﻿Straight Wire or Electric Waves. 51 



Having regard to the fact that R v must not be infinite for 

 ?' = 0, and that R must not be infinite for r=x>, we see that 

 R 1? R must be of the forms 



R^E^mr), Ro=F/t(^r), 



where E and F are constants. Then also 



*?i = E^V(»»'). %=-Fg/ t '(i*r), 



the exponential factor being, for brevity, omitted. 



The boundary conditions at the surface r=za are continuity 

 of LI and of rj ; these require 



Eg(ina) = Fh (i/ca) , 



and the elimination from these of! the ratio E/F gives the 

 relation 



<r?7 g(ina) fcv 2 h{ifca) . 



47rig / (i/ia)~ ph'(iica)' 



Now the form of R shows that l//c is comparable with the 

 change in the value of r requisite to produce a sensible 

 change in the value of R ; this can never be of a smaller 

 order of magnitude than the wave-length in the dielectric, 

 and may be much greater, as in the case when the waves are 

 nearly plane ; thus ica is either smaller than or comparable 

 with the ratio of a to the wave-length, and is therefore small 

 when the wave-length is great compared with the radius of 

 the wire. So tea is small, and we may use for the functions 

 of Ka the approximations obtained above. "We therefore put 



h(i/ca)lh'(iica) = itca log (i#a), 

 and get 



. . . . 2 acrnpgiitia) ... 



(ueay log titeay=z — ~ — %- ' . . . .(b) 



4. 



Slowly Varying Currents. 



In this case na is small, and g{ina)jg r (ina) = — 2/ma, so equa- 

 tion (6) becomes 



(i/ca) 2 \og (ilea) 2 — — iop/irv 2 (7) 



The right side of this equation is small, and an approximate 

 solution may be obtained by the method used in Prof. J. J. 

 Thomson's article on ( Electric Waves ' in the Encyclopaedia 

 Bntannica, or in a paper by Prof. A. Sommerfeld in the 



E 2 



