576 BELL SYSTEM TECHNICAL JOURNAL 



where I is the separation between the axes of the wires. Therefore, 



$= k-^^ p (138) 



Resistance of Nearly Coaxial Tubular Conductors 



When two tubular conductors are not quite coaxial, a proximity 

 effect ^^ appears which disturbs the symmetry of current distribution 

 and therefore somewhat increases their resistance. This effect can 

 be estimated by the following method of successive approximations. 

 To begin with, we assume a symmetrical current distribution in the 

 inner conductor. The magnetic field outside this conductor is then 

 the same as that of a simple source along its axis and can be replaced 

 by an equivalent distribution of sources situated along the axis of the 

 outer conductor. The principal component of this distribution is a 

 simple source of the same strength as the actual source and does not 

 enter into the proximity effect. The next largest component is a 

 double source given by 



„ ioiixll . 



hz = -r COS d, 



// (139) 



He = -?i — 5 COS 9, 



where / is the interaxial separation, r is the distance of a typical point 

 of the field from the axis of the outer conductor, and d is the remaining 

 polar coordinate. 



This field is impressed upon the inner surface of the outer conductor ^^ 

 and the resulting power loss equals, by equation (136), the real part of 



$ = 2^av (^yP^^, rjP, (140) 



where at high frequencies -q = ^iwiijg is simply the intrinsic impedance 

 of the outer conductor.^^ This loss increases the resistance of the 

 outer tube by the amount. 



^Ra 



^#. " (141) 



32 For promixity effect in parallel wires external to each other, the reader is 

 referred to the following papers: John R. Carson [1], C. Manneback [9], S. P. 

 Mead [12]. 



33 The radius of this surface is designated by a. 



34 At low frequencies j? has to be replaced by the radial impedance looking into 

 the shield. 



