CHAP. XI] INDUCTANCE OF TRANSMISSION LINES 191 



conductor, and is always equal to 0.05 perm, per cm., or 0.05 milli- 

 henry per kilometer length of the cable. 



The exact expression for the part of the inductance of the cable 

 L D ' due to the linkage within the outer conductor is given in 

 problem 10 below. The formula is rather complicated for prac- 

 tical use, especially in view of the fact that this part of the induct- 

 ance is comparatively small, because the flux density on the part 

 rs of the curve is small. It is more convenient, therefore, to make 

 simplifying assumptions, when the thickness t of the outer con- 

 ductor is small as compared to 6. Namely, the length of all the 

 paths within the outer conductor may be assumed to be equal 

 to 2^6, so that the permeance of an infinitesimal path of a radius x 

 and thickness dx nearly equals pdx/lnb. Furthermore, the volume 

 of the current in the outer conductor, between the radii 6 and x 

 may be assumed to be proportional to the distance x b, and 

 hence equal to i(x b)/(c b). A line of force of a radius x is linked 

 therefore with the whole current i in the inner conductor and with 

 the above-stated part of the current in the outer conductor, and, 

 since the currents flow in opposite directions, this line of 

 force is linked altogether with the current i(c x)/(c b). Hence, 

 it is linked with n p = (c x)/(c b) turns. Thus, the inductance of 

 the cable, due to the outer partial linkages, is, in the first approxi- 

 mation, 



L D '=fi/(2M 2 ) C\c -x) 2 dx=M/b perm/cm. . (Ill) 



*s\t 



If a closer approximation is desired, it is convenient to expand 

 eq. (114) in ascending powers of t/b, as is explained in problem 11. 

 The result is 



L D '=fot/b[\ fa(t/ b) 2 + j t j i (t/b) 3 . . .] perm/cm. . (112) 



It will thus be seen that eq. (Ill) is an accurate approximation, 

 because eq. (112) contains in the parentheses no term with the 

 first power of the ratio t/b. 



Thus, the total inductance of a concentric cable, / kilometers 

 long, is 



where L D ' is given by eqs. (Ill), (112), or (1 l-h. :i>nling to the 

 accuracy desired. Expressions (1 10) and (1 1 -I rt only at 



1<\\- frequencies, such as are used for power transmission. With 



