542 BELL SYSTEM TECHNICAL JOURNAL 



Also at the surface of separation f)f the two media (3' = 0), II x and II y 

 must be continuous. Equating the values of IIx and Ily at >' = 0, 

 as given by (2), (3) and by (7), (8) and (9), (10), we have 



1 / 



^^VM" + /a./^(M) =2/.e-''^ + 0(M), 



}n.F{n)=2I.e-'^'^-ci>{^), 

 whence 



F{i^)=--7=f=^^^I, (11) 



,^(^) = (VV+|«^),-/,M.2/, (12) 



VM^ + ia + M 



which determines the functions F{p) and 0(m). 



Inserting the value of Fi\i), as given by (11) in (1), the axial electric 

 force E; in the ground (v^O) and therefore the distribution of current 

 density in the ground is expressed as a Fourier integral in terms of the 

 frequency co/27r, the current / in the wire, the height h of the wire 

 above ground, and the conductivity X of the ground. Similarly the 

 insertion of (/)(m), as given by (12) in formulas (7) and (8) gives the 

 magnetic field //.v, Ily in the dielectric. Thus 



/ , , ■ — g>'^^^^+'"COS.\>.^M, Y^O. (13) 



This can be further simplified if we write 

 x' —x\/ a 

 y' = y\' a. 



}l' = h\/' a, 

 whence 



£,= -4a;/y^ {yj]^i-p)e''^\/^^^ COS .v'm ^/m, 3-^0. (14) 



The axial electric force in the dielectric is now to be formulated. 

 This is always derivable from a vector and a scalar potential; thus 



E,= -ii^A,-^V, (15) 



where Az is the xector potential of the axial currents, and V the 

 scalar potential. Consequently, 



