, as regards the axial component of the field, it may be shown that if a cylinder 
is placed with its axis along the direction of an alternating field, the losses in the cylinder 
are one-half the losses which would occur if the cylinder were placed at right angles to 
the field. 
Therefore the loss in an element d\ of one strand due to the fields H,, H 2 is by (49) 
dW = ir 0 d\<l 2 G( 2 )(H B 2 +iH A 2 ). 
Now, as we pass along one strand, the value of 0 increases uniformly as we are rotating 
relative to the field H 2 . Hence the average loss in one strand is got by replacing H A 2 , 
H b 2 by tlieir mean values throughout a complete cycle of 0 ; that is, H A 2 , H B 2 are replaced 
by 
sin 2 a (H/ + 1H/), cos 2 a (H, 2 + fH 2 2 ) +Pf 2 2 
The loss per unit axial length of the stranded wire is thus 
W = -|-r 0 sec aG (z) S 2 {Hj 2 (l — ■\ sin 2 a) + H 2 2 (I — \ sin 2 a)}. 
(20) Since each strand carries the same current, the value of II, at a distance r from 
the axis is 21 r/a 0 2 , where I is the whole current and a 0 the over-all radius of the stranded 
wire, the number of strands being assumed large. 
Again the number of strands crossing an annular belt of width dr in the cross-section 
of the wire is 2 rs dr ja 2 , s being the whole number of strands. The mean value of II 2 
throughout the section is therefore 
81 2 f °r 3 dr/a 0 6 = 21 2 /a 2 = 8l 2 /d 2 , 
Jo 
in which d 0 — 2 a 0 . 
The field H 2 is the same as that for the corresponding solid wire coil, and the mean 
value of H 2 throughout the coil is 4 m„F/D 2 , D being the distance apart of consecutive 
