ELECTROMAGNETIC THEORY OF LINES AND SHIELDS 549 



The attenuation and the phase constants are each equal to V irg^if. 

 The intrinsic propagation constants of metals are large quantities 

 except at low frequencies as the accompanying table indicates. 



Propagation Constant of Commercial Copper 



g = 5.800 10= mhos/cm. 

 H = 0.01257 ixh/cm. 



= V^rgju/ 







1 



10 



100 



10,000 



1,000,000 



100,000,000 



0.0 



0.1513 

 0.4785 

 1.513 

 15.13 

 151.3 

 1513. 



On the other hand, r is very small; if air is the dielectric between the 

 conductors, r is of the order of (l/3)ico 10"^'^. Hence, even at high 

 frequencies T^ is negligibly small by comparison with a^ and we can 

 rewrite (54) as follows: 



d_ 

 dp 



ld_ 

 p dp 



{pH,) 



a-'IK 



(55) 



This is Bessel's equation and its solution can be written down at 



once ^^ as 



H^ = Ah{ap) +BK,{<xp), 



(56) 



where the functions /i(m) and Ki{u) are the modified Bessel functions 

 of the first order and respectively of the first and second kind. For large 

 values of the argument we have approximately 



if.(«)=^/f,e-"(l+|^ 



(57) 



^^ It is interesting to note that in the case of a fairly thin hollow conductor whose 

 inner radius is not too small there exist very simple approximate solutions of (55). 

 Under these circumstances p varies over such a small range that no serious error is 

 introduced in treating the factors (1/p) and p in (55) as constants, and the equation 

 becomes 



, ., — cr n,p, 

 dp- 



which is satisfied by the exponential functions e"'' and e"''''. The larger the value 

 of p and the faster the change in H^ with p, the better is the approximation. 



