278 BELL SYSTEM TECHNICAL JOURNAL 



11. "The Rigorous and Approximate Theories of Electrical Transmission along 



Wires." (Carson, B. S. T. J., Jan., 1928.) 



12. As a general reference, the treatise "Electrical and Optical Wave Motion," by 



Bateman, published by the Cambridge University Press, may be consulted 

 with profit. 



Appendix 



The mode of attack outlined in the latter part of the text will be 

 illustrated by an application to the simplest possible case. 



Let the transmission system consist of a wire of radius a whose 

 axis coincides with the A' axis, and a coaxial c^-linder of internal 

 radius b. Both conductors are supposed to be perfectly conducting, 

 while the dielectric in the space between {a — p — h) \s supposed to 

 be perfect. For this system we know that the principal wave is 

 transmitted without attenuation with the velocity of light c; that is 

 to say, 7 = ioj/c, where co is lir times the frequency. 



We suppose that the system extends for an indefinite distance along 

 the positive X axis so that reflected waves are absent. The principal 

 current and charge waves are then : 



/ = /oe-^^^ Q = Q,e-'^\ (la) 



where /3 denotes co/c, and i = V— 1. From the relation 



C dt dx 



it follows that 



Q= I. (2a) 



Now by definition the retarded potentials are 



$ = j ^e-'^'dv (Scalar), 



A = I -e-'^'dv (Vector), 



where q and u denote the charge and vector current density respec- 

 tively, r is the distance between the contributing element dv and the 

 point at which the potential is to be calculated, and the integration 

 is extended over the entire system of currents and charges. In terms 

 of the retarded potentials the magnetic and electric intensities E and 

 H are given by 



//= curl A, 



(3a) 

 E = — grad 4> — i^A. 



