TRANSMISSION OVER SUlUL-iKINIi CABI.F.S 91 



in finding the particular solution of Maxwell's equations which satis- 

 fies the boundary conditions — continuity of tangential electric and 

 magnetic forces at the surfaces of the conductors. Let the common 

 axis of the conductors coincide with the Z axis of a system of polar 

 coordinates, R, <I>, Z, and let the electric and inagnetic variables in- 

 volve the common factor exp ( — F 2 + ipt), T is therefore the propa- 

 gation factor characterizing transmission, and p is 2ir times the fre- 

 quency. This factor will not be explicitly written in any of the work 

 that follows, but it will be assumed to be incorporated in each of the 

 electric variables so that 



Qz" Of" 



From symmetry, it is evident that the component of electric field 

 intensity in the direction of <^ vanishes, and that the magnetic lines 

 of force are circles lying in planes perpendicular to the axis of the 

 system, and centered on that axis. Also, the axial and radial electric 

 forces are independent of 0. It can be shown that the radial compo- 

 nent of electric field intensity in the conductors is negligibly small com- 

 pared with the axial component. The latter, for a given conductor, 

 is of the form E exp{ — Tz -\- ift), where £ is a solution of the differen- 

 tial equation 



1^ + - 1^ + (n - 47rXMip) £ = 0. (2) 



Here X and m are the electrical conductivity and the magnetic perme- 

 ability of the particular conductor, measured in absolute electro- 

 magnetic units, and £ is a function of r alone. 



For the frequencies in which we are interested it may be shown 

 that T-/4Tr\ijLp is exceedingly small, so that (2) may be written 



9!| + i ?^ _ 4r\tJ.ipE = 0. (3) 



6^ r cr 



We will designate by the subscript j all quantities pertaining to the 

 j'^ conductor, counting from the axis. The solution of (3) for this 

 conductor may then be written 



Ej = AMp,) + BjKoipd, (4) 



where /<, and K^ are Bessel functions of zero order, Aj and Bj are 

 arbitrary constants and 



Pj = ri y/4ir\jiJ.jpi = ra,. 



