810 ELECTRICAL METHODS [Chap. 10 



geologic problems, the theoretical assumption of a simple radiation wave 

 is frequently not justified and all field components must be considered, 

 particularly at points close to the source. Assume a radiating doublet 

 source of pole distance I, in a. medium of the dielectric constant k = 1 , the 

 permeability y = 1, and the conductivity <r = 0. At a point P in the 

 vicinity of this doublet, with the polar coordinates r and cp in reference to 

 the center of the doublet, the magnetic and electrical field intensities due 

 to a periodic charge e = eo sin cot are given by the following equations:^"* 



Bol •_ I %i J 2 



H = — — sin <p [ - (t) cos cot — 0) sin ut 

 re 



E. = 



eol . /c^ . . , c , 2 . A 



= sm <p I — sm cot + - CO cos cot — co sm cot I 



r \r^ r / 



<p I — sm cot H — CO cos cot I , 

 \r^ r / 



) (10-65) 



E, = -2^^ cos 

 r 



where H is the magnetic field, E^ the tangential and Er the radial electrical 

 field, and c is the light velocity. The first term in the electrical field ex- 

 pressions [(c /r ) sin co^] represents the electrostatic component of the 

 doublet, decreasing with the inverse cube of distance. It is independent 

 of frequency and important only in the immediate vicinity of the source. 

 The second term [(c/r)co cosco^] is the induction effect, inversely proportional 

 to the square of distance and proportional to frequency. The third term 

 (co^ sin (Jit) represents the radiation component, which is inversely propor- 

 tional to the first power of distance and directly proportional to the 

 square of the frequency. The last term is the only one important in 

 long-range radio communication. If the distance r has reached the value 

 of X/2ir, the first and second terms (static and inductive) have dropped to 

 the amount of the radiation term at that point. The electrical field 

 strength E^ is of particular interest in the equatorial plane (^ = 90°). 

 With the phase shift ^ = rco/c, the equatorial electrical field strength 



E = 



-y [(72 - A sin CO 6 - ^) + ^ CO COS oiit- 01. (10-66) 



Substituting co = 27rc/X and r/X = x, the relative field in terms of wave 

 length is 



(10-67a) 



"« Hummel, Zeit. Geophys., 6(3-4), 109 (1929). 



